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arXiv · 0912.1758

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

A. S. Fokas·B. Pelloni
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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.

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A. S. Fokas, B. Pelloni. 2009-12-09. Boundary value problems for the elliptic sine-Gordon equation in a semi-strip. https://arxiv.org/abs/0912.1758

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