On the nonlinear Dysthe equation

TL;DR

This paper establishes the local well-posedness of the two-dimensional Dysthe equation, derived from Navier-Stokes, using advanced harmonic analysis techniques, and also presents an ill-posedness result.

Contribution

It provides the first rigorous analytic foundation for the well-posedness of the anisotropic 2D Dysthe equation, addressing technical challenges due to anisotropy.

Findings

Proved local well-posedness using Strichartz and smoothing estimates.

Resolved technical challenges from anisotropy in the equation.

Presented an ill-posedness result for the equation.

Abstract

This work is dedicated to putting on a solid analytic ground the theory of local well-posedness for the two dimensional Dysthe equation. This equation can be derived from the incompressible Navier-Stokes equation after performing an asymptotic expansion of a wavetrain modulation to the fourth order. Recently, this equation has been used to numerically study rare phenomena on large water bodies such as rogue waves. In order to study well-posedness, we use Strichartz, and improved smoothing and maximal function estimates. We follow ideas from the pioneering work of Kenig, Ponce and Vega, but since the equation is highly anisotropic, several technical challenges had to be resolved. We conclude our work by also presenting an ill-posedness result.