Stability for line solitary waves of Zakharov-Kuznetsov equation

TL;DR

This paper investigates the stability and instability of line solitary waves in the two-dimensional Zakharov-Kuznetsov equation on a cylindrical domain, extending stability results known for the Korteweg-de Vries equation.

Contribution

It establishes the orbital, asymptotic, and transverse stability properties of line solitary waves for the Zakharov-Kuznetsov equation using Evans' function and other analytical methods.

Findings

Proves stability of line solitary waves under certain conditions.

Identifies transverse instability of these waves.

Demonstrates asymptotic stability for specific stable waves.

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.