Scattering of the 2D modified Zakharov-Kuznetsov equation

Abstract

We study the modified Zakharov-Kuznetsov equation in dimension $2$ : \[ \partial_t u + \partial_x \left( Δu + u^3 \right) = 0 \] where $u:(t,(x,y))\in \mathbb{R}\times {\mathbb{R}}^2\mapsto u(t,x,y)\in \mathbb{R}$ and $\Delta ={\partial }_x^2+{\partial }_y^2$ is the full Laplacian. We prove that solutions for small and localized initial data scatter for large time. Our proof relies on the method of space-time resonances.

24 pages