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

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.

Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion

“…Recently, the results of the infinite propagation speed for the Camassa-Holm equation and the Degasperis-Procesi equation was extensively established [7,22]. Infinite propagation speed means that they loose instantly the property of having compact x-support.…”

The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise

“…The classical Korteweg-de Vries-type equations and Camassa-Holm-type equations are fundamental models for shallow water waves. Physical structures of such equations have been extensively studied by many authors [35,10,6,20,38,32,39,42,23,24]. In the past few decades remarkable progresses have been made in understanding the Korteweg-de Vries (KdV) equation, it can be considered as a paradigm in nonlinear science and has many applications in weakly nonlinear and weakly dispersive physical systems.…”