  This work is dedicated to putting on a solid analytic ground the theory of
local well-posedness for the two dimensional Dysthe equation. This equation can
be derived from the incompressible Navier-Stokes equation after performing an
asymptotic expansion of a wavetrain modulation to the fourth order. Recently,
this equation has been used to numerically study rare phenomena on large water
bodies such as rogue waves. In order to study well-posedness, we use
Strichartz, and improved smoothing and maximal function estimates. We follow
ideas from the pioneering work of Kenig, Ponce and Vega, but since the equation
is highly anisotropic, several technical challenges had to be resolved. We
conclude our work by also presenting an ill-posedness result.

Analysis of PDEs