Energy scattering of a generalized Davey–Stewartson system in three dimension

Abstract

In this paper, we study the global scattering result of the solution for the generalized Davey–Stewartson system $$\left\{ {\begin{array}{*{20}{c}} {i{\partial _t}u + \Delta u = {{\left| u \right|}^2}u + u{v_{{x_1}}},\left( {t,x} \right) \in \mathbb{R} \times {\mathbb{R}^3},} \\ { - \Delta u = {{\left( {{{\left| u \right|}^2}} \right)}_{{x_1}}}.} \end{array}} \right.$$ The main difficulties are the failure of the interaction Morawetz estimate and the asymmetrical structure of nonlinearity (in particular, the nonlinearity is non-local). To compensate, we utilize the strategy derived from concentration-compactness idea, which was first introduced by Kenig and Merle [Invent. Math., 166, 645–675 (2006)].