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 $\Bbb{R}^3$. 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 $0\leq\theta\leq{\pi\over 3}-\delta$.