In [Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411–1443] A.S. Fokas introduced a novel method for solving a large class of boundary value problems associated with evolution equations. This approach relies on the construction of a so-called global relation: an integral expression that couples initial and boundary data. The global relation can be found by constructing a differential form dependent on some spectral parameter, that is closed on the condition that a given partial dif ferential equation is satisf ied. Such a differential form is said to be fundamental [Quart. J. Mech. Appl. Math. 55 (2002), 457–479]. We give an algorithmic approach in constructing a fundamental k-form associated with a given boundary value problem, and address issues of uniqueness. Also, we extend a result of Fokas and Zyskin to give an integral representation to the solution of a class of boundary value problems, in an arbitrary number of dimensions. We present an extended example using these results in which we construct a global relation for the linearised Navier-Stokes equations.

Open Problem

It is suggested here that an interesting and important line of inquiry is the elaboration of methods of inverse scattering transform (IST) type in contexts where non-homogeneous boundary conditions intercede. The issue, which has practical relevance we indicate by example, appears ripe for development, thanks to recent new ideas interjected into the panoply of IST methodologies. A sketch of the principal steps envisaged in carrying out analysis of boundary-value problems using inverse scattering ideas is provided.

The classical methods for solving initial-boundary-value problems for linear partial differential equations with constant coefficients rely on separation of variables, and specific integral transforms. As such, they are limited to specific equations, with special boundary conditions. Here we review a method introduced by Fokas, which contains the classical methods as special cases. However, this method allows also for the explicit solution of problems for which no classical approach exists. In addition, it is easy to elucidate which boundary-value problems are well posed and which are not. We provide explicit examples of problems posed on the positive half-line and on the finite interval. Some of these examples have solutions obtainable using classical methods, others do not. For the former examples, it is illustrated how the classical methods may be recovered from the more general approach of Fokas.

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas' transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over π.

A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equations in cylindrical coordinates). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via the Fourier or Laplace transforms. Examples include the heat equation and the first and second versions of the Stokes equation for arbitrary initial and boundary data on the half-line. The new method has advantages in comparison with classical methods, such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries. As a result, these solutions are well suited for numerics, allowing the solution to be computed at any point in space and time without the need to time step. Simple deformation of the contours of integration followed by mapping the contours from the complex plane to the real line allow for fast and efficient numerical evaluation of the integrals.

A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann-Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine-Gordon equation in light-cone coordinates, the nonlinear Schrödinger equation and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrödinger equation is derived.

A new method for studying boundary value problems for linear and for integrable nonlinear partial differential equations (PDE's) in two dimensions is reviewed. This method provides a unification as well as a significant extension of the following three seemingly different topics: (a) The classical integral transform method for solving linear PDE's and several of its variations such as the Wiener-Hopf technique. (b) The integral representation of the solution of linear PDE's in terms of the Ehrenpreis fundamental principle. (c) The inverse spectral (scattering) method for solving the initial value problem for nonlinear integrable evolution equations. The detailed implementation of the method is presented for: (a) An arbitrary linear dispersive evolution equation on the half line. (b) The nonlinear Schrödinger equation on the half line. (c) The Laplace, Helmholtz and modified Helmholtz equations in an arbitrary convex polygon. In addition, several other applications are briefly considered. The possible extension of this method to multidimensions is also discussed.

We introduce a new transform method for solving initial‐boundary‐value problems for linear evolution partial differential equations with spatial derivatives of arbitrary order. This method is illustrated by solving several such problems on the half‐line {t > 0, 0 < x < ∞}, and on the quarter‐plane {t > 0, 0 < xj < ∞, j = 1, 2}. For equations in one space dimension this method constructs q(x, t) as an integral in the complex k‐plane involving an x‐transform of the initial condition and a t‐transform of the boundary conditions. For equations in two space dimensions it constructs q(x1, x2, t) as an integral in the complex (k1, k2)‐planes involving an (x1, x2)‐transform of the initial condition, an (x2, t)‐transform of the boundary conditions at x1 = 0, and an (x1, t)‐transform of the boundary conditions at x2 = 0. This method is simple to implement and yet it yields integral representations which are particularly convenient for computing the long time asymptotics of the solution.

Every applied mathematician has used separation of variables. For a given boundary value problem (BVP) in two dimensions, the starting point of this powerful method is the separation of the given PDE into two ODEs. If the spectral analysis of either of these ODEs yields an appropriate transform pair, i.e. a transform consistent with the given boundary conditions, then the given BVP can be reduced to a BVP for an ODE. For simple BVPs it is straightforward to choose an appropriate transform and hence the spectral analysis can be avoided. In spite of its enormous applicability, this method has certain limitations. In particular, it requires the given domain, PDE, and boundary conditions to be separable, and also may not be applicable if the BVP is non-self-adjoint. Furthermore it expresses the solution as either an integral or a series, neither of which are uniformly convergent on the boundary of the domain (for non-vanishing boundary conditions), which renders such expressions unsuitable for numerical computations.

This paper describes a recently introduced transform method that can be applied to certain non-separable and non-self-adjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane that is uniformly convergent on the boundary of the domain. The starting point of the method is to write the PDE as a one parameter family of equations formulated in a divergence form, and this allows one to consider the variables together. In this sense, the method is based on the “synthesis” as opposed to the “separation” of variables. The new method has already been applied to a plethora of BVPs and furthermore has led to the developement of certain novel numerical techniques. However, a large number of related analytical and numerical questions remain open.

This paper illustrates the method by applying it to two particular non-self-adjoint BVPs: one for the linearised KdV equation formulated on the half line, and the other for the Helmholtz equation in the exterior of the disc (the latter is non-self-adjoint due to a radiation condition). The former problem played a crucial role in the development of the new method, whereas the latter problem was instrumental in the full development of the classical transform method.

Although the new method can now be presented using only classical techniques, it actually originated in the theory of certain nonlinear PDEs called integrable, whose crucial feature is the existence of a Lax pair formulation. It is shown here that Lax pairs provide the generalisation of the divergence formulation from a separable linear to an integrable nonlinear PDE.

The unified transform method of A. S. Fokas has led to important new developments, regarding the analysis and solution of various types of linear and nonlinear PDE problems. In this work we use these developments and obtain the solution of time-dependent problems in a straightforward manner and with such high accuracy that cannot be reached within reasonable time by use of the existing numerical methods. More specifically, an integral representation of the solution is obtained by use of the A. S. Fokas approach, which provides the value of the solution at any point, without requiring the solution of linear systems or any other calculation at intermediate time levels and without raising any stability problems. For instance, the solution of the initial boundary value problem with the non-homogeneous heat equation is obtained with accuracy 10⁻¹⁵, while the well-established Crank–Nicholson scheme requires 2048 time steps in order to reach a 10⁻⁸ accuracy.

Chapter 1 of this thesis outlines the main motivation behind the introduction of the FM. Chapter 2 is devoted to the one-dimensional heat equation (1.9), formulated on the half-line as well as on the finite interval. Motivated by a problem which arises in the context of heat conduction, we begin by implementing the FM for the oblique Robin boundary condition (1.16). Furthermore, we investigate certain non-local boundary conditions arising as conservation laws in various physical situations.

Chapter 3 focuses on linear evolution PDEs in two spatial dimensions. In particular, we analyse the two-dimensional heat equation formulated on the quarter-plane with oblique Neumann (non-separable) boundary conditions, and the linearised version of the Kadomtsev-Petviashvili II equation (1.27) posed on the upper half-plane.

The latter problem paves the way to Chapters 4 and 5, which we regard as the core of the thesis. Following a recent breakthrough achieved by Fokas in 2009 regarding the Davey-Stewartson system (1.29) (see [55]), we present a method for the analysis of the nonlinear Kadomtsev-Petviashvili I and Kadomtsev-Petviashvili II equations on the upper half-plane by using an appropriate d-bar formalism (see Appendix 7.3).

We conclude the thesis with an overview of some interesting extensions and open problems that should be investigated in the near future, following the main ideas and the methodology presented in the preceding chapters.

This article presents an overview of a transform method for solving linear and integrable nonlinear partial differential equations. This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method.

In this review I summarise some of the most significant advances of the last decade in the analysis and solution of boundary value problems posed for integrable partial differential equations in two independent variables. These equations arise widely in mathematical physics, and in order to model realistic applications, it is essential to consider bounded domain and inhomogeneous boundary conditions.

I focus specifically on a general and widely applicable approach, usually referred to as the Unified Transform or Fokas Transform, that provides a substantial generalisation of the classical Inverse Scattering Transform. This approach preserves the conceptual efficiency and aesthetic appeal of the more classical transform approaches, but presentsa distinctive and important difference. While the Inverse Scattering Transform follows the ``separation of variables'' philosophy, albeit in a nonlinear setting, the Unified Transform is a based on the idea of synthesis, rather than separation, of variables.

I will outline the main ideas in the case of linear evolution equations, and then illustrate their generalisation to certain nonlinear cases of particular significance.

The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the unified transform introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals introduced by one of the authors. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.

Inverse Scattering methods for solving integrable nonlinear p.d.e. found their limits as soon as one tried to solve with them new boundary value problems. However, some of these problems, e.g. the quarter-plane problem, can be solved (e.g. by Fokas linear methods), for related linear p.d.e., (e.g. LKdV). It is shown here that a nonlinear algebraic inverse scattering method, which we already applied to nonlinear KdV, but with only partial results, gives the full solution of the quarter-plane problem of another linear p.d.e. associated to KdV. The method makes use of generalised Lax equations and their solutions.

It is known that the unified transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coefficient evolution equation on the finite interval or the half-line. In contrast, classical methods such as Fourier series and transform techniques may only be used to solve certain problems. The solution representation obtained by such a classical method is known to be an expansion in the eigenfunctions or generalised eigenfunctions of the self-adjoint ordinary differential operator associated with the spatial part of the initial-boundary value problem. In this work, we emphasise that the unified transform method may be viewed as the natural extension of Fourier transform techniques for non-self-adjoint operators. Moreover, we investigate the spectral meaning of the transform pair used in the new method; we discuss the recent definition of a new class of spectral functionals and show how it permits the diagonalisation of certain non-self-adjoint spatial differential operators.

We describe a solution method for initial-boundary value problems of linear dispersive evolution equations on the half-line and present some results obtained by this method.

In this dissertation we evaluate numerically the solution representations obtained from a recently developed Fokas integral method for solving boundary value problems for linear evolution PDEs. In particular, we consider the case of the linear KdV equation.The Fokas method is quite general and it is therefore of wider interest to assess its competitiveness for numerical purposes. Until now pseudospectral methods have been know to be the most accurate numerical scheme for smooth functions. In the work following, the linear KdV equation will be computed numerically using both a pseudospectral method and the dircet evaluation of the integral representation, and comparisons will be made between these two methods for accuracy and speed of the numerical computation. The nonlinear KdV equation will be looked at using pseudospectral methods and a motivation for a possible hybrid method which would use both the Fokas and pseudospectral methods together will be given.

In this book the solution representations obtained from a recently developed Fokas integral method for solving boundary value problems for linear evolution PDEs are evaluate numerically. In particular, the case of the linear KdV equation is considered. The Fokas method is quite general and it is therefore of wider interest to assess its competitiveness for numerical purposes. Until now pseudospectral methods have been know to be the most accurate numerical scheme for smooth functions. To compare the two methods, the linear KdV equation is computed numerically using both, a pseudospectral method and the direct evaluation of the integral representation. The two methods are compared for accuracy and speed of the numerical computation, showing that for linear evolutionary PDEs the numerical implementation of Fokas method is much faster and more accurate than a pseudospectral method. The nonlinear KdV equation is also looked at using pseudospectral methods and a motivation for a possible hybrid method which would use both the Fokas and pseudospectral methods together is given.

A new transform method recently developed by Fokas, is used for the study of two-point boundary value problems of the form qt(x, t) + Tq(x, t) = 0 for linear evolution partial differential equations of arbitrary order, posed on the finite space domain [0,L]. Here T is an appropriately defined x-differential operator and suitable initial and boundary data is assumed.

The solution representation is expressible as an integral in the complex plane. For problems of odd order such representations are new, while for even orders it is shown that they are equivalent to classical series representations.

Spectral codes are developed for the numerical solution of a variety of illustrative examples, with many different types of boundary conditions. Finally, these codes are generalised and developed for linear third order problems for the solution of two-point boundary value problems for the important nonlinear equation, the Korteweg-deVries equation.

We implement the new transform method for solving boundary value problems developed by Fokas for periodic boundary conditions. The approach presented here is not a replacement for classical methods nor is it necessarily an improvement. However, in addition to establishing that periodic problems can indeed be solve by the new transform method (which enhances further its scope and applicability), our implementation also has the advantage that it yields a new simpler approach to computing the limit from the periodic Cauchy problem to the Cauchy problem on the line

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas' transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over π.

We use the heat equation as an illustrative example to show that the unified method introduced by one of the authors can be employed for constructing analytical solutions for linear evolution partial differential equations in one spatial dimension involving non-separable boundary conditions as well as non-local constraints. Furthermore, we show that for the particular case in which the boundary conditions become separable, the unified method provides an easier way for constructing the relevant classical spectral representations avoiding the classical spectral analysis approach. We note that the unified method always yields integral expressions which, in contrast to the series or integral expressions obtained by the standard transform methods, are uniformly convergent at the boundary. Thus, even for the cases that the standard transform methods can be implemented, the unified method provides alternative solution expressions which have advantages for both numerical and asymptotic considerations. The former advantage is illustrated by providing the numerical evaluation of typical boundary value problems.

A method is introduced for solving boundary‐value problems for linear partial differential equations (PDEs) in convex polygons. It consists of three algorithmic steps.

(1) Given a PDE, construct two compatible eigenvalue equations.

(2) Given a polygon, perform the simultaneous spectral analysis of these two equations. This yields an integral representation in the complex k‐plane of the solution q(x1,x2) in terms of a function q(k), and an integral representation in the (x1, x2)‐plane of q(k) in terms of the values of q and of its derivatives on the boundary of the polygon. These boundary values are in general related, thus only some of them can be prescribed.

(3) Given appropriate boundary conditions, express the part of q(k) involving the unknown boundary values in terms of the boundary conditions. This is based on the existence of a simple global relation formulated in the complex k‐plane, and on the invariant properties of this relation.

As an illustration, the following integral representations are obtained: (a) q(x, t) for a general dispersive evolution equation of order n in a domain bounded by a linearly moving boundary; (b) q(x,y) for the Laplace, modified Helmholtz and Helmholtz equations in a convex polygon. These general formulae and the analysis of the associated global relations are used to discuss typical boundary‐value problems for evolution equations and for elliptic equations.

We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the domain 0 < x < L, 0 < t < T, with L and T positive finite constants. We present a general method for identifying well-posed problems, as well a for constructing an explicit representation of the solution of such problems. This representation has explicit x and t dependence, and it consists of an integral in the k-complex plane and of a discrete sum. As illustrative examples we solve some two-point boundary value problems for the equations iq_t + q_xx = 0 and q_t + q_xxx = 0.

