Nonlinear wave interactions for the Benjamin-Ono equation

TL;DR

This paper investigates the nonlinear interactions of waves in the Benjamin-Ono equation, demonstrating that the flow map lacks uniform continuity on bounded sets in certain Sobolev spaces, which impacts understanding of the equation's well-posedness.

Contribution

It proves the non-uniform continuity of the flow map for the Benjamin-Ono equation in Sobolev spaces with s>0, revealing new limitations in the equation's well-posedness.

Findings

Flow map is not uniformly continuous on bounded sets in H^s(R) for s>0.

Wave interactions at different frequencies influence the flow's regularity.

Results impact the theoretical understanding of the Benjamin-Ono equation's solution behavior.

Abstract

We study the interaction of suitable small and high frequency waves evolving by the flow of the Benjamin-Ono equation. As a consequence, we prove that the flow map of the Benjamin-Ono equation can not be uniformly continuous on bounded sets of H^s(R) for s>0.