On the $L^2$ well-posedness and decay estimate of third order Benjamin-Ono equation

TL;DR

This paper proves $L^2$ well-posedness and dispersive decay estimates for the third order Benjamin-Ono equation by transforming it into an Airy-type equation and employing a nonlinear vector field method.

Contribution

It introduces a novel approach combining normal form and gauge transformation to analyze the third order Benjamin-Ono equation's well-posedness and decay properties.

Findings

Established $L^2$ well-posedness of the third order Benjamin-Ono equation.

Proved dispersive decay estimates for solutions.

Transformed the nonlinear equation into an Airy-type equation using gauge and normal form techniques.

Abstract

We consider the $L^2$ well-posedness of third order Benjamin-Ono equation. We show that by means of a normal form and a gauge transformation, the equation can be changed into an Airy-type equation. A second goal of this work is to establish that the solutions to the nonlinear third order Benjamin-Ono equation problem exhibit a dispersive decay estimate analogue to the corresponding linear associated problem. The key ingredient is the use of a nonlinear vector field method.