We study initial boundary value problems for linear scalar partial differential equations with constant coefficients, with spatial derivatives of {\em arbitrary order}, posed on the domain $\{t>0, 0<x<L\}$. We first show that by analysing the so-called {\em global relation}, which is an algebraic relation defined in the complex $k$-plane coupling all boundary values of the solution, it is possible to identify how many boundary conditions must be prescribed at each end of the space interval in order for the problem to be well posed. We then show that the solution can be expressed as an integral in the complex $k$-plane. This integral is defined in terms of an $x$-transform of the initial condition and a $t$-transform of the boundary conditions. For particular cases, such as the case of periodic boundary conditions, or the case of boundary value problems for {\em second} order PDEs, the integral can be rewritten as an infinite series. However, there exist initial boundary value problems for which the only representation is an integral which {\em cannot} be written as an infinite series. An example of such a problem is provided by the linearised version of the KdV equation. Thus, contrary to common belief, the solution of many linear initial boundary value problems on a finite interval {\em cannot} be expressed in terms of an infinite series.

The so-called unified method expresses the solution of an initial-boundary value problem for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple initial-boundary value problems, which will be referred to as problems of type I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated initial-boundary value problems, which will be referred to as problems of type II, there does not exist a classical transform pair and the solution cannot be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalised eigenfunctions introduced in the sixties by Gel'fand and his co-authors.

Chapter 1 of this thesis outlines the main motivation behind the introduction of the FM. Chapter 2 is devoted to the one-dimensional heat equation (1.9), formulated on the half-line as well as on the finite interval. Motivated by a problem which arises in the context of heat conduction, we begin by implementing the FM for the oblique Robin boundary condition (1.16). Furthermore, we investigate certain non-local boundary conditions arising as conservation laws in various physical situations.

Chapter 3 focuses on linear evolution PDEs in two spatial dimensions. In particular, we analyse the two-dimensional heat equation formulated on the quarter-plane with oblique Neumann (non-separable) boundary conditions, and the linearised version of the Kadomtsev-Petviashvili II equation (1.27) posed on the upper half-plane.

The latter problem paves the way to Chapters 4 and 5, which we regard as the core of the thesis. Following a recent breakthrough achieved by Fokas in 2009 regarding the Davey-Stewartson system (1.29) (see [55]), we present a method for the analysis of the nonlinear Kadomtsev-Petviashvili I and Kadomtsev-Petviashvili II equations on the upper half-plane by using an appropriate d-bar formalism (see Appendix 7.3).

We conclude the thesis with an overview of some interesting extensions and open problems that should be investigated in the near future, following the main ideas and the methodology presented in the preceding chapters.

We present in detail a linear, constant-coefficient initial/boundary value problem for which the classical method of eigenfunction expansions fails. We note that the new method recently introduced by {\it A. Fokas} and {\it B. Pelloni} [IMA J. Appl. Math. 70, No.~4, 564--587 (2005; Zbl 1084.35002)] can be successfully applied to the same problem.

We identify the class of smooth boundary conditions that yield an initial-boundary value problem admitting a unique smooth solution for the case of a dispersive linear evolution PDE of arbitrary order, in one spatial dimension, defined on a finite interval.

This result is obtained by an application of a spectral transform method, introduced by Fokas, which allows us to reduce the problem to the study of the singularities of the set of functions arising as the unique solution of a certain linear system.

We use a spectral transform method to study general boundary-value problems for third-order, linear, evolution partial differential equations with constant coefficients, posed on a finite space domain. We show how this method yields a simple characterization of the discrete spectrum of the associated spatial differential operator, and discuss the obstructions that arise when trying to represent the solution of such a problem as a series of exponential functions.

We first review the theory for second-order two-point boundary-value problems, and present an alternative way to derive the classical series representation, as well as an equivalent integral representation, which generally involves complex contours. We illustrate the advantages of the integral representation by studying in some detail the case where Robin-type boundary conditions are prescribed.

We then consider the third-order case and show that the integral representation is in general not equivalent to a discrete series representation, justifying a posteriori the failure of some of the classical approaches. We illustrate the third-order case in detail, using the example of the equation qt+qxxx=0 for various types of boundary conditions. In contrast with the second-order case, the qualitative properties of the spectrum of the associated spatial differential operator depend in this case not only on the equation but also on the type of boundary conditions. In particular, the solution appears to admit a series representation only when the prescribed boundary conditions couple the two endpoints of the interval.

In this review I summarise some of the most significant advances of the last decade in the analysis and solution of boundary value problems posed for integrable partial differential equations in two independent variables. These equations arise widely in mathematical physics, and in order to model realistic applications, it is essential to consider bounded domain and inhomogeneous boundary conditions.

I focus specifically on a general and widely applicable approach, usually referred to as the Unified Transform or Fokas Transform, that provides a substantial generalisation of the classical Inverse Scattering Transform. This approach preserves the conceptual efficiency and aesthetic appeal of the more classical transform approaches, but presentsa distinctive and important difference. While the Inverse Scattering Transform follows the ``separation of variables'' philosophy, albeit in a nonlinear setting, the Unified Transform is a based on the idea of synthesis, rather than separation, of variables.

I will outline the main ideas in the case of linear evolution equations, and then illustrate their generalisation to certain nonlinear cases of particular significance.

We give a characterisation of the spectral properties of certain linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and complemented with boundary conditions that may make the operator non-selfadjoint.

In particular, we associate the spectral properties of such an operator S with the form of the integral representation of the solution of a corresponding boundary value problem for the PDE q_t(x,t)±iSq=0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we will use its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas, and studied extensively by one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

A new, unified transform method for boundary value problems on linear and integrable nonlinear partial differential equations was recently introduced by Fokas. We consider initial-boundary value problems for linear, constant-coefficient evolution equations of arbitrary order on a finite domain. We use Fokas' method to fully characterise well-posed problems. For odd order problems with non-Robin boundary conditions we identify sufficient conditions that may be checked using a simple combinatorial argument without the need for any analysis. We derive similar conditions for the existence of a series representation for the solution to a well-posed problem.

We also discuss the spectral theory of the associated linear two-point ordinary differential operator. We give new conditions for the eigenfunctions to form a complete system, characterised in terms of initial-boundary value problems.

We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series.

The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.

We discuss initial-boundary value problems of arbitrary spatial order subject to arbitrary boundary conditions. We formalise the concept of the conditioning of such a problem and show that it represents a necessary criterion for well-posedness. The other requirement for well-posedness, the convergence of certain series, is also analysed. We illustrate these results with a full classification of 3rd order problems having non-Robin boundary conditions.

Part of this work is devoted to correcting an oversight in the author's earlier work Well-posed two-point initial-boundary value problems with arbitrary boundary conditions in volume 152 of Math. Proc. Camb. Philos. Soc.

It is known that the unified transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coefficient evolution equation on the finite interval or the half-line. In contrast, classical methods such as Fourier series and transform techniques may only be used to solve certain problems. The solution representation obtained by such a classical method is known to be an expansion in the eigenfunctions or generalised eigenfunctions of the self-adjoint ordinary differential operator associated with the spatial part of the initial-boundary value problem. In this work, we emphasise that the unified transform method may be viewed as the natural extension of Fourier transform techniques for non-self-adjoint operators. Moreover, we investigate the spectral meaning of the transform pair used in the new method; we discuss the recent definition of a new class of spectral functionals and show how it permits the diagonalisation of certain non-self-adjoint spatial differential operators.

We introduce a method of solving initial boundary value problems for linear evolution equations in a time-dependent domain, and we apply it to an equation with dispersion relation ω(k), in the domain l(t)<x<∞, 0<t<T. We show that the solution of this problem admits an integral representation in the complex k plane, involving either an integral of exp[ikx-iω(k)t]ρ(k) along a time-dependent contour, or an integral of exp[ikx-iω(k)t]ρ(k,kbar) over a fixed two-dimensional domain. The functions ρ(k) and ρ(k,kbar) can be computed through the solution of a system of Volterra linear integral equations. This method can be generalized to nonlinear integrable partial differential equations.

We discuss the implementation of a method of solving initial boundary value problems in the case of integrable evolution equations in a time-dependent domain. This method is applied to a dispersive linear evolution equation with spatial derivatives of arbitrary order and to the defocusing nonlinear Schrödinger equation, in the domain l(t)<x<∞, 0<t<T, where l(t) is a given real sufficiently smooth function whose first derivative is monotonic, and T is a fixed positive constant.

We present an algorithm for characterizing the generalized Dirichlet to Neumann map for moving initial-boundary value problems. This algorithm is derived by combining the so-called global relation, which couples the initial and boundary values of the problem, with a new method for inverting certain one-dimensional integrals. This new method is based on the spectral analysis of an associated ordinary differantial equation and on the use of the d-bar formalism. As an illustration, the Neumann boundary value for the linearized Schrödinger equation is determined in terms of the Dirichlet boundary value and of the initial condition.

We study the heat, linear Schrödinger (LS), and linear KdV equations in the domain l(t) < x < ∞,0 < t < T, with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.

The general representation of the solution of the modified Helmholtz equation, qxx(x,y)+qyy(x,y)-4[beta]2q(x,y)=0, has been recently constructed. Here we apply this representation in the wedge 0≤x≤y≤[infinity] and use this solution to find the explicit steady-state of diffusion-limited coalescence, A+A[right harpoon over left]A, on the half-line, with a trap-source at the origin.

A new transform method for solving boundary value problems for linear and integrable nonlinear partial differential equations recently introduced in the literature is used here to obtain the solution of the modified Helmholtz equation q(xx)(x,y)+q(yy)(x,y)-4 beta(2)q(x,y)=0 in the triangular domain 0≤x≤L-y≤L, with mixed boundary conditions. This solution is applied to the problem of diffusion-limited coalescence, A+A<==>A, in the segment (-L/2,L/2), with traps at the edges.

We study the modified Helmholtz equation in a semi-strip with Poincaré type boundary conditions. On each side of the semi-strip the boundary conditions involve two parameters and one real-valued function. Using a new transform method recently introduced in the literature we show that the above boundary-value problem is equivalent to a $2\times 2$-matrix Riemann-Hilbert (RH) problem. If the six parameters specified by the boundary conditions satisfy certain algebraic relations this RH problem can be solved in closed form. For certain values of the parameters the solution is not unique, furthermore in some cases the solution exists only under certain restrictions on the functions specifying the boundary conditions. The asymptotics of the solution at the corners of the semi-strip is investigated. In the case that the $2\times 2$ RH problem cannot be solved in closed form, the Carleman-Vekua method for regularising it is illustrated by analysing in detail a particular case.

We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a non-local equation in the Fourier space that relates the known boundary data to the unknown boundary values. Assuming that the global relation is satisfied in the weakest possible sense, i.e. in a distributional sense, we prove there exist solutions to Dirichlet, Neumann and Robin boundary value problems with distributional boundary data. We also show that the analysis of the global relation characterises in a straightforward manner the possible existence of both integrable and non-integrable corner-singularities.

In this paper, we address some of the rigorous foundations of the Fokas method, confining attention to boundary value problems for linear elliptic partial differential equations on bounded convex domains. The central object in the method is the global relation, which is an integral equation in the spectral Fourier space that couples the given boundary data with the unknown boundary values. Using techniques from complex analysis of several variables, we prove that a solution to the global relation provides a solution to the corresponding boundary value problem, and that the solution to the global relation is unique. The result holds for any number of spatial dimensions and for a variety of boundary value problems.

We provide a new approach to studying the Dirichlet-Neumann map for Laplace's equation on a convex polygon using Fokas' unified method for boundary value problems. By exploiting the complex analytic structure inherent in the unified method, we provide new proofs of classical results using mainly complex analytic techniques. The analysis takes place in a Banach space of complex valued, analytic functions and the methodology is based on classical results from complex analysis. Our approach gives way to new numerical treatments of the underlying boundary value problem and the associated Dirichlet-Neumann map. Using these new results we provide a family of well-posed weak problems associated with the Dirichlet-Neumann map, and prove relevant coercivity estimates so that standard techniques can be applied.

A transform method for determining the flow generated by the singularities of Stokes flow in a two-dimensional channel is presented. The analysis is based on a general approach to biharmonic boundary value problems in a simply connected polygon formulated by Crowdy & Fokas in this journal. The method differs from a traditional Fourier transform approach in entailing a simultaneous spectral analysis in the independent variables both along and across the channel. As an example application, we find the evolution equations for a circular treadmilling microswimmer in the channel correct to third order in the swimmer radius. Significantly, the new transform method is extendible to the analysis of Stokes flows in more complicated polygonal microchannel geometries.

A new method for solving the biharmonic equation in an arbitrary convex polygon with arbitrary linear boundary conditions is applied to a class of mixed boundary-value problems involving a semi-infinite strip. Emphasis is placed on boundary-value problems for which an explicit solution can be constructed. A variety of mixed boundary-value problems are shown to admit explicit solutions. This class includes, but is not limited to, the canonical problems of the elastostatic semi-strip. New integral representations of generalizations of these canonical problems are derived where the sidewall boundary conditions are inhomogeneous.

In 1750 D'Alembert demonstrated how a linear partial differential equation can be solved via separation of variables, a method that decomposes a PDE into a set of ODEs. This method was the basis for the development of many branches of contemporary analysis, from function spaces to spectral analysis of operators and the theory of special functions. A condition for the method of separation of variables to work is the existence of a coordinate system that fits the boundary of the fundamental domain and at the same time it separates the PDE. It is remarkable that two and a half centuries later a generalization is introduced that has its origin in the analysis of non-linear integrable equations. In the present work, this promising new transform method is outlined and applied to particular boundary value problems. A crucial part of the method is the introduction of a global relation which, if properly used, can provide the missing boundary data in a very elegant and effective way. We show how this can be used to generate separable solutions of partial differential equations even when no system, that fits the geometry of the fundamental domain, is available. This is shown for the case of the Dirichlet problem for the modified Helmholtz equation in the interior of an equilateral triangle. Furthermore, the connection of the Fokas method to the classical moment problem is investigated. It is shown that, in this case, the global relation is decomposed into a sequence of global relations, directly associated with the Fourier coefficients of the Dirichlet and Neumann boundary values.

In his deep and prolific investigations of heat diffusion, Lamé was led to the investigation of the eigenvalues and eigenfunctions of the Laplace operator in an equilateral triangle. In particular, he derived explicit results for the Dirichlet and Neumann cases using an ingenious change of variables. The relevant eigenfunctions are a complicated infinite series in terms of his variables.

Here we first show that boundary-value problems with simple boundary conditions, such as the Dirichlet and the Neumann problems, can be solved in an elementary manner. In particular, the unknown Neumann and Dirichlet boundary values can be expressed in terms of a Fourier series for the Dirichlet and the Neumann problems, respectively. Our analysis is based on the so-called global relation, which is an algebraic equation coupling the Dirichlet and the Neumann spectral values on the perimeter of the triangle.

As Lamé correctly pointed out, infinite series are inadequate for expressing the solution of more complicated problems such as mixed boundary-value problems. In this paper we show, further utilizing the global relation, that such problems can be solved in terms of generalized Fourier integrals.

