New construction of algebro-geometric solutions to the Camassa-Holm equation and their numerical evaluation

TL;DR

This paper introduces a new method for constructing algebro-geometric solutions to the Camassa-Holm equation using theta functions and Fay's identities, analyzing their properties and numerical behavior.

Contribution

It provides an independent derivation of solutions via theta functions, studies their reality and smoothness conditions, and explores their numerical evaluation including soliton and cuspon limits.

Findings

Solutions derived using Fay's identities are valid and explicit.

Reality and smoothness conditions depend on the hyperelliptic surface topology.

Numerical examples demonstrate soliton and cuspon behaviors.

Abstract

An independent derivation of solutions to the Camassa-Holm equation in terms of multi-dimensional theta functions is presented using an approach based on Fay's identities. Reality and smoothness conditions are studied for these solutions from the point of view of the topology of the underlying real hyperelliptic surface. The solutions are studied numerically for concrete examples, also in the limit where the surface degenerates to the Riemann sphere, and where solitons and cuspons appear.