B\"acklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity

TL;DR

This paper develops a compact parametric form for smooth multisoliton solutions of the modified Camassa-Holm equation with cubic nonlinearity, using Bäcklund transformations and asymptotic analysis to explore their properties and conservation laws.

Contribution

It introduces a novel Bäcklund transformation linking the mCH equation to a shallow-water wave model, enabling explicit multisoliton solutions and analysis of their properties.

Findings

Smoothness depends on amplitude parameters satisfying certain conditions

At a critical parameter value, solutions become singular and differ from peakons

Asymptotic analysis confirms the solitonic phase shifts and properties

Abstract

We present a compact parametric representation of the smooth bright multisolution solutions for the modified Camassa-Holm (mCH) equation with cubic nonlinearity. We first transform the mCH equation to an associated mCH equation through a reciprocal transformation and then find a novel B\"acklund transformation between solutions of the associated mCH equation and a model equation for shallow-water waves (SWW) introduced by Ablowitz {\it at al}. We combine this result with the expressions of the multisoliton solutions for the SWW and modified Korteweg-de Vries equations to obtain the multisoliton solutions of the mCH equation. Subsequently, we investigate the properties of the one- and two-soliton solutions as well as the general multisoliton solutions. We show that the smoothness of the solutions is assured only if the amplitude parameters of solitons satisfy certain conditions. We also find that at a critical value of the parameter beyond which the solution becomes singular, the soliton solution exhibits a different feature from that of the peakon solution of the CH equation. Then, by performing an asymptotic analysis for large time, we obtain the formula for the phase shift and confirm the solitonic nature of the multisoliton solutions. Last, we use the B\"acklund transformation to derive an infinite number of conservation laws of the mCH equation.