Large data scattering for the defocusing supercritical generalized KdV equation

TL;DR

This paper proves that solutions to the defocusing supercritical generalized KdV equation with initial data in H^1 are global and scatter, extending understanding of long-term behavior in supercritical regimes.

Contribution

It establishes global existence and scattering for supercritical gKdV equations in H^1, using a compactness method inspired by Kenig and Merle.

Findings

Solutions are global in time for initial data in H^1.

Solutions scatter in H^1, indicating dispersion over time.

Method adapts compactness techniques to supercritical gKdV context.

Abstract

We consider the defocusing supercritical generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u-\partial_x(u^{k+1})=0$, where $k>4$ is an even integer number. We show that if the initial data $u_0$ belongs to $H^1$ then the corresponding solution is global and scatters in $H^1$. Our method of proof is inspired on the compactness method introduced by C. Kenig and F. Merle.