Chapter 1: Introduction

S., Fokas, A.; Pelloni, B.

doi:10.1137/1.9781611973822.ch1

Cited by 6 publications

(2 citation statements)

References 38 publications

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“…See for example Dyson [39], Miura [74], Zakharov and Shabat [107,108], Ablowitz et al [4], Fokas and Ablowitz [43], Mumford [77], Pöppe [86,87,88], Pöppe and Sattinger [89], Bauhardt and Pöppe [9] and Nijhoff et al [78,79]. There has also been a resurgence of interest in such linearisation approaches, see for example Fokas and Pelloni [44],…”

Section: Introductionmentioning

confidence: 99%

The Algebraic Structure of the Non-Commutative Nonlinear Schrodinger and Modified Korteweg-De Vries Hierarchy

Malham¹,

Blower

2023

Preprint

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Section: Introductionmentioning

confidence: 99%

The Algebraic Structure of the Non-Commutative Nonlinear Schrodinger and Modified Korteweg-De Vries Hierarchy

Malham¹,

Blower

2023

Preprint

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“…This method, proposed by Fokas and extensively developed by him, myself and others, has produced a substantial body of results and, unexpectedly, has led to a new perspective and results on the theory of boundary value problems also for linear PDEs in two independent variables. For an introduction to this theory and a summary of the main results, see [30] (see also [20] for a review of linear boundary value problems and [39] for recent applications and advances).…”

Section: Introductionmentioning

confidence: 99%

Advances in the study of boundary value problems for nonlinear integrable PDEs

Pelloni

2015

Nonlinearity

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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 presents a 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.

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Progress in Initial-Boundary Value Problems for Nonlinear Evolution Equations and the Fokas Method

Himonas

2023

Springer Proceedings in Complexity

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Chapter 1: Introduction

Cited by 6 publications

References 38 publications

The Algebraic Structure of the Non-Commutative Nonlinear Schrodinger and Modified Korteweg-De Vries Hierarchy

The Algebraic Structure of the Non-Commutative Nonlinear Schrodinger and Modified Korteweg-De Vries Hierarchy

Advances in the study of boundary value problems for nonlinear integrable PDEs

Progress in Initial-Boundary Value Problems for Nonlinear Evolution Equations and the Fokas Method

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