A Rigidity Property for the Novikov Equation and the Asymptotic Stability of Peakons

Abstract

We consider weak solutions of the Novikov equation that lie in the energy space $$H^1$$ with non-negative momentum densities. We prove that a special family of such weak solutions, namely the peakons, is $$H^1$$ -asymptotically stable. Such a result is based on a rigidity property of the Novikov solutions which are $$H^1$$ -localized and the corresponding momentum densities are localized to the right, which extends the earlier work of Molinet (Arch Ration Mech Anal 230:185–230, 2018; Nonlinear Anal Real World Appl 50:675–705, 2019) for the Camassa–Holm and Degasperis–Procesi peakons. The main new ingredients in our proof consist of exploring the uniform in time exponential decay property of the solutions from the localization of the $$H^1$$ energy and redesigning the localization of the total mass from the finite speed of propagation property of the momentum densities.