Global Well-Posedness and Long-time Asymptotics for the Defocussing Davey-Stewartson II Equation in $H^{1,1}(R^2)$

Abstract

We show that the inverse scattering map for the linear system associated with the defocussing Davey-Stewartson II equation is locally Lipschitz continuous with locally Lipschitz continuous inverse on $H^{1,1}(R^2)$. From the inverse scattering method we then obtain global well-posedness for the defocussing Davey-Stewartson II equation. We show that these global solutions are dispersive by computing their leading asymptotic behavior as $t \rightarrow \infty$ in terms of an associated linear problem.