  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.
