Space-time resonances

TL;DR

This paper explains the space-time resonances method, a versatile analytical tool for studying global existence in nonlinear dispersive equations with small initial data, by combining classical resonance concepts with wave packet propagation.

Contribution

It introduces and elucidates the space-time resonances method, highlighting its general applicability to nonlinear dispersive equations in the whole space.

Findings

Provides a clear exposition of the space-time resonances technique

Demonstrates the method's effectiveness in understanding global existence

Highlights the method's generality for dispersive PDEs

Abstract

This article is a short exposition of the space-time resonances method. It was introduced by Masmoudi, Shatah, and the author, in order to understand global existence for nonlinear dispersive equations, set in the whole space, and with small data. The idea is to combine the classical concept of resonances, with the feature of dispersive equations: wave packets propagate at a group velocity which depends on their frequency localization. The analytical method which follows from this idea turns out to be a very general tool.