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

Baoqiang Xia; Zhijun Qiao; Jibin Li

paper

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

arXiv:1205.2028

Abstract

In this paper, we study an integrable system with both quadratic and cubic nonlinearity: $m_{t} = b u_{x} + 1/2 k_{1} [m (u^{2} - u_{x}^{2})]_{x} + 1/2 k_{2} (2 m u_{x} + m_{x} u)$ , $m = u - u_{xx}$ , where $b$ , $k_{1}$ and $k_{2}$ are arbitrary constants. This model is kind of a cubic generalization of the Camassa-Holm (CH) equation: $m_{t} + m_{x} u + 2 m u_{x} = 0$ . 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 $b \neq = 0$ 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.

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

Abstract

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