On the elliptic sinh-Gordon equation with integrable boundary conditions II -- Baker-Akhiezer theory

TL;DR

This paper analyzes finite-type solutions of the elliptic sinh-Gordon equation with Durham boundary conditions, establishing rationality criteria and spectral curve conditions necessary for solutions to satisfy boundary constraints.

Contribution

It introduces necessary rationality criteria and spectral curve conditions for finite-type solutions under Durham boundary conditions, extending the understanding beyond periodic cases.

Findings

Rationality criteria analogous to periodic solutions are necessary for boundary conditions.

Additional spectral curve point criteria are required for boundary satisfaction.

Special cases exist where criteria are sufficient for boundary conditions.

Abstract

We study finite-type solutions of the elliptic sinh-Gordon equation along the strip $\Bbb{R}\times[0,L]$ with Durham conditions on each boundary component. We determine necessary rationality criteria for these conditions to be satisfied on both components. These rationality criteria are analogous to the necessary and sufficient criteria for finite-type solutions to be periodic. However, in contrast to the periodic case, they are not in themselves sufficient for Durham conditions to be satisfied on both boundary components: certain additional criteria must also be satisfied at 4 specific points of the spectral curve. Nevertheless, in the special case where the Durham conditions imposed on the two boundary components are complementary to one another in a sense that we will clarify, these rationality criteria are, indeed, sufficient. This study is a necessary preliminary for any general construction of finite-type solutions of the elliptic sinh-Gordon equation subject to Durham boundary conditions.