A novel method for solving boundary value problems for linear partial differential equations in convex polygons was developed by A.S. Fokas in the late 1990s. A spectrally accurate numerical implementation for the case of the Laplace equation led the way for the present implementation for both the Helmholtz equation and the modified Helmholtz equation. We obtain again spectrally accurate numerical solutions for Dirichlet to Neumann maps, without any discretization of the domain interior. The eigenvalue problem is also examined using the present Legendre expansion method. A key feature of the current approach is that all integrals that arise in the map problem can be evaluated in closed form, reducing it to the solution of a small overdetermined linear system of equations.

Novel integral formulae for harmonic functions in rectangular domains are presented. This representations are analytic in the complex C-plane, displaying strong decay as the complex variable tends to infinity and are therefore suitable for numerical computations and asymptotic analysis of the solution. The analysis is based on the new method introduced by Fokas and his collaborators, yielding novel formulae even for simple problems that can be solved by the method of the classical transforms. This is achieved by implementing the global relation, which is an integral relation connecting the boundary values of the solution with the normal derivative of the solution on the boundary, in appropriate subdomains of the fundamental domain. Furthermore, a changing-type boundary value problem is solved.

A recently discovered transform approach allows a large class of PDEs to be solved in terms of boundary and/or contour integrals. We introduce here a spectrally accurate numerical discretization of this approach for the case of Laplace's equation on a polygonal domain, and compare it against an also spectrally accurate implementation of the traditional boundary integral formulation.

A new approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was introduced in Fokas [A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 53 (1997) 1411-1443]. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet to Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. This is based on the analysis of the so-called global relation, an equation which couples known and unknown components of the derivative on the boundary and which is valid for all values of a complex parameter k. A collocation-type numerical method for solving the global relation for the Laplace equation in an arbitrary bounded convex polygon was introduced in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465-483]. Here, by choosing a different set of the "collocation points" (values for k), we present a significant improvement of the results in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465-483]. The new collocation points lead to well-conditioned collocation methods. Their combination with sine basis functions leads to a collocation matrix whose diagonal blocks are point diagonal matrices yielding efficient implementation of iterative methods; numerical experimentation suggests quadratic convergence. The choice of Chebyshev basis functions leads to higher order convergence, which for regular polygons appear to be exponential.

A new method for analyzing initial-boundary value problems for linear and integrable nonlinear partial differential equations (PDEs) has been introduced by one of the authors.

For linear PDEs this method yields analytical solutions for certain problems that apparently cannot be solved by classical methods such as Green's function representations, classical transforms and the method of images. Even for problems that can be solved by classical methods, the new method has advantages such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries.

Here we review the numerical implementation of the new method to evolution and elliptic PDEs.

A new numerical method for solving linear elliptic boundary value problems with constant coefficients in a polygonal domain is introduced. This method produces a generalized Dirichlet-Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicular to this direction is computed without solving on the interior of the domain. If desired, the solution on the interior can then be computed via an integral representation.

The key to the method is a "global condition" which couples known and unknown components of the derivative on the boundary and which is valid for all values of a complex parameter k. This condition has been solved recently analytically for several equations on simple domains. In this paper, first the previous analytical result is strengthened, and then a numerical method is introduced for solving the global condition for the Laplace equation on an arbitrary bounded convex polygon. Numerical results demonstrate the applicability and convergence of the method; however, a rigorous proof of convergence remains open. Extensions to other problems are also discussed.

The recent numerical implementation by Fornberg and collaborators of the so-called unified method to linear elliptic PDEs in polygonal domains involves the computation of the finite Fourier transform of the Legendre polynomials. A variation of this approach, introduced by two of the authors, also involves the same computation. Here, instead of expressing the finite Fourier transform of the Legendre polynomials in terms of Bessel functions J_{n+1/2}, we employ an explicit formula in terms of exponentials. We illustrate the usefulness of this formula, which is considerably cheaper to use, by implementing the unified method to the modified Helmholtz equation in the interior of a square. For completeness we present an explicit formula for the finite Fourier transform of all Jacobi polynomials.

Let (x,y) satisfy the Laplace equation in an arbitrary convex polygon. By performing the spectral analysis of the equation [mu]z - ik[mu] = qx - iqy, z = x + y, which involves solving a scalar Riemann-Hilbert (RH) problem, we construct an integral representation in the complex k-pla q(x,y) in terms of a function [rho](k). It has been recently shown that the function [rho](k) can be expressed in terms of the given boundary conditions by solving a matrix RH problem. Here we show that this method is also useful for solving problems in a non-convex polygon. We also recall that for simple polygons it is possible to bypass the above integral representation and to solve the Laplace equation by formulating a RH problem in the complex z-plane.

Let q(x, y) satisfy a boundary value problem for the Laplace equation in an arbitrary convex polygon with n sides. An integral representation in the complex k‐plane is given for q(x, y) in terms of n functions ρj(k), j = 1, ..., n. The function ρj consists of an integral over the jth side involving both qx and qy, thus each ρj involves one unknown boundary value. The functions ρj are not independent but they satisfy the important global relation that their sum vanishes. The solution of a given boundary value problem reduces to the analysis of this single relation for the n unknown ρj. For a general polygon with general Poincaré boundary conditions, this gives rise to a matrix Riemann-Hilbert problem. In this paper it is shown that for simple polygons and for a large class of boundary conditions, the above Riemann-Hilbert problem (a) can either be reduced to a triangular RH problem which can be solved in closed form or (b) can be bypassed, and the ρj can be obtained using only algebraic manipulations. As an illustration of these 'triangular' and 'algebraic' cases we solve the Laplace equation in the quarter‐plane, the semi‐infinite strip and the right isosceles triangle with certain Poincaré boundary conditions. These boundary value problems, which include the Dirichlet and the Neumann problems as particular cases, cannot be solved by conformal mappings.

A method is introduced for solving boundary‐value problems for linear partial differential equations (PDEs) in convex polygons. It consists of three algorithmic steps.

(1) Given a PDE, construct two compatible eigenvalue equations.

(2) Given a polygon, perform the simultaneous spectral analysis of these two equations. This yields an integral representation in the complex k‐plane of the solution q(x1,x2) in terms of a function q(k), and an integral representation in the (x1, x2)‐plane of q(k) in terms of the values of q and of its derivatives on the boundary of the polygon. These boundary values are in general related, thus only some of them can be prescribed.

(3) Given appropriate boundary conditions, express the part of q(k) involving the unknown boundary values in terms of the boundary conditions. This is based on the existence of a simple global relation formulated in the complex k‐plane, and on the invariant properties of this relation.

As an illustration, the following integral representations are obtained: (a) q(x, t) for a general dispersive evolution equation of order n in a domain bounded by a linearly moving boundary; (b) q(x,y) for the Laplace, modified Helmholtz and Helmholtz equations in a convex polygon. These general formulae and the analysis of the associated global relations are used to discuss typical boundary‐value problems for evolution equations and for elliptic equations.

We discuss two different approaches for the analysis of the Poisson and of the non-homogeneous biharmonic equations in two dimensions. The first approach yields the solution as an integral in the complex $z$-plane (the physical plane), involving explicitly the given boundary conditions. The second approach yields an integral in the complex $k$-plane (the Fourier plane), involving the Fourier transforms of the given boundary conditions. For simple boundary value problems, such as certain problems formulated in the half complex plane, the first approach is easier. However, for more complicated problems, such as those formulated in the interior of an equilateral triangle, it appears that only the second approach can be used. Furthermore, the second approach also seems more efficient for numerical computations.

In this work we derive the structural properties of the Collocation coefficient matrix associated with the Dirichlet-Neumann map for Laplace's equation on a square domain. The analysis is independent of the choice of basis functions and includes the case involving the same type of boundary conditions on all sides, as well as the case where different boundary conditions are used on each side of the square domain. Taking advantage of said properties, we present efficient implementations of direct factorization and iterative methods, including classical SOR-type and Krylov subspace (Bi-CGSTAB and GMRES) methods appropriately preconditioned, for both Sine and Chebyshev basis functions. Numerical experimentation, to verify our results, is also included.

Analytic expressions for the Fourier transforms of the Chebyshev and Legendre polynomials are derived, and the latter is used to find a new representation for the half-order Bessel functions. The numerical implementation of the so-called unified method in the interior of a convex polygon provides an example of the applicability of these analytic expressions.

Integral representations for the solutions of the Laplace and modified Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the solution and its normal derivative on the boundary, and for a well-posed boundary-value problem (BVP) one of these functions is unknown. Determining the Neumann data from the Dirichlet data is known as constructing the Dirichlet-to-Neumann map. A new transform method was introduced in Fokas (1997, Proc. R. Soc. Lond. A, 53, 1411-1443) for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs). For linear PDEs this method can be considered as the analogue of the Green's function approach in the Fourier plane. In this method the Dirichlet-to-Neumann map is characterized by a certain equation, the so-called global relation, which is formulated in the complex k-plane, where k denotes the complex extension of the spectral (Fourier) variable. Here we solve the global relation numerically for the Laplace and modified Helmholtz equations in a convex polygon. This is achieved by evaluating the gobal relation at a properly chosen set of points in the spectral (Fourier) plane, which is why this method has been called a 'spectral collocation method'. Numerical experiments suggest that the method inherits the order of convergence of the basis used to expand the unknown functions, namely, exponential for a polynomial basis such as Chebyshev, and algebraic for a Fourier basis. However, the condition number of the associated linear system is much higher for a polynomial basis than for a Fourier one.

A new transform method for investigating boundary value problems in two dimensions was introduced by Fokas recently, which provides an effective approach to solve the linear elliptic PDEs in convex polygonal. In this paper, the feasibility and effectiveness of this method have been analyzed, which provide a reference for the use of Fokas transform. As an illustration, the boundary value problem for the modified Helmholtz equation in a wedge domain is discussed.

We study the high-frequency asymptotics for the solution of the modified Helmholtz equation, in a quarter plane and subject to a specific Neumann condition, as an illustration of the procedure to extract the asymptotics via Fokas' transform method. The approach is based on Fokas' integral representation of the solution. Full asymptotic expansions of the solution are obtained by using Watson's lemma and the method of steepest descents for definite integrals

We introduce some novel formulae in the theory of Segre quaternions, namely a Dbar (or Ponpeiu-Borel) formula as well as certain variations of the fundamental theorem of calculus. The latter enable us to obtain the solution to a boundary value problem of a four dimensional Laplace equation by using spectral analysis.

We study a system of conserved quantities of the periodic Klein–Gordon equation. We obtain these quantities by means of a perturbation construction from a Lax pair of the periodic sine–Gordon equation. We show that for a suitable choice of values of the spectral parameter, our conserved quantities have simple expressions in terms of the Fourier coefficients of the initial data. Moreover, they turn out to be the action variables. This provides an interesting illustration of the role of the spectral parameter. Our perturbation construction also provides a new Lax pair for the Klein-Gordon equation, and our action variables arise from this Lax pair. This turns out to be a special case (k = 2) of a more general Lax pair for a certain k-jet system of the sine-Gordon equation. The structure algebra of this Lax pair is the algebra $\mathcal{T}\mathcal{A}_k$ of upper triangular k × k block Toeplitz matrices whose blocks are elements of $\mathfrak {s} \mathfrak {u}(2)$. The Ad-invariant functions on the Lie group $\mathcal{T}\mathcal{G}_k$ corresponding to the Lie algebra $\mathcal{T}\mathcal{A}_k$ are needed for the construction of the integrals. These functions are not given by the spectra of the matrices. They have to be constructed by other means.

This thesis is concerned with new analytical and numerical methods for solving boundary value problems for the 2nd order linear elliptic PDEs of Poisson, Helmholtz, and modified Helmholtz in two dimensions.

In 1967 a new method called the Inverse Scattering Transform (IST) method was introduced to solve the initial value problem of certain non-linear PDEs (so-called "integrable" PDEs) including the celebrated Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equation. The extension of the IST method from initial value problems to boundary value problems (BVPs) was achieved by Fokas in 1997 when a unified method for solving BVPs for integrable nonlinear and linear PDEs was introduced. This thesis applies "the Fokas method" to the basic elliptic PDEs in two dimensions.

It is perhaps suprising that ideas from the theory of integrable nonlinear PDEs can be used to obtain new results in the classical theory of linear PDEs. In fact, the new method has a beautiful connection with the classical integral representations of the solutions of these PDEs due to Green. Indeed, this thesis shows that the Fokas method provides the analogue of Green's integral representation (IR) in the transform, or spectral, space. Both Irs contain boundary values which are not given as boundary conditions, and the main difficulty with BVPs is determining these unknown boundary values. In addition to the novel IR, the Fokas method provides a relation coupling the transforms of both the known and unknown boundary values known as "the global relation", which is then used to determine the contribution of the unknown boundary values to the solution.

One of the conclusions of this thesis is that the new method (applied to these 2nd order linear elliptic PDEs) does three things: (a) solves certain BVPs which cannot be solved by classical techniques, (b) yields novel expressions for the solutions of BVPs which have both analytical and computational advantages over the classical ones, and (c) provides an alternative, simpler, method for obtaining the classical solutions.

Chapter 2 is about the novel integral representations. Chapter 3 is about the global relation. In Chapter 4, a variety of boundary value problems in the separable domains of the half plane, quarter plane and the exterior of the circle are solved. In Chapter 5, boundary value problems are solved in a non-separable domain, the interior of a right isosceles triangle. Just as Green's integral representation gives rise to a numerical method for solving these PDEs (the boundary integral method), the Fokas method can also be used to design new numerical schemes; Chapter 6 presents these for the Laplace and modfied Helmholtz equation in the interior of a convex polygon.

A new and novel approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was recently introduced. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet-Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. For this computation, a collocation-type numerical method has been recently developed. Here, we study the collocation's coefficient matrix properties. We prove that, for the Laplace's equation on regular polygon domains with the same type of boundary conditions on each side, the collocation matrix is Block Circulant, independently of the choice of basis functions. This leads to the deployment of the FFT for the solution of the associated collocation linear system, yielding significant computational savings. Numerical experiments are included to demonstrate the efficiency of the whole computation.

A new method for solving boundary-value problems (BVPs) for linear and certain nonlinear PDEs was introduced by one of the authors in the late 1990s. For linear PDEs, this method constructs novel integral representations (IRs) that are formulated in the Fourier (transform) space. In a previous paper, a simplified way of obtaining these representations was presented. In the current paper, first, the second ingredient of the new method, namely the derivation of the so-called 'global relation' (GR) - an equation involving transforms of the boundary values - is presented. Then, using the GR as well as the IR derived in the previous paper, certain BVPs in polar coordinates are solved. These BVPs elucidate the fact that this method has substantial advantages over the classical transform method.

Every applied mathematician has used separation of variables. For a given boundary value problem (BVP) in two dimensions, the starting point of this powerful method is the separation of the given PDE into two ODEs. If the spectral analysis of either of these ODEs yields an appropriate transform pair, i.e. a transform consistent with the given boundary conditions, then the given BVP can be reduced to a BVP for an ODE. For simple BVPs it is straightforward to choose an appropriate transform and hence the spectral analysis can be avoided. In spite of its enormous applicability, this method has certain limitations. In particular, it requires the given domain, PDE, and boundary conditions to be separable, and also may not be applicable if the BVP is non-self-adjoint. Furthermore it expresses the solution as either an integral or a series, neither of which are uniformly convergent on the boundary of the domain (for non-vanishing boundary conditions), which renders such expressions unsuitable for numerical computations.

