Asymptotic stability of peakons for the Novikov equation

Abstract

The Novikov equation is an integrable Camassa-Holm type equation with a cubic nonlinearity. One of the most important features of this equation is the existence of peaked traveling waves, also called peakons. This paper aims to prove the asymptotic stability of peakon solutions under H1(R)-perturbations satisfying that their associated momentum density defines a non-negative Radon measure. Motivated by Molinet's work [24–26], we shall first prove a Liouville property for H1(R) global solutions belonging to a certain class of almost localized functions. More precisely, we show that such solutions have to be a peakon. The main novelty in our analysis in comparison to the Camassa-Holm equation comes from the fact that in our present case the momentum is not a conserved quantity and may be unbounded along the trajectory. In this regard, to prove the Liouville property, we used a new Lyapunov functional not related to the (not conserved) momentum of the equation.