Abstract
We investigate the orbital stability of the peakons for a generalized Camassa-Holm equation (gCH). Using variable transformation, a planar dynamical system is obtained from the gCH equation. It is shown that the planar system has two heteroclinic cycles which correspond two peakon solutions. We then prove that the peakons for the gCH equation are orbitally stable by using the method of Constantin and Strauss.
Highlights
Open AccessIn recent years, there has been great interest in the nonlinearly dispersive equations for model breaking waves
A planar dynamical system is obtained from the generalized Camassa-Holm equation (gCH) equation
We prove that the peakons for the gCH equation are orbitally stable by using the method of Constantin and Strauss
Summary
There has been great interest in the nonlinearly dispersive equations for model breaking waves. The complete classification of gCH Equation (2) conservation laws has been given [11]. The peakons were proved to be orbitally stable by Constantin and Strauss in [12]. The approach in [13] was extended to prove the orbital stability of the peakons for the other nonlinear wave equations [14]-[25]. Recio obtained an interesting generalization of the Camassa-Holm and FORQ/modified Camassa-Holm equations by deriving the most general subfamily of peakon equations that possess the Hamiltonian structure shared by the Camassa-Holm and FORQ/modified Camassa-Holm equations. We will prove the orbital stability of peakons of the gCH Equation (9).
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