Dispersion Estimates for Third Order Equations in Two Dimensions

Two-dimensional deep water waves and some problems in nonlinear optics can be described by various third order dispersive equations, modifying and generalizing the KdV as well as nonlinear Schrödinger equations. We classify all third order polynomials up to certain transformations and study the pointwise decay for the fundamental solutions, ∫_ℝ2e ^{itp(ξ)+ix·ξ} dξ for all third order polynomials p in two variables. We deduce the corresponding Strichartz estimates. These estimates imply global existence and uniqueness for the Shrira system.