This paper describes a recently introduced transform method that can be applied to certain non-separable and non-self-adjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane that is uniformly convergent on the boundary of the domain. The starting point of the method is to write the PDE as a one parameter family of equations formulated in a divergence form, and this allows one to consider the variables together. In this sense, the method is based on the “synthesis” as opposed to the “separation” of variables. The new method has already been applied to a plethora of BVPs and furthermore has led to the developement of certain novel numerical techniques. However, a large number of related analytical and numerical questions remain open.

This paper illustrates the method by applying it to two particular non-self-adjoint BVPs: one for the linearised KdV equation formulated on the half line, and the other for the Helmholtz equation in the exterior of the disc (the latter is non-self-adjoint due to a radiation condition). The former problem played a crucial role in the development of the new method, whereas the latter problem was instrumental in the full development of the classical transform method.

Although the new method can now be presented using only classical techniques, it actually originated in the theory of certain nonlinear PDEs called integrable, whose crucial feature is the existence of a Lax pair formulation. It is shown here that Lax pairs provide the generalisation of the divergence formulation from a separable linear to an integrable nonlinear PDE.

The so-called unified method expresses the solution of an initial-boundary value problem for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple initial-boundary value problems, which will be referred to as problems of type I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated initial-boundary value problems, which will be referred to as problems of type II, there does not exist a classical transform pair and the solution cannot be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalised eigenfunctions introduced in the sixties by Gel'fand and his co-authors.

We give a characterisation of the spectral properties of certain linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and complemented with boundary conditions that may make the operator non-selfadjoint.

In particular, we associate the spectral properties of such an operator S with the form of the integral representation of the solution of a corresponding boundary value problem for the PDE q_t(x,t)±iSq=0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we will use its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas, and studied extensively by one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the unified transform introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals introduced by one of the authors. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.

A new, unified transform method for boundary value problems on linear and integrable nonlinear partial differential equations was recently introduced by Fokas. We consider initial-boundary value problems for linear, constant-coefficient evolution equations of arbitrary order on a finite domain. We use Fokas' method to fully characterise well-posed problems. For odd order problems with non-Robin boundary conditions we identify sufficient conditions that may be checked using a simple combinatorial argument without the need for any analysis. We derive similar conditions for the existence of a series representation for the solution to a well-posed problem.

We also discuss the spectral theory of the associated linear two-point ordinary differential operator. We give new conditions for the eigenfunctions to form a complete system, characterised in terms of initial-boundary value problems.

It is known that the unified transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coefficient evolution equation on the finite interval or the half-line. In contrast, classical methods such as Fourier series and transform techniques may only be used to solve certain problems. The solution representation obtained by such a classical method is known to be an expansion in the eigenfunctions or generalised eigenfunctions of the self-adjoint ordinary differential operator associated with the spatial part of the initial-boundary value problem. In this work, we emphasise that the unified transform method may be viewed as the natural extension of Fourier transform techniques for non-self-adjoint operators. Moreover, we investigate the spectral meaning of the transform pair used in the new method; we discuss the recent definition of a new class of spectral functionals and show how it permits the diagonalisation of certain non-self-adjoint spatial differential operators.

A general method for studying boundary value problems for linear and for integrable nonlinear partial differential equations in two dimensions was introduced in Fokas, 1997. For linear equations in a convex polygon (Fokas and Kapaev (2000) and (2003), and Fokas (2001)), this method: (a) expresses the solution in the form of an integral (generalized inverse Fourier transform) in the complex -plane involving a certain function (generalized direct Fourier transform) that is defined as an integral along the boundary of the polygon, and (b) characterizes a generalized Dirichlet-to-Neumann map by analyzing the so-called global relation. For simple polygons and simple boundary conditions, this characterization is explicit. Here, we extend the above method to the case of elliptic partial differential equations in the exterior of a convex polygon and we illustrate the main ideas by studying the Laplace equation in the exterior of an equilateral triangle.

Regarding (a), we show that whereas is identical with that of the interior problem, the contour of integration in the complex -plane appearing in the formula for depends on . Regarding (b), we show that the global relation is now replaced by a set of appropriate relations which, in addition to the boundary values, also involve certain additional unknown functions. In spite of this significant complication we show that, for certain simple boundary conditions, the exterior problem for the Laplace equation can be mapped to the solution of a Dirichlet problem formulated in the interior of a convex polygon formed by three sides.

By employing conformal mappings, it is possible to express the solution of certain boundary value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identities for special functions. A convenient tool for deriving this type of identities is the so-called \emph{global relation}, which has appeared recently in a wide range of boundary value problems. As a concrete application, we analyze the Neumann boundary value problem for the Laplace equation in the exterior of the so-called Hankel contour, which is the contour that appears in the definition of both the gamma and the Riemann zeta functions. By utilizing the explicit solution of this problem, we derive a plethora of novel identities involving the hypergeometric function.

The Unified Transform provides a novel method for analyzing boundary value problems for linear and for integrable nonlinear PDEs. The numerical implementation of this method to linear elliptic PDEs formulated in the {\it interior} of a polygon has been investigated by several authors (see the article by Iserles, Smitheman, and one of the authors in this book). Here, we show that the Unified Transform also yields a novel numerical technique for computing the solution of linear elliptic PDEs in the {\it exterior} of a polygon. One of the advantages of this new technique is that it actually yields directly the scattering amplitude. Details are presented for the modified Helmholtz equation in the exterior of a square.

The phenomenon of stimulated Raman scattering (SRS) can be described by three coupled PDEs which define the pump electric field, the Stokes electric field, and the material excitation as functions of distance and time. In the transient limit these equations are integrable, i.e., they admit a Lax pair formulation. Here we study this transient limit. The relevant physical problem can be formulated as an initial-boundary value (IBV) problem where both independent variables are on a finite domain. A general method for solving IBV problems for integrable equations has been introduced recently. Using this method we show that the solution of the equations describing the transient SRS can be obtained by solving a certain linear integral equation. It is interesting that this equation is identical to the linear integral equation characterizing the solution of an IBV problem of the sine-Gordon equation in light-cone coordinates. This integral equation can be solved uniquely in terms of the values of the pump and Stokes fields at the entry of the Raman cell. The asymptotic analysis of this solution reveals that the long-distance behavior of the system is dominated by the underlying self-similar solution which satisfies a particular case of the third Painlevé transcendent. This result is consistent with both numerical simulations and experimental observations. We also discuss briefly the effect of frequency mismatch between the pump and the Stokes electric fields.

It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrödinger equation on the half-line can be expressed through the solution of a 2×2 matrix Riemann–Hilbert problem. This problem is specified by the spectral functions {a(k),b(k)} which are defined in terms of the initial condition q(x,0)=q₀(x), and by the spectral functions {A(k),B(k)} which are defined in terms of the specified boundary condition q(0,t)=g₀(t) and the unknown boundary value q_x(0,t)=g₁(t). Furthermore, it has been shown that given q₀ and g₀, the function g₁ can be characterized through the solution of a certain ''global relation'' coupling q₀, g₀, g₁, and Φ(t,k), where Φ satisfies the t-part ofthe associated Lax pair evaluated at x=0. We show here that, by using a Gelfand–Levitan–Marchenko triangular representation of Φ, the global relation can be explicitly solved for g₁.

An initial boundary-value problem for the modified Korteweg-de Vries equation on the half-line, $0<x<\infty$, $t>0$, is analysed by expressing the solution $q(x,t)$ in terms of the solution of a matrix Riemann-Hilbert (RH) problem in the complex $k$-plane. This RH problem has explicit $(x,t)$ dependence and it involves certain functions of $k$ referred to as the spectral functions. Some of these functions are defined in terms of the initial condition $q(x,0)=q_0(x)$, while the remaining spectral functions are defined in terms of the boundary values $q(0,t)=g_0(t)$, $q_x(0,t)=g_1(t)$, and $q_{xx}(0,t)=g_2(t)$. The spectral functions satisfy an algebraic global relation which characterizes, say, $g_2(t)$ in terms of $\{q_0(x),g_0(t),g_1(t)\}$. It is shown that for a particular class of boundary conditions, the linearizable boundary conditions, all the spectral functions can be computed from the given initial data by using algebraic manipulations of the global relation; thus, in this case, the problem on the half-line can be solved as efficiently as the problem on the whole line.

This paper is concerned with the direct and inverse scattering problems for compatible differential equations connected with the nonlinear Schrödinger equation (NLSE) on the semi-axis. The corresponding initial boundary value problem (x,t∈ℝ₊) was studied recently by Fokas and Its. They found that the key to this problem is to linearize the initial boundary value problem using a Riemann-Hilbert problem. The main goal of this paper is to obtain characteristic properties of the scattering data for compatible differential equations. Our approach uses the transformation operators for both x- and t-equations. For Schwartz type initial and boundary functions we obtain the characteristic properties (A1)-(A5) of the scattering data and derive the so-called xt- and t-integral equations of Marchenko type. The xt-integral equations guarantee the existence of the solution of the NLSE and give an expression of the solution with given scattering data. In turn, the t-integral equations guarantee that one can recover from the scattering data the boundary Dirichlet data v(t) and the corresponding Neumann data w(t) consistent with the given initial function u(x).

This article is about the focusing nonlinear Schrödinger equation on the half-line. The initial function vanishes at infinity while boundary data are local perturbations of periodic or quasi-periodic (finite-gap) functions. We study the corresponding scattering problem for the Zakharov-Shabat compatible differential equations, the representation of the solution of the nonlinear Schrödinger equation in the quarter of the (x,t)-plane through functions, which satisfy Marchenko integral equations. We use this formalism to investigate the asymptotic behavior of the solution for large time. We prove that under certain conditions a periodic (quasi-periodic) behavior at infinity of boundary data generates an unbounded train of asymptotic solitons running away from the boundary. The asymptotics of the solution shows that boundary data with periodic behavior as time tends to infinity generates a train of such asymptotic solitons even in the case when the initial function is identically zero.

In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in some half-strip $(0<x<\infty)\times[0,T]$, or in the quarter plane $(0<x<\infty)\times(0<t<\infty)$. We suppose that this solution has a $C^{\infty}$ initial function vanishing as $x\to\infty$, and $C^{\infty}$ boundary values, vanishing as $t\to\infty$ when $T=\infty$. We study the corresponding scattering problem for the compatible Zakharov-Shabat system of differential equations associated with the mKdV equation and obtain a representation of the solution of the mKdV equation through Marchenko integral equations of the inverse scattering method. The kernel of these equations is valid only for $x\geq 0$ and it takes into account all specific properties of the pair of compatible differential equations in the chosen half-strip or in the quarter plane. The main result is the collection A-B-C of characteristic properties of the scattering functions given in the paper.

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic. The main goal of this paper is to introduce a Riemann-Hilbert problem whose solution gives the solution of our initial-boundary-value problem. This is a preliminary step to obtain uniform long-time asymptotics for the solution of this equation using both the stationary phase and the Deift-Zhou methods.

The long time behaviour of solutions of (Dirichlet) initial boundary value problems for the focusing nonlinear Schrödinger equation in the case of (asymptotically) periodic boundary conditions of special (one-frequency) structure is considered both theoretically and numerically. The results of numerical simulations are shown to confirm the theoretical description in the parameter ranges where the assumption about the one-frequency character of the Neumann boundary values is (theoretically) admissible, whereas in the other ranges they suggest a more complicated description of the behaviour of these values.

The authors further develop the nonlinear steepest descent method of [5] and [6] to give a full description of the collisionless shock region for solutions of the KdV equation with decaying initial data.

In this announcement we present a general and new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, in evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg de Vries (MKdV) equation, $$y_t-6y^2y_x+y_{xxx}=0,\qquad -\infty<x<\infty,\ t>0, y(x,t=0)=y_0(x),$$ but it will be clear immediately to the reader with some experience in the field, that the method extends naturally and easily to the general class of wave equations solvable by the inverse scattering method, such as the KdV, nonlinear Schrödinger (NLS), and Boussinesq equations, etc., and also to ``integrable'' ordinary differential equations such as the Painlevé transcendents.

The inverse spectral method is a nonlinear Fourier transform method for solving certain equations. Here, we emphasize that such transforms should be considered in their own right. We also elucidate further the connection between the Fourier transform and inverse spectral methods by establishing that linear equations can also be solved through the inverse spectral method.

Boundary value problems for integrable nonlinear evolution PDEs, like the modified KdV equation, formulated on the half-line can be analyzed by the so-called unified transform method. For the modified KdV equation, this method yields the solution in terms of the solution of a matrix Riemann-Hilbert problem uniquely determined in terms of the initial datum q(x,0), as well as of the boundary values {q(0, t),qx(0, t),qxx(0, t)}. For the Dirichlet problem, it is necessary to characterize the unknown boundary values qx(0, t) and qxx(0, t) in terms of the given data q(x, 0) and q(0, t). It is shown here that in the particular case of a vanishing initial datum and of a sine wave as Dirichlet datum, qx(0, t) and qxx(0, t) can be computed explicitly at least up to third order in a perturbative expansion and that at least up to this order, these functions are asymptotically periodic for large t.

The solution of the initial-boundary-value problem of integrabale nonlinear evolution equations, with the spatial variable on a half-infinite line, can be reduced to the solution of a linear intregral equation. The asymptotic analysis of this equation for large t shows how the boundary conditions can generate solitons.

We consider the sine-Gordon equation in laboratory coordinates with both x and t in [0, ∞). We assume that u(x, 0), u_t(x, 0), u(0, t) are given, and that they satisfy u(x, 0)→2πq, u_t(x, 0)→0, for large x, u(0, t)→2πp for large t, where q, p are integers. We also assume that u_x(x, 0), u_t(x, 0), u_t(0, t), u(0, t)-2πp, u(x, 0)-2πq ∈ L₂. We show that the solution of this initial-boundary value problem can be reduced to solving a linear integral equation which is always solvable. The asymptotic analysis of this integral equation for large t shows how the boundary conditions can generate solitons.

We consider the nonlinear Schrödinger (NLS) equation in the variable $q(x,t)$ with both $x$ and $t$ in $[ {0,\infty } )$. We assume that $q(x,0) = u(x)$ and $q(0,t) = v(t)$ are given, that $u(0) = v(0)$, and that $u(x)$ and $v(t)$ as well as their first two derivatives belong to $L_1 \cap L_2 (\mathbb{R}^ + )$. We show that the solution of this initial-boundary value problem can be reduced to solving a Riemann-Hilbert (RH) problem in the complex $k$-plane with jumps on $\operatorname{Im} (k^2 ) = 0$. This RH problem is equivalent to a linear integral equation which has a unique global solution. This linear integral equation is uniquely defined in terms of certain functions (scattering data) $b(k)$ and $c(k)$. The function $b(k)$ can be effectively computed in terms of $u(x)$. However, although the analytic properties of $c(k)$ are completely determined, the relationship between $c(k)$, $u(x)$ and $v(t)$ is highly nonlinear. In spite of this difficulty, we can give an effective description of the asymptotic behavior of $q(x,t)$ for large $t$. In particular, we show that as $t \to \infty $, solitons are generated moving away from the boundary. In addition, our formalism can be used to generate effectively pairs of functions $q(0,t)$ and $q_x (0,t)$ compatible with a given $q(x,0)$ as well as to determine the associated $q(x,t)$. It is important to emphasize that the analysis of this problem, in addition to techniques of exact integrability, requires the essential use of general partial differential equations (PDE) techniques.

