The periodic Camassa-Holm equation by the Riemann-Hilbert problem approach

TL;DR

This paper develops a Riemann-Hilbert problem approach to analyze the periodic Camassa-Holm equation, providing a new framework for solving and understanding its solutions under periodic boundary conditions.

Contribution

It introduces a Riemann-Hilbert problem formalism for the periodic Camassa-Holm equation, linking initial data to spectral functions for solution representation.

Findings

Formulation of Riemann-Hilbert problem for periodic Camassa-Holm

Representation of solutions via spectral functions

Foundation for further analytical and numerical studies

Abstract

This work addresses the development of the Riemann-Hilbert problem (RHP) formalism (the Fokas method) for the Camassa-Holm equation under periodic boundary conditions. Particularly, we present a representation of the solution to this problem in terms of the solution of the associated Riemann-Hilbert problem, the data for which are determined by the initial data for the problem in terms of the associated spectral functions.