Initial-Boundary Value Problems for Linear and Soliton PDEs

TL;DR

This paper develops a spectral method for solving initial-boundary value problems for linear and nonlinear dispersive PDEs, including Schrödinger equations, by eliminating unknown boundary values through functional space restrictions.

Contribution

It introduces a novel spectral approach that handles boundary conditions for integrable PDEs, expanding solution techniques for dispersive wave equations.

Findings

Effective solution method for linear Schrödinger equations on various domains.

Extension of the spectral method to nonlinear Schrödinger equations.

Demonstrated applicability through illustrative examples.

Abstract

Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based on the elimination of the unknown boundary values by proper restrictions of the functional space and of the spectral variable complex domain. Illustrative examples include the linear Schroedinger equation on compact and semicompact n-dimensional domains and the nonlinear Schroedinger equation on the semiline.