Assuming that the solution q ( x , t ) of the nonlinear Schrödinger equation on the half-line exists, it has been shown in Fokas (2002 Commun. Math. Phys. 230 1-39) that q ( x , t ) can be represented in terms of the solution of a matrix Riemann-Hilbert (RH) problem formulated in the complex k -plane. The jump matrix of this RH problem has explicit x , t dependence and it is defined in terms of the scalar functions { a ( k ), b ( k ), A ( k ), B ( k )} referred to as spectral functions. The functions a ( k ) and b ( k ) are defined in terms of q 0 ( x ) = q ( x ,0), while the functions A ( k ) and B ( k ) are defined in terms of g 0 ( t ) = q (0, t ) and g 1 ( t ) = q x (0, t ). The spectral functions are not independent but they satisfy an algebraic global relation . Here we first prove that if there exist spectral functions satisfying this global relation, then the function q ( x , t ) defined in terms of the above RH problem exists globally and solves the nonlinear Schrödinger equation, and furthermore q ( x , 0) = q 0 ( x ), q (0, t ) = g 0 ( t ) and q x (0, t ) = g 1 ( t ). We then show that, given appropriate initial and boundary conditions, it is possible to construct such spectral functions through the solution of a nonlinear Volterra integral equation whose solution exists globally. We also show that for a particular class of boundary conditions it is possible to bypass this nonlinear equation and to compute the spectral functions using only the algebraic manipulation of the global relation; thus for this particular class of boundary conditions, which we call linearizable , the problem on the half-line can be solved as effectively as the problem on the line. An example of a linearizable boundary condition is q x (0, t ) - ρ q (0, t ) = 0 where ρ is a real constant.

We study the zero-dispersion limit for certain initial boundary value problems for the defocusing nonlinear Schrödinger (NLS) equationand for the Korteweg-de Vries (KdV)equation with dominant surface tension. These problems are formulated on the half-line and they involve linearisable boundaryconditions.

We analyze initial-boundary value problems for an integrable generalization of the nonlinear Schrödinger equation formulated on the half-line. In particular, we investigate the so-called linearizable boundary conditions, which in this case are of Robin type. Furthermore, we use a particular solution to verify explicitly all the steps needed for the solution of a well-posed problem.

There exists a distinctive class of physically significant evolution PDEs in one spatial dimension which can be treated analytically. A prototypical example of this class (which is referred to as integrable) is the Korteweg-de Vries equation. Integrable PDEs on the line can be analysed by the so-called inverse scattering transform (IST) method. A particularly powerful aspect of the IST is its ability to predict the large t behaviour of the solution. Namely, starting with initial data u(x, 0), IST implies that the solution u(x, t) asymptotes to a collection of solitons as t → ∞, x/t = O(1); moreover, the shapes and speeds of these solitons can be computed from u(x, 0) using only linear operations. One of the most important developments in this area has been the generalization of the IST from initial to initial-boundary value (IBV) problems formulated on the half-line. It can be shown that again u(x, t) asymptotes into a collection of solitons, where now the shapes and the speeds of these solitons depend both on u(x, 0) and on the given boundary conditions at x = 0. A major complication of IBV problems is that the computation of the shapes and speeds of the solitons involves the solution of a nonlinear Volterra integral equation. However, for a certain class of boundary conditions, called linearizable, this complication can be bypassed and the relevant computation is as effective as in the case of the problem on the line. Here, after reviewing the general theory for KdV, we analyse three different types of linearizable boundary conditions. For these cases, the initial conditions are (a) restrictions of one- and two-soliton solutions at t = 0; (b) profiles of certain exponential type and (c) box-shaped profiles. For each of these cases, by computing explicitly the shapes and the speeds of the asymptotic solitons, we elucidate the influence of the boundary.

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant, and on the computation of the large $k$ asymptotics of the eigenfunctions defining the relevant spectral functions.

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we first present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation and on the introduction of the so-called Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the spectral functions. We then concentrate on the physically significant case of $t$-periodic Dirichlet boundary data. After presenting certain heuristic arguments which suggest that the Neumann boundary values become periodic as $t\to\infty$, we show that for the case of the NLS with a sine-wave as Dirichlet data, the asymptotics of the Neumann boundary values can be computed explicitly at least up to third order in a perturbative expansion and indeed at least up to this order are asymptotically periodic.

A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann-Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine-Gordon equation in light-cone coordinates, the nonlinear Schrödinger equation and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrödinger equation is derived.

A new method for studying boundary value problems for linear and for integrable nonlinear partial differential equations (PDE's) in two dimensions is reviewed. This method provides a unification as well as a significant extension of the following three seemingly different topics: (a) The classical integral transform method for solving linear PDE's and several of its variations such as the Wiener-Hopf technique. (b) The integral representation of the solution of linear PDE's in terms of the Ehrenpreis fundamental principle. (c) The inverse spectral (scattering) method for solving the initial value problem for nonlinear integrable evolution equations. The detailed implementation of the method is presented for: (a) An arbitrary linear dispersive evolution equation on the half line. (b) The nonlinear Schrödinger equation on the half line. (c) The Laplace, Helmholtz and modified Helmholtz equations in an arbitrary convex polygon. In addition, several other applications are briefly considered. The possible extension of this method to multidimensions is also discussed.

A rigorous methodology for the analysis of initial-boundary value problems on the half-line, is applied to the nonlinear §(NLS), to the sine-Gordon (sG) in laboratory coordinates, and to the Korteweg-deVries (KdV) with dominant surface tension. Decaying initial conditions as well as a smooth subset of the boundary values are given, where n=2 for the NLS and the sG and n=3 for the KdV. For the NLS and the KdV equations, the initial condition q(x,0) = q 0 (x) as well as one and two boundary conditions are given respectively; for the sG equation the initial conditions q(x,0) = q 0 (x), q t (x,0) = q 1 (x), as well as one boundary condition are given. The construction of the solution q(x,t) of any of these problems involves two separate steps: (a) Given decaying initial conditions define the spectral (scattering) functions {a(k),b(k)}. Associated with the smooth functions , define the spectral functions {A(k),B(k)}. Define the function q(x,t) in terms of the solution of a matrix Riemann-Hilbert problem formulated in the complex k-plane and uniquely defined in terms of the spectral functions {a(k),b(k),A(k),B(k)}. Under the assumption that there exist functions such that the spectral functions satisfy a certain global algebraic relation, prove that the function q(x,t) is defined for all , it satisfies the given nonlinear PDE, and furthermore that . (b) Given a subset of the functions as boundary conditions, prove that the above algebraic relation characterizes the unknown part of this set. In general this involves the solution of a nonlinear Volterra integral equation which is shown to have a global solution. For a particular class of boundary conditions, called linearizable, this nonlinear equation can be bypassed and {A(k),B(k)} can be constructed using only the algebraic manipulation of the global relation. For the NLS, the sG, and the KdV, the following particular linearizable cases are solved: , respectively, where χ is a real constant.

A rigorous methodology for the analysis of initial boundary value problems on the half-line, 0 < x < ∞, t > 0, for integrable nonlinear evolution PDEs has recently appeared in the literature. As an application of this methodology the solution q(x, t) of the sine-Gordon equation can be obtained in terms of the solution of a 2 × 2 matrix Riemann-Hilbert problem. This problem is formulated in the complex k-plane and is uniquely defined in terms of the so-called spectral functions a(k), b(k), and B(k)/A(k). The functions a(k) and b(k) can be constructed in terms of the given initial conditions q(x, 0) and q_t(x, 0) via the solution of a system of two linear ODEs, while for arbitrary boundary conditions the functions A(k) and B(k) can be constructed in terms of the given boundary condition via the solution of a system of four nonlinear ODEs. In this paper, we analyse two particular boundary conditions: the case of constant Dirichlet data, q(0, t) = χ, as well as the case when q_x(0, t), sin (q(0, t)/2), and cos(q(0, t)/2) are linearly related by two constants χ₁ and χ₂. We show that for these particular cases, the system of the above nonlinear ODEs can be avoided, and B(k)/A(k) can be computed explicitly in terms of { a(k), b(k), χ} and {a(k), b(k), χ₁, χ₂}, respectively. Thus, these 'linearizable' initial boundary value problems can be solved with absolutely the same level of efficiency as the classical initial value problem of the line.

Let q(x,t) satisfy a nonlinear integrable evolution PDE whose highest spatial derivative is of order n. An initial boundary value problem on the half-line for such a PDE is at least linearly well-posed if one prescribes initial conditions, as well as N boundary conditions at x = 0, where for n even N equals n/2 and for n odd, depending on the sign of the highest derivative, N equals either n−1/2 or n+1/2. For example, for the nonlinear Schrödinger (NLS) and the sine-Gordon (sG), N = 1, while for the modified Korteweg-deVries (mKdV) N = 1 or N = 2 depending on the sign of the third derivative. Constructing the generalized Dirichlet-to-Neumann map means determining those boundary values at x = 0 that are not prescribed as boundary conditions in terms of the given initial and boundary conditions. A general methodology is presented that constructs this map in terms of the solution of a system of two nonlinear ODEs. This formulation implies that for the focusing NLS, for the sG, and for the two focusing versions of the mKdV, this map is global in time. It appears that this is the first time in the literature that such a characterization for nonlinear PDEs is explicitly described. It is also shown here that for particular choices of the boundary conditions the above map can be linearized.

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q_t and q_xxx have the same sign (KdVI) or two boundary conditions if q_t and q_xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q_x(0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q_x(0,t),q_xx(0,t)} and {q_xx(0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ^(t)(t,k), where Φ^(t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation for Φ^(t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation for Φ^(t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ^(t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation.

We present a Riemann-Hilbert problem formalism for the initial-boundary value problem for the three-wave equation:
\[p_{ij,t}-\frac{b_i-b_j}{a_i-a_j}p_{ij,x}+\sum_k(\frac{b_k-b_j}{a_k-a_j}-\frac{b_i-b_k}{a_i-a_k})p_{ik}p_{kj}=0,\quad i,j,k=1,2,3.\]
on the half-line.

We present a Riemann-Hilbert problem formalism for the initial-boundary value problem for the Sasa-Satsuma(SS) equation: $iq_T+\frac{1}{2}q_{XX}+|q|^2q+i\eps (q_{XXX}+6|q|^2q_X+3q(|q|^2)_X)=0$ on the half-line. And we also analysis the global relation in this paper.

We consider the initial-boundary-value problem on the half-line for the Korteweg-de Vries equation $\begin{equation}\left.\begin{aligned} u_{t} + uu_{x}+u_{xxx} = 0,\;\;\;\;\;t > 0,\;\;\;x > 0, \\ u(x,0) = u_{0}(x),\;\;\;\;\;x > 0 \\ u(0,t) = 0,\;\;\;\;\;t > 0. \end{aligned}\right\}$ We prove that if the initial data u₀ \in H₁^0,2 \cap H₂^0,3/2 and the norm |u₀|_H1^0,2 + |u₀|_H2^0,3/2 are sufficiently small, where $H_{p}^{s,k} = \lbrace f \epsilon L^{2}; {\|f\|}{H_p^{s,k}} = \|\langle x \rangle ^k \langle i\delta_x \rangle ^s f{\|Lp\|} < \infty\rbrace$, $\langle x \rangle = \sqrt{1 + x^{2}}$, then there exists a unique solution u \in C([0, ∞), H₂^0,1)\cap L^∞ (0,∞, H₂^0,3/2) \cap C((0, ∞), H₂^2,0) \cap L^∞ (0, ∞, H₂^3,0) of the initial boundary-value problem. Moreover, we proved that there exists a constant B such that the solution has the following asymptotics, $u(x,t)= t^{-1} B \frac{x}{3\sqrt{t} Ai({x\over{3\sqrt{t}}}) + O(t^{-1-{\delta/3}} max(1, {x\over{3 \sqrt{t}}}))$ for t \to ∞ uniformly with respect to x > 0, where $0 < \delta < \frac{1}{2}$, Ai(q) is the Airy function defined by Ai(q) = \int ^i∞_-i∞e^-z3+zqdz.

There exists a particular class of boundary value problems for integrable nonlinear evolution equations formulated on the half-line, called linearizable. For this class of boundary value problems, the Fokas method yields a formalism for the solution of the associated initial-boundary value problem, which is as efficient as the analogous formalism for the Cauchy problem. Here, we employ this formalism for the analysis of several concrete initial-boundary value problems for the nonlinear Schrödinger equation. This includes problems involving initial conditions of a hump type coupled with boundary conditions of Robin type.

We are studying the semiclassical limit of the 1+1 dimensional integrable nonlinear Schrödinger equation with defocusing cubic nonlinearity on the half line. Our analysis relies on the recent theory of Fokas et al., which reduces boundary value problems for soliton equations to Riemann-Hilbert factorization problems. We employ the method of nonlinear steepest descent to asymptotically deform the given Riemann-Hilbert problem to an explicilty solvable one.

The concept of the nonholonomic deformation formulated recently for the Ablowitz-Kaup-Newell-Segur family is extended to the Kaup-Newell class. Applying this construction we discover a novel integrable mixed twofold hierarchy related to the deformed derivative nonlinear Schrödinger (DNLS) equation and found the exact soliton solutions exhibiting unusual accelerating motion for both its field and the perturbing functions. Extending the idea of deformation the integrable perturbation of the gauge related Chen-Lee-Liu DNLS equation is constructed together with its soliton solution. We show that the recently proposed Lenells-Fokas (LF) equation falls in the deformed DNLS hierarchy, sharing the accelerating soliton and other unusual features. Higher order integrable deformations of the LF and the DNLS equations are proposed.

We analyze the derivative nonlinear Schrödinger equation on the half-line using the Fokas method. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann–Hilbert problem formulated in the plane of the complex spectral parameter ζ. The jump matrix has explicit x,t dependence and is given in terms of the spectral functions a(ζ), b(ζ) (obtained from the initial data q₀(x)=q(x,0)) as well as A(ζ), B(ζ) (obtained from the boundary values g₀(t)=q(0,t) and g₁(t)=q_x(0,t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q₀(x),g₀(t),g₁(t)} such that there exist spectral functions satisfying the global relation, we show that the function q(x,t) defined by the above Riemann–Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.

We consider initial-boundary value problems for the derivative nonlinear Schrödinger (DNLS) equation on the half-line x>0. In a previous work, we showed that the solution q(x,t) can be expressed in terms of the solution of a Riemann–Hilbert problem with jump condition specified by the initial and boundary values of q(x,t). However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.

