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

Shaojie Yang; Tianzhou Xu

doi:10.3934/dcds.2018016

Article Contents

2018, Volume 38, Issue 1: 329-341. Doi: 10.3934/dcds.2018016

This issue Previous Article Vanishing viscosity limit of the rotating shallow water equations with far field vacuum Next Article Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one

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

Shaojie Yang^, and
Tianzhou Xu

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

* Corresponding author: ysjmath@163.com
* Corresponding author: ysjmath@163.com

Received: April 30, 2017

Revised: June 30, 2017

Published: October 2017

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Abstract

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.

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Mathematics Subject Classification: Primary:35G25, 35L05, 37J15.

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Table 1. 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

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Table 2. The adjoint representation

$Ad(\exp(\epsilon V_i))V_j$	$V_1$	$V_2$	$V_3$
$V_1$	$V_1$	$e^\epsilon V_2$	$V_3$
$V_2$	$V_1-\epsilon V_2$	$V_2$	$V_3$
$V_3$	$V_1$	$V_2$	$V_3$

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