Scattering of the 2D modified Zakharov-Kuznetsov equation

Abstract

We study the modified Zakharov-Kuznetsov equation in dimension $2$ : \[ \partial_t u + \partial_x \left( \Delta u + u^3 \right) = 0 \] 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.