The Dysthe equation is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. After reviewing the derivation of the Dysthe and related equations, we will focus on 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 $H^s(\mathbb R^2)$, $s\gt 0$.

Our results extend to the finite depth version of the Dysthe equation.

This talk is based on a joint work with Razvan Mosincat (University of Bergen) and Jean-Claude Saut (Université Paris-Saclay).