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

We consider the quadratic Zakharov-Kuznetsov equation $${\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_x\mathrm{\Delta }u+{\mathrm{\partial }}_x{u}^2=0$$ on ${\mathbb{R}}^3$. A solitary wave solution is given by $Q(x-t,y,z)$, where $Q$ is the ground state solution to $-Q+\mathrm{\Delta }Q+Q^2=0$. 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 $t\to \mathrm{\infty }$, 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$0 \leq \theta \leq \frac{\pi}{3}-\delta$.