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.
The peakons and mulit-peakons for a generalized cubic–quintic Camassa–Holm type equation have been obtained by Weng et al. (Monatsh Math, 2022. https://doi.org/10.1007/s00605-022-01699-w). In this paper, by constructing certain Lyapunov functionals, we prove that the peakons were orbitally stable in the energy space. Furthermore, using energy argument and combining the method of the orbital stability of peakons with monotonicity of the local energy norm, we also prove that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space.
Non-transversal diffusion seems similar to the Arnold diffusion, but it is a different phenomenon since it could be applied to integrable systems in contrast to the Arnold diffusion, which is a phenomenon that only takes place in non integrable systems. In this paper, the existence of the non-transversal heteroclinic chain in the quintic nonlinear Schrödinger equation is studied. Based on a finite dimensional system of ordinary differential equations, the Toy model system associated with the quintic nonlinear Schrödinger equation, we construct the non-transversal heteroclinic chain of the Toy model system. The explicit expression for the dynamics of the heteroclinic is given.
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