Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation

Abstract

We consider the quadratic Zakharov-Kuznetsov equation $${\partial }_{t}u+{\partial }_x\Delta u+{\partial }_x{u}^2=0$$ on ${\mathbb{R}}^3$. A solitary wave solution is given by $Q(x-t,y,z)$, where $Q$ is the ground state solution to $-Q+\Delta Q+Q^2=0$. We prove the asymptotic stability of these solitary wave solutions. Specifically, we show that initial data close to $Q$ in the energy space, evolves to a solution that, as $t\to \infty$, converges to a rescaling and shift of $Q(x-t,y,z)$ in $L^2$ in a rightward shifting region $x>\delta t-\tan \theta \sqrt{y^2+z^2}$ for $0\le \theta \le \frac{\pi }{3}-\delta$.