We present an approach for analyzing initial-boundary value problems for integrable equations whose Lax pairs involve 3×3 matrices. Whereas initial value problems for integrable equations can be analyzed by means of the classical Inverse Scattering Transform (IST), the presence of a boundary presents new challenges. Over the last fifteen years, an extension of the IST formalism developed by Fokas and his collaborators has been successful in analyzing boundary value problems for several of the most important integrable equations with 2×2 Lax pairs, such as the Korteweg–de Vries, the nonlinear Schrödinger, and the sine-Gordon equations. In this paper, we extend these ideas to the case of equations with Lax pairs involving 3×3 matrices.

Chapter 1 of this thesis outlines the main motivation behind the introduction of the FM. Chapter 2 is devoted to the one-dimensional heat equation (1.9), formulated on the half-line as well as on the finite interval. Motivated by a problem which arises in the context of heat conduction, we begin by implementing the FM for the oblique Robin boundary condition (1.16). Furthermore, we investigate certain non-local boundary conditions arising as conservation laws in various physical situations.

Chapter 3 focuses on linear evolution PDEs in two spatial dimensions. In particular, we analyse the two-dimensional heat equation formulated on the quarter-plane with oblique Neumann (non-separable) boundary conditions, and the linearised version of the Kadomtsev-Petviashvili II equation (1.27) posed on the upper half-plane.

The latter problem paves the way to Chapters 4 and 5, which we regard as the core of the thesis. Following a recent breakthrough achieved by Fokas in 2009 regarding the Davey-Stewartson system (1.29) (see [55]), we present a method for the analysis of the nonlinear Kadomtsev-Petviashvili I and Kadomtsev-Petviashvili II equations on the upper half-plane by using an appropriate d-bar formalism (see Appendix 7.3).

We conclude the thesis with an overview of some interesting extensions and open problems that should be investigated in the near future, following the main ideas and the methodology presented in the preceding chapters.

In this review I summarise some of the most significant advances of the last decade in the analysis and solution of boundary value problems posed for integrable partial differential equations in two independent variables. These equations arise widely in mathematical physics, and in order to model realistic applications, it is essential to consider bounded domain and inhomogeneous boundary conditions.

I focus specifically on a general and widely applicable approach, usually referred to as the Unified Transform or Fokas Transform, that provides a substantial generalisation of the classical Inverse Scattering Transform. This approach preserves the conceptual efficiency and aesthetic appeal of the more classical transform approaches, but presentsa distinctive and important difference. While the Inverse Scattering Transform follows the ``separation of variables'' philosophy, albeit in a nonlinear setting, the Unified Transform is a based on the idea of synthesis, rather than separation, of variables.

I will outline the main ideas in the case of linear evolution equations, and then illustrate their generalisation to certain nonlinear cases of particular significance.

We show that by deforming the Riemann-Hilbert (RH) formalism associated with certain linear PDEs and using the so-called dressing method, it is possible to derive in an algorithmic way nonlinear integrable versions of these equations. In the usual Dressing Method, one first postulates a matrix RH problem and then constructs dressing operators. Here we present an algorithmic construction of matrix Riemann-Hilbert (RH) problems appropriate for the dressing method as opposed to postulating them ad hoc. Furthermore, we introduce two mechanisms for the con- struction of the relevant dressing operators: The first uses operators with the same dispersive part, but with different decay at infinity, while the second uses pairs of operators corresponding to different Lax pairs of the same linear equation. As an application of our approach, we derive the NLS, derivative NLS, KdV, modified KdV and sine-Gordon equations.

This work is devoted to an integrable generalization of the nonlinear Schrödinger equation proposed by Fokas and Lenells. I discuss the relationships between this equation and other integrable models. Using the reduction of the Fokas–Lenells equation to the already known ones I obtain the N-dark soliton solutions.

The paper deals with a problem of developing an inverse-scattering based formalism for solving problems for the cubic nonlinear (or the modified Korteweg-de Vries (KdV)) equations: qt+qxxx+6q2qx=0, 0=xtqt+qxxx-6q2qx=0, with the given initial and boundary conditions: q(x,0)=q(x),q(0,t)=p(t), p(t)?L1(-8,8). The relation between the solution of the initial-boundary value problem (1), (3), (4) and that of the KdV equation on the half-line is shown. The Cauchy problem for the cubic nonlinear equation: qt+qxxx-6|q|2qx=0, 0=xtxt<8.

The paper deals with the problems for the sine-Gordon and sinh-Gordon equations on a half-line:



with the given initial and boundary conditions:



The Dirichlet initial-boundary-value problem (IBVP) (1), (3), (4) is associated with the scattering problem (SP) for the system



J = diag(1, − 1), u = (u₁, u₂), with the boundary condition:



The Dirichlet IBVP (2)-(4) is associated with the SP for the system:


with the boundary condition (6).

We apply a formalism of the direct and inverse SP (5), (6) ((7), (6)) to investigate the considered IBVPs. The difficulty associated with these problems is that the time dependence of the scattering data set s of the SP is determined by unknown boundary values (BVs) at x = 0 of the Jost solutions of (5) ((7)) and more, the evolution equations for BVs contain unknown boundary data (BD). We overcome this difficulty in the following way. We show that the evolution of the BD is described by a linear Volterra integral equation, the solution of which is found in terms of the given (3) and (4). Then the BVs are calculated in terms of (3) and (4). We find the necessary and sufficient conditions on the quantities of the data set s which ensure a unique solution of the inverse problem. Therefore, the antiderivative of the recovered potential is the solution of the considered IBVP.

The numerical approximation of one-dimensional cubic nonlinear Schrödinger equations on the whole real axis is studied in this paper. Based on the work of A. Boutet de Monvel, A.S. Fokas and D. Shepelsky [Lett. Math. Phys., 65(3): 199-212, 2003], a kind of exact nonreflecting boundary conditions are derived on the artificially introduced boundary points. The related numerical issues are discussed in detail. Several numerical tests are performed to demonstrate the behaviour of the proposed scheme.

Absorbing boundary conditions (ABCs) are generally required for simulating waves in unbounded domains. As one of those approaches for designing ABCs, perfectly matched layer (PML) has achieved great success for both linear and nonlinear wave equations. In this paper we apply PML to the nonlinear Schrödinger wave equations. The idea involved is stimulated by the good performance of PML for the linear Schrödinger equation with constant potentials, together with the time-transverse invariant property held by the nonlinear Schrödinger wave equations. Numerical tests demonstrate the effectiveness of our PML approach for both nonlinear Schrödinger equations and some Schrödinger-coupled systems in each spatial dimension.

Let q(x,t) satisfy an integrable nonlinear evolution PDE on the interval 0<x<L, and let the order of the highest x-derivative be n. For a problem to be at least linearly well-posed one must prescribe N boundary conditions at x=0 and n-N boundary conditions at x=L, where if n is even, N=n/2, and if n is odd, N is either (n−1)/2 or (n+1)/2, depending on the sign of ∂nxq. For example, for the sine-Gordon (sG) equation one must prescribe one boundary condition at each end, while for the modified Korteweg-de Vries (mKdV) equations involving qt+qxxx and qt−qxxx one must prescribe one and two boundary conditions, respectively, at x=0. We will refer to these two mKdV equations as mKdV-I and mKdV-II, respectively.

Here we analyze the Dirichlet problem for the sG equation, as well as typical boundary value problems for the mKdV-I and mKdV-II equations. We first show that the unknown boundary values at each end (for example, qx(0,t) and qx(L,t) in the case of the Dirichlet problem for the sG equation) can be expressed in terms of the given initial and boundary conditions through a system of four nonlinear ODEs. We then show that q(x,t) can be expressed in terms of the solution of a 2×2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x,t) dependence in the form of an exponential; for example, for the case of the sG this exponential is exp {i(k−1/k)x+i(k+1/k)t}. Furthermore, the relevant jump matrices are explicitly given in terms of the spectral functions {a(k),b(k)}, {A(k),B(k)}, and , which in turn are defined in terms of the initial conditions, of the boundary values of q and of its x-derivatives at x=0, and of the boundary values of q and of its x-derivatives at x=L, respectively. This Riemann-Hilbert problem has a global solution.

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann–Hilbert problem in the complex k-plane. This Riemann–Hilbert problem has explicit (x,t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of the initial condition q(x,0)=q₀(x), while the remaining spectral functions are defined in terms of two sets of boundary values. We show that the spectral functions satisfy an algebraic “global relation” that characterize the boundary values in spectral terms.

On analyse l'équation de "Korteweg-de Vries modifiée" sur un intervalle borné (0,L), avec conditions aux limites en t=0 et en x=0,L, en exprimant sa solution q(x,t) en termes de la solution d'un problème de Riemann-Hilbert associé. Ce problème est défini par des fonctions spectrales déterminées par les conditions aux limites. Nous explicitons la relation globale qui reflète en termes de ces fonctions spectrales la compatibilité des conditions aux limites.

Let q(x, t) satisfy the Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation on the finite interval, 0 < x < L, with q₀(x) = q(x, 0), g₀(t) = q(0, t), f₀(t) = q(L, t). Let g₁(t) and f₁(t) denote the unknown boundary values q_x(0, t) and q_x(L, t), respectively. We first show that these unknown functions can be expressed in terms of the given initial and boundary conditions through the solution of a system of nonlinear ODEs. For the focusing case it can be shown that this system has a global solution. It appears that this is the first time in the literature that such a characterization is explicitly described for a nonlinear evolution PDE defined on the interval; this result is the extension of the analogous result of [4] and [20] from the half-line to the interval. We then show that q(x, t) can be expressed in terms of the solution of a 2 × 2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x, t) dependence in the form exp[2ikx + 4ik²t], and it has jumps across the real and imaginary axes. The relevant jump matrices are explicitly given in terms of the spectral data {a(k), b(k)}, {A(k), B(k)}, and , which in turn are defined in terms of q₀(x), {g₀(t), g₁(t)}, and {f₀(t), f₁(t)}, respectively.

Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving six scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other four depend on all boundary values. The most difficult step of the new method is the characterization of the latter four spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data. We present two different characterizations of this problem. One is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant, and on the computation of the large $k$ asymptotics of the eigenfunctions defining the relevant spectral functions. The other is based on the analysis of the global relation and on the introduction of the so-called Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the relevant spectral functions. We also show that these two different characterizations are equivalent and that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained recently for the case of boundary value problems formulated on the half-line.

We use the Fokas method to analyze the derivative nonlinear Schr\"odinger (DNLS) equation $iq_t(x,t)=-q_{xx}(x,t)+(r q^2)_x$ on the interval $[0,L]$. Assuming that the solution $q(x,t)$ exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $\x$. This problem has explicit $(x,t)$ dependence, and it has jumps across $\{\x \in \C|\im{\x^4}=0 \}$. The relevant jump matrices are explicitly given in terms of the spectral functions $\{a(\x),b(\x)\},\{A(\x),B(\x)\}$, and $\{\ca({\x}),\cb(\x)\}$, which in turn are defined in terms of the initial data $q_0(x)=q(x,0)$, the boundary data $g_0(t)=q(0,t),g_1(t)=q_x(0,t)$, and another boundary values $f_0(t)=q(L,t),f_1(t)=q_x(L,t)$. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

We use the Fokas method to analyze the derivative nonlinear Schr¨odinger (DNLS) equation iq_t(x, t) = −q_xx(x, t)+(rq²)_x on the interval [0, L]. Assuming that the solution q(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit (x, t) dependence, and it has jumps across {ξ ∈ C|Imξ⁴ = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a(ξ), b(ξ)}, {A(ξ), B(ξ)}, and {A(ξ), B(ξ)}, which in turn are defined in terms of the initial data q₀(x) = q(x, 0), the boundary data g₀(t) = q(0, t), g₁(t) = q_x(0, t), and another boundary values f₀(t) = q(L, t), f₁(t) =q_x(L, t). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann–Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space–time variables. For an appropriate set of initial and boundary data, we derive the k-dependent “spectral functions” which guarantee the uniqueness of Riemann–Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given.

The initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval of the spatial variable is considered. Using the method of simultaneous spectral analysis of the associated Lax pair, this problem is mapped into a Riemann-Hilbert problem formulated in the complex plane of the spectral parameter, but with explicit dependence on the space-time variables appearing in the Korteweg-de Vries equation. It is shown that, under certain conditions, the solution of this Riemann–Hilbert is uniquely determined by certain functions of the spectral parameter, which are defined by the initial and boundary data of the original problem. In turn, the solution of the Riemann-Hilbert problem provides the solution of the initial-boundary value problem for the Korteweg-de Vries equation, for which an integral representation is derived.

Chapter 1 of this thesis outlines the main motivation behind the introduction of the FM. Chapter 2 is devoted to the one-dimensional heat equation (1.9), formulated on the half-line as well as on the finite interval. Motivated by a problem which arises in the context of heat conduction, we begin by implementing the FM for the oblique Robin boundary condition (1.16). Furthermore, we investigate certain non-local boundary conditions arising as conservation laws in various physical situations.

Chapter 3 focuses on linear evolution PDEs in two spatial dimensions. In particular, we analyse the two-dimensional heat equation formulated on the quarter-plane with oblique Neumann (non-separable) boundary conditions, and the linearised version of the Kadomtsev-Petviashvili II equation (1.27) posed on the upper half-plane.

The latter problem paves the way to Chapters 4 and 5, which we regard as the core of the thesis. Following a recent breakthrough achieved by Fokas in 2009 regarding the Davey-Stewartson system (1.29) (see [55]), we present a method for the analysis of the nonlinear Kadomtsev-Petviashvili I and Kadomtsev-Petviashvili II equations on the upper half-plane by using an appropriate d-bar formalism (see Appendix 7.3).

We conclude the thesis with an overview of some interesting extensions and open problems that should be investigated in the near future, following the main ideas and the methodology presented in the preceding chapters.

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n-N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient an in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.

In this review I summarise some of the most significant advances of the last decade in the analysis and solution of boundary value problems posed for integrable partial differential equations in two independent variables. These equations arise widely in mathematical physics, and in order to model realistic applications, it is essential to consider bounded domain and inhomogeneous boundary conditions.

I focus specifically on a general and widely applicable approach, usually referred to as the Unified Transform or Fokas Transform, that provides a substantial generalisation of the classical Inverse Scattering Transform. This approach preserves the conceptual efficiency and aesthetic appeal of the more classical transform approaches, but presentsa distinctive and important difference. While the Inverse Scattering Transform follows the ``separation of variables'' philosophy, albeit in a nonlinear setting, the Unified Transform is a based on the idea of synthesis, rather than separation, of variables.

I will outline the main ideas in the case of linear evolution equations, and then illustrate their generalisation to certain nonlinear cases of particular significance.

The stationary, axisymmetric reduction of the vacuum Einstein equations, the so-called Ernst equation, is an integrable nonlinear PDE in two dimensions. There now exists a general method for analyzing boundary value problems for integrable PDEs, and this method consists of two steps: (a) Construct an integral representation of the solution characterized via a matrix Riemann-Hilbert (RH) problem formulated in the complex $k$-plane, where $k$ denotes the spectral parameter of the associated Lax pair. This representation involves, in general, some unknown boundary values, thus the solution formula is {\it not} yet effective. (b) Characterize the unknown boundary values by analyzing a certain equation called the {\it global relation}. This analysis involves, in general, the solution of a nonlinear problem; however, for certain boundary value problems called linearizable, it is possible to determine the unknown boundary values using only linear operations. Here, we employ the above methodology for the investigation of certain boundary value problems for the elliptic version of the Ernst equation. For this problem, the main novelty is the occurence of the spectral parameter in the form of a square root and this necessitates the introduction of a two-sheeted Riemann surface for the formulation of the relevant RH problem. As a concrete application of the general formalism, it is shown that the particular boundary value problem corresponding to the physically significant case of a rotating disk is a linearizable boundary value problem. In this way the remarkable results of Neugebauer and Meinel are recovered.

