Boundary value problems for the elliptic sine-Gordon equation in a semi-strip

TL;DR

This paper develops a method to solve boundary value problems for the elliptic sine-Gordon equation in a semi-strip using a Riemann-Hilbert problem approach, explicitly handling boundary conditions and providing unique solutions.

Contribution

It introduces a generalized inverse scattering method for boundary value problems of the elliptic sine-Gordon equation, expressing solutions via a Riemann-Hilbert problem with explicit jump matrices.

Findings

Solution expressed as a 2x2 matrix Riemann-Hilbert problem.

Explicit formulas for jump matrices in special boundary conditions.

Proved the Riemann-Hilbert problem has a unique solution.

Abstract

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 $2\times 2$ matrix Riemann-Hilbert problem formulated in terms of both the Dirichlet and the Neumann boundary values on the boundary of a semistrip. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of the semistrip and constant along the bounded side; in this particular case we show that the "jump matrices" of the above Riemann-Hilbert problem can be expressed explicitly in terms of the width of the semistrip and the constant value of the solution along the bounded side. This Riemann-Hilbert problem has a unique solution.