Integrable system with peakon, weak kink, and kink-peakon interactional solutions

TL;DR

This paper introduces an integrable nonlinear system extending the Camassa-Holm equation, exploring peakon, kink, and interaction solutions, and analyzing their dynamics and properties.

Contribution

It presents a new integrable model with quadratic and cubic nonlinearities, including explicit peakon, kink, and interaction solutions, and investigates their dynamics and collisions.

Findings

Existence of peakon and multi-peakon solutions.

Explicit two-peakon dynamical system and collision analysis.

Discovery of weak kink and kink-peakon interaction solutions.

Abstract

In this paper, we study an integrable system with both quadratic and cubic nonlinearity: $m_t=bu_x+1/2k_1[m(u^2-u^2_x)]_x+1/2k_2(2m u_x+m_xu)$, $m=u-u_{xx}$, where $b$, $k_1$ and $k_2$ are arbitrary constants. This model is kind of a cubic generalization of the Camassa-Holm (CH) equation: $m_t+m_xu+2mu_x=0$. The equation is shown integrable with its Lax pair, bi-Hamiltonian structure, and infinitely many conservation laws. In the case of $b=0$, the peaked soliton (peakon) and multi-peakon solutions are studied. In particular, the two-peakon dynamical system is explicitly presented and their collisions are investigated in details. In the case of $b\neq0$ and $k_2=0$, the weak kink and kink-peakon interactional solutions are found. Significant difference from the CH equation is analyzed through a comparison. In the paper, we also study all possible smooth one-soliton solutions for the system.