Asymptotic stability of peakons for the two-component Novikov equation

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.