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2 by 2 matrix Riemann-Hilbert problem whose "jump matrix" depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however a major difficulty for this problem is the existence of non-integrable singularities of the function q_y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann-Hilbert problem to an equivalent modified Riemann-Hilbert problem, we show that the solution can be expressed in terms of a 2 by 2 matrix Riemann-Hilbert problem whose jump matrix depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h. The determination of the function h remains open.

We analyse the Dirichlet problem for the elliptic sine Gordon equation in the upper half plane. We express the solution $q(x,y)$ in terms of a Riemann-Hilbert problem whose jump matrix is uniquely defined by a certain function $b(\la)$, $\la\in\R$, explicitly expressed in terms of the given Dirichlet data $g_0(x)=q(x,0)$ and the unknown Neumann boundary value $g_1(x)=q_y(x,0)$, where $g_0(x)$ and $g_1(x)$ are related via the global relation $\{b(\la)=0$, $\la\geq 0\}$. Furthermore, we show that the latter relation can be used to characterise the Dirichlet to Neumann map, i.e. to express $g_1(x)$ in terms of $g_0(x)$. It appears that this provides the first case that such a map is explicitly characterised for a nonlinear integrable {\em elliptic} PDE, as opposed to an {\em evolution} PDE.

We solve a class of boundary value problems for the stationary axisymmetric Einstein equations corresponding to a disk of dust rotating uniformly around a central black hole. The solutions are given explicitly in terms of theta functions on a family of hyperelliptic Riemann surfaces of genus 4. In the absence of a disk, they reduce to the Kerr black hole. In the absence of a black hole, they reduce to the Neugebauer-Meinel disk.

We consider boundary value problems for the elliptic sine-Gordon equation posed in the half plane y > 0. This problem was considered in Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197-201) using the classical inverse scattering transform approach. Given the limitations of this approach, the results obtained rely on a nonlinear constraint on the spectral data derived heuristically by analogy with the linearized case.

We revisit the analysis of such problems using a recent generalization of the inverse scattering transform known as the Fokas method, and show that the nonlinear constraint of Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197-201) is a consequence of the so-called global relation. We also show that this relation implies a stronger constraint on the spectral data, and in particular that no choice of boundary conditions can be associated with a decaying (possibly mod 2π) solution analogous to the pure soliton solutions of the usual, time-dependent sine-Gordon equation.

We also briefly indicate how, in contrast to the evolutionary case, the elliptic sine-Gordon equation posed in the half plane does not admit linearisable boundary conditions.

We study a system of conserved quantities of the periodic Klein–Gordon equation. We obtain these quantities by means of a perturbation construction from a Lax pair of the periodic sine–Gordon equation. We show that for a suitable choice of values of the spectral parameter, our conserved quantities have simple expressions in terms of the Fourier coefficients of the initial data. Moreover, they turn out to be the action variables. This provides an interesting illustration of the role of the spectral parameter. Our perturbation construction also provides a new Lax pair for the Klein-Gordon equation, and our action variables arise from this Lax pair. This turns out to be a special case (k = 2) of a more general Lax pair for a certain k-jet system of the sine-Gordon equation. The structure algebra of this Lax pair is the algebra $\mathcal{T}\mathcal{A}_k$ of upper triangular k × k block Toeplitz matrices whose blocks are elements of $\mathfrak {s} \mathfrak {u}(2)$. The Ad-invariant functions on the Lie group $\mathcal{T}\mathcal{G}_k$ corresponding to the Lie algebra $\mathcal{T}\mathcal{A}_k$ are needed for the construction of the integrals. These functions are not given by the spectra of the matrices. They have to be constructed by other means.

We present the solution of the classical problem of the heat equation formulated in the interior of an equilateral triangle with Dirichlet boundary conditions. This solution is expressed as an integral in the complex Fourier space, i.e., the complex k₁ and k₂ planes, involving appropriate integral transforms of the Dirichlet boundary conditions. By choosing Dirichlet data so that their integral transforms can be computed explicitly, we show that the solution is expressed in terms of an integral whose integrand decays exponentially as . Hence, it is possible to evaluate this integral numerically in an efficient and straightforward manner. Other types of boundary value problems, including the Neumman and Robin problems, can be solved similarly.

We introduce a new transform method for solving initial‐boundary‐value problems for linear evolution partial differential equations with spatial derivatives of arbitrary order. This method is illustrated by solving several such problems on the half‐line {t > 0, 0 < x < ∞}, and on the quarter‐plane {t > 0, 0 < xj < ∞, j = 1, 2}. For equations in one space dimension this method constructs q(x, t) as an integral in the complex k‐plane involving an x‐transform of the initial condition and a t‐transform of the boundary conditions. For equations in two space dimensions it constructs q(x1, x2, t) as an integral in the complex (k1, k2)‐planes involving an (x1, x2)‐transform of the initial condition, an (x2, t)‐transform of the boundary conditions at x1 = 0, and an (x1, t)‐transform of the boundary conditions at x2 = 0. This method is simple to implement and yet it yields integral representations which are particularly convenient for computing the long time asymptotics of the solution.

This thesis: (a) presents the solution of several boundary value problems (BVPs) for the Laplace and the modified Helmholtz equations in the interior of an equilateral triangle; (b) presents the solution of the heat equation in the interior of an equilateral triangle; (c) computes the eigenvalues and eigenfunctions of the Laplace operator in the interior of an equilateral triangle for a variety of boundary conditions; (d) discusses the solution of several BVPs for the non-linear Schrödinger equation on the half line. In 1967 the Inverse Scattering Transform method was introduced; this method can be used for the solution of the initial value problem of certain integrable equations including the celebrated Korteweg-de Vries and nonlinear Schrödinger equations. The extension of this method from initial value problems to BVPs was achieved by Fokas in 1997, when a unified method for solving BVPs for both integrable nonlinear PDEs, as well as linear PDEs was introduced. This thesis applies "the Fokas method" to obtain the results mentioned earlier. For linear PDEs, the new method yields a novel integral representation of the solution in the spectral (transform) space; this representation is not yet effective because it contains certain unknown boundary values. However, the new method also yields a relation, known as "the global relation", which couples the unknown boundary values and the given boundary conditions. By manipulating the global relation and the integral representation, it is possible to eliminate the unknown boundary values and hence to obtain an effective solution involving only the given boundary conditions. This approach is used to solve several BVPs for elliptic equations in two dimensions, as well as the heat equation in the interior of an equilateral triangle. The implementation of this approach: (a) provides an alternative way for obtaining classical solutions; (b) for problems that can be solved by classical methods, it yields novel alternative integral representations which have both analytical and computational advantages over the classical solutions; yields solutions of BVPs that apparently cannot be solved by classical methods. In addition, a novel analysis of the global relation for the Helmholtz equation provides a method for computing the eigenvalues and the eigenfunctions of the Laplace operator in the interior of an equilateral triangle for a variety of boundary conditions. Finally, for the nonlinear Schrödinger on the half line, although the global relation is in general rather complicated, it is still possible to obtain explicit results for certain boundary conditions, known as "linearizable boundary conditions". Several such explicit results are obtained and their significance regarding the asymptotic behavior of the solution is discussed.

Chapter 1 of this thesis outlines the main motivation behind the introduction of the FM. Chapter 2 is devoted to the one-dimensional heat equation (1.9), formulated on the half-line as well as on the finite interval. Motivated by a problem which arises in the context of heat conduction, we begin by implementing the FM for the oblique Robin boundary condition (1.16). Furthermore, we investigate certain non-local boundary conditions arising as conservation laws in various physical situations.

Chapter 3 focuses on linear evolution PDEs in two spatial dimensions. In particular, we analyse the two-dimensional heat equation formulated on the quarter-plane with oblique Neumann (non-separable) boundary conditions, and the linearised version of the Kadomtsev-Petviashvili II equation (1.27) posed on the upper half-plane.

The latter problem paves the way to Chapters 4 and 5, which we regard as the core of the thesis. Following a recent breakthrough achieved by Fokas in 2009 regarding the Davey-Stewartson system (1.29) (see [55]), we present a method for the analysis of the nonlinear Kadomtsev-Petviashvili I and Kadomtsev-Petviashvili II equations on the upper half-plane by using an appropriate d-bar formalism (see Appendix 7.3).

We conclude the thesis with an overview of some interesting extensions and open problems that should be investigated in the near future, following the main ideas and the methodology presented in the preceding chapters.

The recently developed Fokas method for solving two-dimensional Boundary Value Problems (BVP) via the use of global relations is utilized to solve axisymmetric problems in three dimensions. In particular, novel integral representations for the interior and exterior Dirichlet and Neumann problems for the sphere are derived, which recover and improve the already known solutions of these problems. The BVPs considered in this paper can be classically solved using either the finite Legendre transform or the Mellin-sine transform (which can be derived from the classical Mellin transform in a way similar to the way that the sine transform can be derived from the Fourier transform). The Legendre transform representation is uniformly convergent at the boundary, but it involves a series that is not useful for many applications. The Mellin-sine transform involves of course an integral but it is not uniformly convergent at the boundary. In this paper: (a) The Legendre transform representation is rederived in a simpler approach using algebraic manipulations instead of solving ODEs. (b) An integral representation, different that the Mellin-sine transform representation is derived which is uniformly convergent at the boundary. Furthermore, the derivation of the Fokas approach involves only algebraic manipulations, instead of solving an ordinary differential equation.

The axisymmetric irrotational Stokes' flow for a spherical shell is analysed by means of the recently developed Fokas method via the use of global relations. Alternative series and new integral representations concerning a system of concentric spheres, yielding, by a limiting procedure, the Dirichlet or Neumann problems for the interior and the exterior of a sphere, are presented. The boundary value problems considered can be classically solved using either the finite Gegenbauer transform or the Mellin transform. Application of the Gegenbauer transform yields a series representation which is uniformly convergent at the boundary, but not convenient for many applications. The Mellin transform, on the other hand, furnishes an integral representation which is not uniformly convergent at the boundary. Here, by algebraic manipulations of the global relation: (i) a Gegenbauer series representation is derived in a simpler manner, instead of solving ODEs and (ii) an alternative integral representation, different from the Mellin transform representation is derived which is uniformly convergent at the boundary.

A new method for investigating boundary value problems in two dimensions has recently been introduced by one of the authors. The main achievement of this method is that it yields explicit integral (as oppose to series) representations for a variety of boundary value problems. In addition, this method also provides an alternative, apparently simpler, approach for deriving those solution representations that are traditionally constructed by the method of images and of classical integral transforms. Here, we implement this latter approach to boundary value problems formulated in spherical coordinates. In particular, we do the following: (a) We derive the classical Poisson integral formula for the solutions of the Dirichlet problem for the Poisson equation in the interior of a sphere, the analogous formula for the Neumann problem, and the generalizations of these formulae in $n$ dimensions. (b) We derive the solutions of various boundary value problems for the inhomogeneous Helmholtz equation in the interior of a sphere. (c) We solve the Dirichlet problem for the Laplace equation in the interior of a spherical sector.

The classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit nonlocal equation for the wave height and for the velocity potential evaluated on the free surface. Here we first extend this formalism to n-dimensions, n>2, and then derive rigorously the linear limit of these equations. Furthermore, for n=2, we generalise the relevant formulation to the case of constant vorticity and to the case where the free surface is described by a multi-valued function. Also, in the latter case we derive a sequence of Hamiltonian systems, hence providing an approximation in the asymptotic limit of certain physical small parameters.

The classical equations of water waves are reformulated as a system of two equations, one of which is an explicit non-local equation, for the wave height and for the velocity potential evaluated on the free surface. Evaluation of the velocity potential as a function of the depth is not required in order to calculate the wave height and the velocity potential on the free surface. The non-local system yields integral relations related to mass and centre of mass, and is shown to reduce to known asymptotic limits in shallow and deep water. Included in these asymptotic reductions are the Boussinesq, Benney-Luke and nonlinear Schrödinger equations. Two-dimensional lumps with sufficient surface tension are obtained numerically. The extension of this non-local formulation to the case of a variable bottom is also presented.

High-order asymptotic series are obtained for two- and three-dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known sech² solution of the Korteweg-de Vries equation; in three dimensions, the first term is the rational lump solution of the Kadomtsev-Petviashvili equation I. The two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large-amplitude waves and waves close to the solitary version of Stokes' extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two-dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension. Finally, the above asymptotic method is applied to the Benney-Luke (BL) equation, and the resulting asymptotic series (to three terms) is verified to agree with the solitary-wave solution of the BL equation.

We present a new spectral method to simulate numerically the waterwave problem in a channel for a fluid of finite or infinite depth. It is formulated in terms of the free surface elevation η and the velocity potential . The numerical method is based on the reduction of this problem to a lower-dimensional computation involving surface variables alone. To accomplish this, we describe the Taylor expansion of the Dirichlet Neumann operator in homogeneous powers of the surface elevation η. Each term is a concatenation of Fourier multipliers with powers of η and its derivatives and is valid uniformly in wavenumber. These are easily calculated using the fast Fourier transform. The method is illustrated by computing the long time evolution of modulated wave packets and of approximations to the Stokes steady wave train. By imposing a surface pressure we observe surface steepening in large amplitude evolution, and wake and bow wave development for flows with a close to critical Froude number. Finally, we give an example of nonlinear evolution of the distribution of energy among normal modes.

Euler's equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this paper, we discuss the stability of periodic travelling wave solutions to the full set of nonlinear equations via a non-local formulation of the water wave problem, modified from that of Ablowitz, Fokas & Musslimani (J. Fluid Mech., vol. 562, 2006, p. 313), restricted to a one-dimensional surface. Transforming the non-local formulation to a travelling coordinate frame, we obtain a new formulation for the stationary solutions in the travelling reference frame as a single equation for the surface in physical coordinates. We demonstrate that this equation can be used to numerically determine non-trivial travelling wave solutions by exploiting the bifurcation structure of this new equation. Specifically, we use the continuous dependence of the amplitude of the solutions on their propagation speed. Finally, we numerically examine the spectral stability of the periodic travelling wave solutions by extending Fourier–Floquet analysis to apply to the associated linear non-local problem. In addition to presenting the full spectrum of this linear stability problem, we recover past well-known results such as the Benjamin–Feir instability for waves in deep water. In shallow water, we find different instabilities. These shallow water instabilities are critically related to the wavelength of the perturbation and are difficult to find numerically. To address this problem, we propose a strategy to estimate a priori the location in the complex plane of the eigenvalues associated with the instability.

