A coupling problem for entire functions and its application to the long-time asymptotics of integrable wave equations

TL;DR

This paper introduces a new method based on a coupling problem for entire functions to analyze the long-time behavior of integrable wave equations with purely discrete spectra, exemplified by the dispersionless Camassa-Holm equation.

Contribution

It develops a novel coupling problem framework as an alternative to Riemann-Hilbert problems for discrete spectrum cases in integrable systems.

Findings

Successfully applied to the dispersionless Camassa-Holm equation

Provides a new analytical tool for purely discrete spectrum problems

Advances understanding of long-time asymptotics in integrable wave equations

Abstract

We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case when the underlying isospectral problem has purely discrete spectrum. To this end, we introduce a natural coupling problem for entire functions, which serves as a replacement for the usual Riemann-Hilbert problem, which does not apply in these cases. As a prototypical example, we investigate the long-time asymptotics of the dispersionless Camassa-Holm equation.