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

Yun Wang; Lixin Tian

paper

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

arXiv:1811.05843

Abstract

In this paper, we study the following generalized Camassa-Holm equation with both cubic and quadratic nonlinearities: $m_{t} + k_{1} (3 u u_{x} m + u^{2} m_{x}) + k_{2} (2 m u_{x} + m_{x} u) = 0, m = u - u_{xx},$ 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.