Nonlinear smoothing and unconditional uniqueness for the Benjamin-Ono equation in weighted Sobolev spaces

TL;DR

This paper proves unconditional uniqueness for the Benjamin-Ono equation in weighted Sobolev spaces by demonstrating Lipschitz continuity and nonlinear smoothing effects after a gauge transform.

Contribution

It introduces a novel approach using gauge transform to establish unconditional uniqueness and nonlinear smoothing for the Benjamin-Ono equation.

Findings

Flow is Lipschitz continuous after gauge transform

Nonlinear smoothing effect is established

Unconditional uniqueness is proved in weighted Sobolev spaces

Abstract

We consider the Benjamin-Ono equation on the real line for initial data in weighted Sobolev spaces. After the application of the gauge transform, the flow is shown to be Lipschitz continuous and to present a nonlinear smoothing effect. As a consequence, unconditional uniqueness for the Benjamin-Ono equation is proved.