the main interaction region. In this region, various decay estimates (in time or in space) become equivalent which gives more flexibility. Since for cubic nonlinearities, one only need a small improvement on the decay estimates, this region can be controlled by exploiting the “hyperbolic” structure of the phase: it cannot be stationary in all directions.

Our strategy is as follows: In Section \[section2\], we reformulate into a quasilinear scalar dispersive equation for a complex-valued unknown, by diagonalizing the linearized system. We then prove our main theorem about the new equation, Theorem \[MainThm2\] assuming two propositions. This implies Theorem \[MainThm\] In Section \[LocalExistence\], we prove our first proposition, Proposition \[LocExLemma\] which gives a good local existence theory with energy estimates suitable for our analysis. In Section \[Bootstrap\] we prove the second proposition, Proposition \[Norm\], which implies the decay of solutions needed for the energy estimates. This is done by first using a normal form transformation and then following the strategy explained in points $(1)$, $(2)$ and $(3)$ above. Finally in Section \[technical\], we collect various technical estimates needed in the analysis.

Main definitions and propositions {#section2}
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We first remark that since we start with irrotational initial velocity, the velocity remains 