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This paper focuses on the Dysthe equation which is a higher orderapproximation of the water waves system in the modulation (Schr\"{o}dinger)regime and in the infinite depth case. We first review the derivation of theDysthe and related equations. Then we study the initial-value problem. We provea 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 mapcannot be $C^3$ continuous below $L^2(\mathbb R^2)$. Our analysis relies onlinear and bilinear Strichartz estimates in the context of the Fourierrestriction norm method. Moreover, since we are at a critical level, we need towork in the framework of the atomic space $U^2_S$ and its dual $V^2_S $ ofsquare bounded variation functions. We also prove that the initial-valueproblem is locally well-posed in $H^s(\mathbb R^2)$, $s>0$. Our results extendto the finite depth version of the Dysthe equation.

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