Advances in the study of boundary value problems for nonlinear integrable PDEs. (12th January 2015)

• Record Type: Journal Article
• Authors: Pelloni, Beatrice
• Is Part Of: Nonlinearity. Volume 28:Number 2(2015:Feb.)
• Page Start: R1
• Page End: R38
• Publication Date: 2015-01-12
• Subjects: partial differential equations -- integral transforms -- Riemann–Hilbert problems -- boundary value problems; 35 : 02 -- 35E05; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/28/2/R1
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this review I summarize some of the most significant advances of the last decade in the analysis and solution of boundary value problems for integrable partial differential equations (PDEs) 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 generalization 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 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 generalization to certain nonlinear cases of particular significance.