Explicit Solutions to Boundary Problems for 2+1-Dimensional Integrable Systems

TL;DR

This paper develops a method to explicitly solve boundary problems for 2+1-dimensional integrable systems like the KP equation and Toda lattice, expanding the class of solvable boundary conditions.

Contribution

It introduces a new approach for obtaining explicit solutions to boundary problems in higher-dimensional integrable models, demonstrating its effectiveness through examples.

Findings

Existence of a large set of integrable boundary problems

Development of a method for explicit solutions

Successful application to specific examples

Abstract

Nonlinear integrable models with two spatial and one temporal variables: Kadomtsev-Petviashvili equation and two-dimensional Toda lattice are investigated on the subject of correct formulation for boundary problem that can be solved within the framework of the Inverse Scattering Problem method. It is shown that there exists a large set of integrable boundary problems and various curves can be chosen as boundary contours for them. We develop a method for obtaining explicit solutions to integrable boundary problems and its effectiveness is illustrated by series of examples.