Existence of peakons for a cubic generalization of the Camassa-Holm equation

TL;DR

This paper investigates a cubic generalization of the Camassa-Holm equation, demonstrating the existence of peakon solutions, including single-peaked solitons and periodic peakons, through analytical methods.

Contribution

The paper introduces a new cubic generalization of the Camassa-Holm equation and proves the existence of peakon solutions within this model.

Findings

Existence of single-peaked solitons.

Presence of periodic peakons.

The model combines features of Novikov and Camassa-Holm equations.

Abstract

In this paper, we study the following generalized Camassa-Holm equation with both cubic and quadratic nonlinearities: $$ m_{t}+k_{1}(3uu_{x}m+u^2m_{x})+k_{2}(2mu_{x}+m_{x}u)=0, \quad m=u-u_{xx}, $$ which is presented as a linear combination of the Novikov equation and the Camassa-Holm equation with constants $k_{1}$ and $k_{2}$. The model is a cubic generalization of the Camassa-Holm equation. It is shown that the equation admits single-peaked soliton and periodic peakons.