Asymptotic stability of peakons for the two-component Novikov equation

Author:

He Cheng1,Li Ze1,Luo Ting2,Qu Changzheng1ORCID

Affiliation:

1. School of Mathematics and Statistics, Ningbo University 1 , Ningbo 315211, People’s Republic of China

2. School of Mathematical Sciences, Zhejiang Normal University 2 , Jinhua 321004, People’s Republic of China

Abstract

We study the asymptotic stability of peaked solitons under H1 × H1-perturbations of the two-component Novikov equation involving interaction between two components. This system, as a two-component generalization of the Novikov equation, is a completely integrable system which has Lax pair and bi-Hamiltonian structure. Interestingly, it admits the two-component peaked solitons with different phases, which are the weak solutions in the sense of distribution and lie in the energy space H1 × H1. It is shown that the peakons are asymptotically stable in the energy space H1 × H1 with non-negative momentum density by establishing a rigidity theorem for H1 × H1-almost localized solutions. Our proof generalizes the arguments for studying the Camassa-Holm and Novikov equations. There are three new ingredients in our proof. One is a new characteristic describing interaction of the two-components; the second is new additional conserved densities for establishing the main inequalities; while the third one is a new Lyapunov functional used to overcome the difficulty caused by the loss of momentum.

Funder

National Natural Science Foundation of China

Natural Science Foundation of Zhejiang Province

Publisher

AIP Publishing

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