# American Institute of Mathematical Sciences

January  2018, 38(1): 329-341. doi: 10.3934/dcds.2018016

## Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system

 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

* Corresponding author: ysjmath@163.com

Received  May 2017 Revised  July 2017 Published  September 2017

In this paper, we study symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system. Lie symmetry analysis and similarity reductions are performed, some invariant solutions are also discussed. Then prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. Furthermore, we show that the system exhibits unique continuation if the initial momentum $m_0$ and $n_0$ are positive.

Citation: Shaojie Yang, Tianzhou Xu. Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 329-341. doi: 10.3934/dcds.2018016
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##### References:
The commutation table of Lie algebra
 $[V_i,V_j]$ $V_1$ $V_2$ $V_3$ $V_1$ 0 $-V_2$ 0 $V_2$ $V_2$ 0 0 $V_3$ 0 $0$ 0
 $[V_i,V_j]$ $V_1$ $V_2$ $V_3$ $V_1$ 0 $-V_2$ 0 $V_2$ $V_2$ 0 0 $V_3$ 0 $0$ 0
 $Ad(\exp(\epsilon V_i))V_j$ $V_1$ $V_2$ $V_3$ $V_1$ $V_1$ $e^\epsilon V_2$ $V_3$ $V_2$ $V_1-\epsilon V_2$ $V_2$ $V_3$ $V_3$ $V_1$ $V_2$ $V_3$
 $Ad(\exp(\epsilon V_i))V_j$ $V_1$ $V_2$ $V_3$ $V_1$ $V_1$ $e^\epsilon V_2$ $V_3$ $V_2$ $V_1-\epsilon V_2$ $V_2$ $V_3$ $V_3$ $V_1$ $V_2$ $V_3$
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