Asymptotic soliton-like and asymptotic peakon-like solutions of the modified Camassa-Holm equation with variable coefficients and singular perturbation

TL;DR

This paper develops a new technique for constructing asymptotic soliton-like and peakon-like solutions to a generalized modified Camassa-Holm equation with variable coefficients and singular perturbation, demonstrating its effectiveness through examples.

Contribution

It introduces a novel method for asymptotic peakon-like solution construction and provides a general scheme for any order approximation, enhancing understanding of wave solutions in complex equations.

Findings

Successfully constructed asymptotic soliton-like solutions.

Developed a technique for asymptotic peakon-like solutions.

Validated the method with illustrative examples.

Abstract

The paper deals with the construction of the asymptotic soliton-like and the asymptotic peakon-like solutions to the modified Camassa-Holm equation with variable coefficicents and a singular perturbation. This equation is a generalization of the well known modified Camassa-Holm equation which is integrable system and in addition to the soliton solutions the equation has the peakon solutions. The novelty of the ideas of this paper lies in the development of a technique for constructing asymptotic peakon-like solutions. In the paper a general scheme of finding asymptotic approximation of any order is presented and accuracy of the asymptotic approximation is found. The obtained results are illustrated by examples both the soliton-like and the peakon-like solutions. For the examples the equations for the phase function as well as the main and the first terms of the soliton-like and peakon-like solutions are found. Moreover, for different values of a small parameter the graphs that demonstrate kind of the solutions are presented. The considered examples demonstrate that for an adequate description of the wave process it is enough obtain the main and the first terms of correspond asymptotic solutions. The results also confirm that the proposed technique can be used for constructing asymptotic wave-like solutions of other equations.