A priori bounds and equicontinuity of orbits for the intermediate long wave equation

TL;DR

This paper establishes uniform bounds and equicontinuity properties for solutions of the intermediate long wave equation in certain Sobolev spaces, using a Lax pair formulation.

Contribution

It provides the first uniform-in-time a priori bounds and equicontinuity results for solutions on both the line and circle for the intermediate long wave equation.

Findings

Proves uniform $H^s$ bounds for solutions with $-rac12<s extless=0$

Shows the set of orbits from bounded equicontinuous sets remains bounded and equicontinuous in $H^s$

Identifies a Lax pair formulation for the equation

Abstract

We prove uniform-in-time a priori $H^s$ bounds for solutions to the intermediate long wave equation posed both on the line and on the circle, covering the range $-\frac12<s\leq0$. Additionally, we prove that the set of orbits emanating from a bounded and equicontinuous set in $H^s$ is also bounded and equicontinuous in $H^s$. Our proof is based on the identification of a suitable Lax pair formulation for the intermediate long wave equation.