Global well-posedness and scattering for the Dysthe equation in $L^2(\mathbb R^2)$

TL;DR

This paper establishes global well-posedness and scattering for the Dysthe equation in the critical space L^2(R^2), using advanced harmonic analysis techniques, and extends results to the finite depth case.

Contribution

It proves the first sharp global well-posedness and scattering results for the Dysthe equation in L^2(R^2), including the finite depth scenario.

Findings

Global well-posedness and scattering in L^2(R^2)

Local well-posedness in H^s(R^2) for s>0

Extension of results to finite depth Dysthe equation

Abstract

This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schr\"{o}dinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space $L^2(\mathbb R^2)$. This result is sharp in view of the fact that the flow map cannot be $C^3$ continuous below $L^2(\mathbb R^2)$. Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space $U^2_S$ and its dual $V^2_S $ of square bounded variation functions. We also prove that the initial-value problem is locally well-posed in $H^s(\mathbb R^2)$, $s>0$. Our results extend to the finite depth version of the Dysthe equation.