Testing by wave packets and modified scattering in nonlinear dispersive pde's

TL;DR

This paper discusses the use of wave packet testing to analyze modified scattering phenomena in nonlinear dispersive PDEs, especially in borderline decay cases, providing a versatile approach for various dispersion relations.

Contribution

The authors introduce and explain how the testing by wave packets method can be applied to a broad class of nonlinear dispersive PDEs with general dispersion relations.

Findings

Effective capture of asymptotic equations for nonlinear flows.

Simplified analysis of modified scattering mechanisms.

Applicable to problems with general dispersion relations.

Abstract

Modified scattering phenomena are encountered in the study of global properties for nonlinear dispersive partial differential equations in situations where the decay of solutions at infinity is borderline and scattering fails just barely. An interesting example is that of problems with cubic nonlinearities in one space dimension. The method of testing by wave packets was introduced by the authors as a tool to efficiently capture the asymptotic equations associated to such flows, and thus establish the modified scattering mechanism in a simpler, more efficient fashion, and at lower regularity. In these expository notes we describe how this method can be applied to problems with general dispersion relations.