On Genus-Two Solutions for the ILW equation

TL;DR

This paper establishes the existence of genus-two theta function solutions for the ILW equation using Krichever's construction, providing a numerical example and exploring related Riemann surface properties.

Contribution

It demonstrates the existence of genus-two solutions for the ILW equation via theta functions and Krichever's method, extending the understanding of integrable systems.

Findings

Existence of genus-two theta function solutions for ILW.

Numerical example illustrating the solutions.

Connection between Riemann surfaces and Abelian integrals.

Abstract

The existence of theta function solutions of genus two for the ILW equation is established. A numerical example is also presented. The method basically goes along with the Krichever's construction of theta function solutions for soliton equations, such as the KP equation. This idea leads us to a question whether a Riemann surface exists which allows a peculiar Abelian integral of the third kind. The answer is affirmative at least for genus-two curves.