INV

Abstract

The Davey-Stewartson II equation and the Novikov-Veselov equations are nonlinear dispersive equations in two dimensions, respectively describing the motion of surface waves in shallow water and geometrical optics in nonlinear media. Both are integrable by the $\overline{\partial}$-method of inverse scattering, and may be considered respective analogues of the cubic nonlinear Schrodinger equation and the KdV equation in one dimension. We will prove global well-posedness for the defocussing DS II equation in the space $H^{1,1}(R^2)$ consisting of $L^2$ functions with $\nabla u$ and $(1+|\, \cdot \,|) u(\, \cdot \, )$ square-integrable. Using the same scattering and inverse scattering maps, we will also show that the inverse scattering method yields global, smooth solutions of the Novikov-Veselov equation for initial data of conductivity type, solving an open problem posed recently by Lassas, Mueller, Siltanen, and Stahel.