Darboux transformation and multi-soliton solutions of the Camassa-Holm equation and modified Camassa-Holm equation

TL;DR

This paper introduces a simplified Darboux transformation method to derive multi-soliton solutions for the Camassa-Holm and modified Camassa-Holm equations by mapping them to a negative order KdV equation and then inverting the transformation.

Contribution

A new, simplified approach using Darboux transformation for solving multi-soliton solutions of CH and MCH equations via reciprocal transformation to NKdV.

Findings

Successfully derived multi-soliton solutions for CH and MCH equations.

Simplified the existing method for solving these equations.

Validated the approach through explicit solution construction.

Abstract

In this paper, we propose a new approach to calculate multi-soliton solutions of Camassa-Holm (CH) equation and modified Camassa-Holm (MCH) equation with aid of Darboux transformation (DT). The new approach simplifies the approach presented in {\it Proc. R. Soc. Lond. A} {\bf 460} 2617-2627 (2004). We first map the CH and MCH equation to a negative order KdV (NKdV) equation by a reciprocal transformation. Then we proceed to apply the DT to solve the NKdV equation in the usual way. Finally we invert the reciprocal transformation to recover the solutions of the CH equation and MCH equation.