Stability for line solitary waves of Zakharov-Kuznetsov equation

Abstract

In this paper, we consider the stability for line solitary waves of the two dimensional Zakharov-Kuznetsov equation on $\mathbb{R}\times {\mathbb{T}}_L$ which is one of a high dimensional generalization of Korteweg-de Vries equation , where ${\mathbb{T}}_L$ is the torus with the period $2\pi L$. The orbital and asymptotic stability of the one soliton of Korteweg-de Vries equation on the energy space has been proved by Benjamin, Pego and Weinstein and Martel and Merle. We regard the one soliton of Korteweg-de Vries equation as a line solitary wave of Zakharov-Kuznetsov equation on $\mathbb{R}\times {\mathbb{T}}_L$. We prove the stability and the transverse instability of the line solitary waves of Zakharov-Kuznetsov equation by applying Evans' function method and the argument of Rousset and Tzvetkov. Moreover, we prove the asymptotic stability for the orbitally stable line solitary wave of Zakharov-Kuznetsov equation by using the argument of Martel and Merle, a Liouville type theorem and a corrected virial type estimate.