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 R×TL which is one of a high dimensional generalization of Korteweg–de Vries equation, where TL is the torus with the 2πL period. The orbital and asymptotic stability of the one soliton of Korteweg–de Vries equation on the energy space was proved by Benjamin [2], Pego and Weinstein [41] and Martel and Merle [30]. We regard the one soliton of Korteweg–de Vries equation as a line solitary wave of Zakharov–Kuznetsov equation on R×TL. We prove the stability and the transverse instability of the line solitary waves of Zakharov–Kuznetsov equation by applying the method of Evans' function and the argument of Rousset and Tzvetkov [44]. Moreover, we prove the asymptotic stability for orbitally stable line solitary waves of Zakharov–Kuznetsov equation by using the argument of Martel and Merle [30–32] and a Liouville type theorem. If L is the critical period with respect to a line solitary wave, the line solitary wave is orbitally stable. However, since this line solitary wave is a bifurcation point of the stationary equation, the linearized operator of the stationary equation is degenerate. Because of the degeneracy of the linearized operator, we can not show the Liouville type theorem for the line solitary wave by using the usual virial type estimate. To show the Liouville type theorem for the line solitary wave, we modify a virial type estimate.