This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrödinger) 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 $U_S^2$ and its dual $V_S^2$ 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.

本文关注Dysthe方程，它是水波系统在调制（Schrödinger）体制和无限深度情况下的高阶近似。我们首先回顾一下Dysthe和相关方程的推导。然后我们研究初始值问题。我们证明了小数据的全局适定性和关键空间中的散射结果${\mathrm{大号}}^{\mathrm{2个}}（{\mathbb{[R}}^{\mathrm{2个}}）$。鉴于流图不能被$C^3$ 连续下面 ${\mathrm{大号}}^{\mathrm{2个}}（{\mathbb{[R}}^{\mathrm{2个}}）$。我们的分析在傅立叶约束范数方法的背景下依赖于线性和双线性Strichartz估计。此外，由于我们处于临界水平，因此我们需要在原子空间的框架内工作${ü}_{\mathrm{小号}}^{\mathrm{2个}}$ 及其双重 ${\mathrm{伏特}}_{\mathrm{小号}}^{\mathrm{2个}}$平方有界变化函数的集合。我们还证明了初始值问题在$H^s（{\mathbb{[R}}^{\mathrm{2个}}）$， $s>0$。我们的结果扩展到Dysthe方程的有限深度版本。