The Zakharov-Kuznetsov equation in weighted Sobolev spaces

In this work we consider the initial value problem (IVP) associated to the two dimensional Zakharov-Kuznetsov equation $$\begin{array}{rl}{u}_t+{\mathrm{\partial }}_x^{3}u+{\mathrm{\partial }}_x{\mathrm{\partial }}_y^{2}u+u{\mathrm{\partial }}_{x}u& =0,(x,y)\in {\mathbb{R}}^2,t\in \mathbb{R},\\ u(x,y,0)& =u_0(x,y)\mathrm{.}\end{array}\}$$ We study the well-posedness of the IVP in the weighted Sobolev spaces $$H^s({\mathbb{R}}^2)\cap L^2((1+x^2+y^2{)}^{r}dxdy),$$ with $s,r\in \mathbb{R}$.