Abstract
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 L2(R2). This result is sharp in view of the fact that the flow map cannot be C3 continuous below L2(R2). 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 US2 and its dual VS2 of square bounded variation functions. We also prove that the initial-value problem is locally well-posed in Hs(R2), s>0. Our results extend to the finite depth version of the Dysthe equation.
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