In Ablowitz, Fokas & Musslimani (J. Fluid Mech., vol. 562, 2006, pp. 313–343) a novel formulation was proposed for water waves in three space dimensions. In the flat-bottom case, this formulation consists of the Bernoulli equation, as well as of a non-local equation. The variable-bottom case, which now involves two non-local equations, was outlined but not explored in the above paper. Here, the variable-bottom formulation is addressed in more detail. First, it is shown that in the weakly nonlinear, weakly dispersive regime, the above system of three equations can be reduced to a system of two equations. Second, by combining the novel non-local formulation of the above authors with conformal mappings, it is shown that in the two-dimensional case, it is possible to obtain a system of two equations without any asymptotic approximations. Furthermore, for the weakly nonlinear, weakly dispersive regime, the nonlinear equations are simpler than the equations obtained without conformal mappings, since they contain lower order derivatives for the terms involving the bottom variable.

(Received September 08 2011)

(Reviewed December 21 2011)

(Accepted January 03 2012)

(Online publication February 24 2012)

In an attempt to enhance the accessibility of the beautiful theory of quaternions, we first revisit this theory emphasising that it provides the proper generalisation of the theory of complex analysis. In particular, we discuss the quaternionic generalisations of the following fundamental complex analytic notions: analytic functions, Cauchy's Theorem, Cauchy's integral formula, Taylor series, Laurent series, Residue Theorem, the Pompeiu (or Cauchy-Green, or Dbar) formula, and the Plemelj-Sokhotzki formulae. We then present two applications of the theory of quaternions, which provide generalisations of the analogous complex analytic applications: (a) the solution of certain boundary value problems for the Poisson equation in four dimensions; (b) the explicit computation of certain three dimensional real integrals.

We first revisit Quaternionic Analysis in ℝ³ and discuss a representation of a quaternion-valued function in terms of two real harmonic functions. Then, we present certain integral representations of the solutions of elliptic Partial Differential Equations (PDEs) in three dimensions, such as the Poisson, inhomogeneous Biharmonic and Navier equations, and consider some relevant boundary value problems. Also, we present a physical application of an important problem in complex and quaternionic analysis called the Dbar problem, thereby elucidating the physical significance of the associated formalism.

We introduce some novel formulae in the theory of Segre quaternions, namely a Dbar (or Ponpeiu-Borel) formula as well as certain variations of the fundamental theorem of calculus. The latter enable us to obtain the solution to a boundary value problem of a four dimensional Laplace equation by using spectral analysis.

We bring together commutative quaternions, functions of two complex variables and spectral analysis to: (i) introduce some novel formulae for commutative quaternions; (ii) present a new application of this theory, namely the solution of boundary value problems. We first consider functions of two complex variables and derive an analogue of the wellknown Dbar formula appearing in complex analysis. We then focus on the subset of holomorphic functions to prove the fundamental theorem of calculus. Finally, we use this theorem to solve boundary value problems for a quaternionic generalization of the Laplace equation. The relevant domains are Cartesian products of convex polygons and the solution is obtained by spectral analysis in analogy with the analysis of two-dimensional problems, see e.g. [A.S. Fokas and A.A. Kapaev, On a transform approach for the Laplace equation in a polygon, IMA J Appl. Math 68 (2003), p. 355, A.S. Fokas and D.A. Pinotsis, The Dbar formalism for certain non homogeneous linear elliptic equations in two dimensions, Eur. J. Appl. Math. 17(3) (2006), pp. 323–346]. These results could provide a first step towards the construction of nonlinear integrable equations using commutative quaternions.

We study the equations governing a fluid-loaded plate. We first reformulate these equations as a system of two equations, one of which is an explicit non-local equation for the wave height and the velocity potential on the free surface. We then concentrate on the linearized equations and show that the problems formulated either on the full or the half-line can be solved by employing the unified approach to boundary value problems introduced by one of the authors in the late 1990s. The problem on the full line was analysed by Crighton & Oswell (Crighton & Oswell 1991 Phil. Trans. R. Soc. Lond. A 335, 557-592 (doi:10.1098/rsta.1991.0060)) using a combination of Laplace and Fourier transforms. The new approach avoids the technical difficulty of the a priori assumption that the amplitude of the plate is in L_dt¹(R⁺) and furthermore yields a simpler solution representation that immediately implies that the problem is well-posed. For the problem on the half-line, a similar analysis yields a solution representation, which, however, involves two unknown functions. The main difficulty with the half-line problem is the characterization of these two functions. By employing the so-called global relation, we show that the two functions can be obtained via the solution of a complex-valued integral equation of the convolution type. This equation can be solved in a closed form using the Laplace transform. By prescribing the initial data η₀ to be in H_〈5〉⁵(R⁺), or equivalently four times continuously differentiable with sufficient decay at infinity, we show that the solution depends continuously on the initial data, and, hence, the problem is well-posed.

We consider the motion of a collection of fluid loaded elastic plates, situated horizontally in an infinitely long channel. We use a new, unified approach to boundary value problems, introduced by A.S. Fokas in the late 1990s, and show the problem is equivalent to a system of one-parameter integral equations. We give a detailed study of the linear problem, providing explicit solutions and well-posedness results in terms of standard Sobolev spaces. We show that the associated Cauchy problem is completely determined by a matrix, which depends solely on the mean separation of the plates and the horizontal velocity of each of the driving fluids. This matrix corresponds to the infinitesimal generator of the C₀-semigroup for the evolution equations in Fourier space. By analyzing the properties of this matrix, we classify necessary and sufficient conditions for which the problem is asymptotically stable.

We introduce a new approach to constructing analytic solutions of the linear PDEs describing elastodynamics. This approach is illustrated for the case of a homogeneous isotropic half-plane body satisfying arbitrary initial conditions and Lamb's boundary conditions. A particular case of this problem, namely the case of homogeneous initial conditions, was first solved by Lamb using the Fourier-Laplace transform. The solution of the general problem can also be expressed in terms of the Fourier transform, but this representation involves transforms of unknown boundary values. This necessitates the formulation and solution of a cumbersome auxiliary problem, which expresses the unknown boundary values in terms of the Laplace transform of the given boundary data. The new approach, which is applicable to arbitrary initial and boundary conditions, bypasses the above auxiliary problem and expresses the solutions directly in terms of the given initial and boundary data.

A new approach was recently introduced by the authors for constructing analytic solutions of the linear PDEs describing elastodynamics. Here, this approach is applied to the case of a homogeneous isotropic half-space body satisfying arbitrary initial conditions and Lamb's boundary conditions. A particular case of this problem, namely the case of homogeneous initial conditions and normal point load boundary conditions, was first solved by Lamb using the Fourier-Laplace transform. The general problem solved here can also be analysed via the Fourier transform, but in this case, the solution representation involves transforms of \textit{unknown} boundary values; this necessitates the formulation and solution of a cumbersome auxiliary problem, which expresses the unknown boundary values in terms of the Laplace transform of the given boundary data. The new approach, which is applicable to arbitrary initial and boundary conditions, bypasses the above auxiliary problem and expresses the solutions directly in terms of the given initial and boundary conditions.

We develop the Riemann–Hilbert (RH) approach to scattering problems in elastic media. The approach is based on the RH method introduced in the 1990s by Fokas (A unified approach to boundary value problems, CBMS-SIAM, 2008) for studying boundary problems for linear and integrable nonlinear PDEs. A suitable Lax pair formulation of the elastodynamic equation is obtained. The integral representations derived from this Lax pair are applied to Rayleigh wave propagation in an elastic half space and quarter space. The latter problem is reduced to the analysis of a certain underdetermined RH problem. We show that the problem can be re-formulated as a well-determined vector Riemann–Hilbert problem with a shift posed on a torus.

We present some new formulae for the solution of boundary value problems for a two-dimensional isotropic elastic body. In particular, using the so-called Dbar formalism and the method introduced by Fokas (2008) [4], we obtain integral representations of the Fourier type for the displacements appearing in boundary value problems formulated in an arbitrary convex polygon with n sides.

A new method for the solution of initial-boundary value problems for evolution PDEs recently introduced by Fokas is generalised to multidimensions. Also the relation of this method with the method of images and with the classical integral transforms is discussed. The new method is easy to implement, yet it is applicable to problems for which the classical approaches apparently fail. As illustrative examples, initial-boundary value problems for the diffusion-convection equation in one and higher dimensions, as well as for the linearised Korteweg-de Vries equation with the space variables on the half-line are solved. The suitability of the new method for the analysis of the the long-time asymptotics is ellucidated.

The KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics, where it describes waves in shallow water. It provides a multidimensional generalisation of the renowned KdV equation. In this work, we employ a novel approach recently introduced by one of the authors in connection with the Davey–Stewartson equation (Fokas (2009) [13]), in order to analyse the initial-boundary value problem for the KPII equation formulated on the half-plane. The analysis makes crucial use of the so-called d-bar formalism, as well as of the so-called global relation. A novel feature of boundary as opposed to initial value problems in 2+1 is that the d-bar formalism now involves a function in the complex plane which is discontinuous across the real axis.

A new method for the solution of initial-boundary value problems for \textit{linear} and \textit{integrable nonlinear} evolution PDEs in one spatial dimension was introduced by one of the authors in 1997 \cite{F1997}. This approach was subsequently extended to initial-boundary value problems for evolution PDEs in two spatial dimensions, first in the case of linear PDEs \cite{F2002b} and, more recently, in the case of integrable nonlinear PDEs, for the Davey-Stewartson and the Kadomtsev-Petviashvili II equations on the half-plane (see \cite{FDS2009} and \cite{MF2011} respectively). In this work, we study the analogous problem for the Kadomtsev-Petviashvili I equation; in particular, through the simultaneous spectral analysis of the associated Lax pair via a d-bar formalism, we are able to obtain an integral representation for the solution, which involves certain transforms of all the initial and the boundary values, as well as an identity, the so-called global relation, which relates these transforms in appropriate regions of the complex spectral plane.

Dromions are exponentially localized coherent structures supported by nonlinear integrable evolution equations in two spatial dimensions. In the study of initial-value problems on the plane, such solutions occur only if one imposes nontrivial boundary conditions at infinity, a situation of dubious physical significance. However, it is established here that dromions appear naturally in the study of boundary-value problems. In particular, it is shown that the long time asymptotics of the solution of the Davey–Stewartson I equation in the quarter plane with arbitrary initial conditions and with zero Dirichlet boundary conditions is dominated by dromions. The case of nonzero Dirichlet boundary conditions is also discussed.

The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.

The Davey-Stewartson (DS) equation is a nonlinear integrable evolution equation in two spatial dimensions. It provides a multidimensional generalisation of the celebrated nonlinear Schrödinger (NLS) equation and it appears in several physical situations. The implementation of the Inverse Scattering Transform (IST) to the solution of the initial-value problem of the NLS was presented in 1972, whereas the analogous problem for the DS equation was solved in 1983. These results are based on the formulation and solution of certain classical problems in complex analysis, namely of a Riemann Hilbert problem (RH) and of either a d-bar or a non-local RH problem respectively. A method for solving the mathematically more complicated but physically more relevant case of boundary-value problems for evolution equations in one spatial dimension, like the NLS, was finally presented in 1997, after interjecting several novel ideas to the panoply of the IST methodology. Here, this method is further extended so that it can be applied to evolution equations in two spatial dimensions, like the DS equation. This novel extension involves several new steps, including the formulation of a d-bar problem for a sectionally non-analytic function, i.e. for a function which has different non-analytic representations in different domains of the complex plane. This, in addition to the computation of a d-bar derivative, also requires the computation of the relevant jumps across the different domains. This latter step has certain similarities (but is more complicated) with the corresponding step for those initial-value problems in two dimensions which can be solved via a non-local RH problem, like KPI.

Chapter 1 of this thesis outlines the main motivation behind the introduction of the FM. Chapter 2 is devoted to the one-dimensional heat equation (1.9), formulated on the half-line as well as on the finite interval. Motivated by a problem which arises in the context of heat conduction, we begin by implementing the FM for the oblique Robin boundary condition (1.16). Furthermore, we investigate certain non-local boundary conditions arising as conservation laws in various physical situations.

Chapter 3 focuses on linear evolution PDEs in two spatial dimensions. In particular, we analyse the two-dimensional heat equation formulated on the quarter-plane with oblique Neumann (non-separable) boundary conditions, and the linearised version of the Kadomtsev-Petviashvili II equation (1.27) posed on the upper half-plane.

The latter problem paves the way to Chapters 4 and 5, which we regard as the core of the thesis. Following a recent breakthrough achieved by Fokas in 2009 regarding the Davey-Stewartson system (1.29) (see [55]), we present a method for the analysis of the nonlinear Kadomtsev-Petviashvili I and Kadomtsev-Petviashvili II equations on the upper half-plane by using an appropriate d-bar formalism (see Appendix 7.3).

We conclude the thesis with an overview of some interesting extensions and open problems that should be investigated in the near future, following the main ideas and the methodology presented in the preceding chapters.

A method is presented which enables one to obtain and solve certain classes of nonlinear differential-difference equations. The introduction of a new discrete eigenvalue problem allows the exact solution of the self-dual network equations to be found by inverse scattering. The eigenvalue problem has as its singular limit the continuous eigenvalue equations of Zakharov and Shabat. Some interesting differences arise both in the scattering analysis and in the time dependence from previous work.

The conceptual analogy between Fourier analysis and the exact solution to a class of nonlinear differential-difference equations is discussed in detail. We find that the dispersion relation of the associated linearized equation is prominent in developing a systematic procedure for isolating and solving the equation. As examples, a number of new equations are presented. The method of solution makes use of the techniques of inverse scattering. Soliton solutions and conserved quantities are worked out.

We present a method to solve initial-boundary-value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A S Fokas to solve initial-boundary-value problems for linear and integrable nonlinear partial differential equations via an extension of the inverse scattering transform. The method takes advantage of the Lax pair formulation for both linear and nonlinear equations, and is based on the simultaneous spectral analysis of both parts of the Lax pair. A key role is also played by the global algebraic relation that couples all known and unknown boundary values. Even though additional technical complications arise in discrete problems compared to continuum ones, we show that a similar approach can also solve initial-boundary-value problems for linear and integrable nonlinear differential-difference equations. We demonstrate the method by solving initial-boundary-value problems for the discrete analogue of both the linear and the nonlinear Schrödinger equations, comparing the solution to those of the corresponding continuum problems. In the linear case we also explicitly discuss Robin-type boundary conditions not solvable by Fourier series. In the nonlinear case, we also identify the linearizable boundary conditions, we discuss the elimination of the unknown boundary datum, we obtain explicitly the linear and continuum limit of the solution, and we write the soliton solutions.

We solve the initial-boundary value problem (IBVP) for the Ablowitz–Ladik system on the natural numbers with certain linearizable boundary conditions. We do so by employing a nonlinear method of images, namely, by extending the scattering potential to all integers in such a way that the extended potential satisfies certain symmetry relations. Using these extensions and the solution of the initial value problem (IVP), we then characterize the symmetries of the discrete spectrum of the scattering problem, and we show that discrete eigenvalues in the IBVP appear in octets, as opposed to quartets in the IVP. Furthermore, we derive explicit relations between the norming constants associated with symmetric eigenvalues, and we identify a new kind of linearizable IBVP. Finally, we characterize the soliton solutions of these IBVPs, which describe the soliton reflection at the boundary of the lattice.