Stability of peakons for the generalized modified Camassa-Holm equation
Abstract
In this paper, we study orbital stability of peakons for the generalized modified Camassa-Holm (gmCH) equation, which is a natural higher-order generalization of the modified Camassa-Holm (mCH) equation, and admits Hamiltonian form and single peakons. We first show that the single peakon is the usual weak solution of the PDEs. Some sign invariant properties and conserved densities are presented. Next, by constructing the corresponding auxiliary function and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions, the orbital stability of single peakon of the gmCH equation is verified. We introduce a new approach to prove the key inequality, which is different from that used for the mCH equation. This extends the result on the stability of peakons for the mCH equation (Comm. Math. Phys., 322:967-997, 2013) successfully to the higher-order case, and is helpful to understand how higher-order nonlinearities affect the dispersion dynamics.
Keywords
Cite
@article{arxiv.1804.04885,
title = {Stability of peakons for the generalized modified Camassa-Holm equation},
author = {Zihua Guo and Xiaochuan Liu and Xingxing Liu and Changzheng Qu},
journal= {arXiv preprint arXiv:1804.04885},
year = {2018}
}
Comments
A gap in the proof of Lemma 3.2 is overcome in this new version. Moreover, our new approach is still valid for even power index. Thus we obtain the stability of peakons for the generalized modified Camassa-Holm equation with arbitrary order nonlinearity