Scattering of the 2D modified Zakharov-Kuznetsov equation

TL;DR

This paper proves that solutions to the 2D modified Zakharov-Kuznetsov equation with small, localized initial data exhibit scattering behavior over large times, using the space-time resonance method.

Contribution

It establishes scattering results for the 2D modified Zakharov-Kuznetsov equation, a novel application of the space-time resonance method in this context.

Findings

Solutions with small initial data scatter as time goes to infinity

The space-time resonance method effectively analyzes the equation's long-term behavior

The results extend understanding of dispersive PDEs in two dimensions

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.