The lifespan of small data solutions for Intermediate Long Wave equation (ILW)

TL;DR

This paper investigates the long-time behavior of solutions to the ILW equation, establishing global well-posedness at low regularity and demonstrating dispersion of small localized solutions up to cubic timescales.

Contribution

It proves global well-posedness for ILW at lower regularity and analyzes the dispersion of small localized solutions over extended times.

Findings

Global well-posedness in L^2 for ILW

Solutions disperse up to cubic timescale for small localized data

Advances understanding of ILW long-time dynamics

Abstract

This article represents a first step toward understanding the long-time dynamics of solutions for the Intermediate Long Wave equation (ILW). While this problem is known to be both completely integrable and globally well-posed in $H^{\frac{3}{2}}$, much less seems to be known concerning its long-time dynamics. Here we prove well-posedness at much lower regularity, namely an $L^2$ global well-posedness result. Then we consider the case of small and localized data and show that the solutions disperse up to cubic timescale.