Long time behavior of solutions to the mKdV

TL;DR

This paper investigates the long-term behavior of solutions to the mKdV equation on the real line, establishing global existence and asymptotics for small initial data without depending on integrability, using wave packet methods.

Contribution

It introduces a novel approach using wave packet testing to analyze the long-time dynamics of mKdV solutions without relying on integrability.

Findings

Proves global existence for small, smooth, decaying initial data.

Derives modified asymptotics for solutions over long times.

Addresses the asymptotic completeness problem for mKdV.

Abstract

In this paper we consider the long time behavior of solutions to the modified Korteweg-de Vries equation on R. For sufficiently small, smooth, decaying data we prove global existence and derive modified asymptotics without relying on complete integrability. We also consider the asymptotic completeness problem. Our result uses the method of testing by wave packets, developed in the work of Ifrim and Tataru on the 1d cubic nonlinear Schr\"odinger and 2d water wave equations.