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

Abstract

In this paper, we study the following generalized Camassa-Holm equation with both cubic and quadratic nonlinearities: $$m_t+k_1(3u{u}_{x}m+u^2{m}_x)+k_2(2m{u}_x+m_{x}u)=0,m=u-u_{xx},$$ 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.

References in corpus (4)

• An integrable shallow water equation with peaked solitons
• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• The Camassa-Holm equation as a geodesic flow on the diffeomorphism group
• On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity