In the theory of water waves (esp. surface waves), the $2D$ generalization of the usual cubic 1D Schr\"odinger equation turns out to be the Davey-Stewartson equation.
In Chapter I, we study the scattering for a class of nonlinear Davey-Stewartson equations with three nonlinearities. We proved that their scattering operator exists in $H^1$.
In Chapter II, We generalize its nonlinearity from the cubic case to the $p$-th power cases. Through considering the Cauchy problem for the generalized Davey-Stewartson equation in $\Sigma(\Bbb{R}^n) := \{u\in H^1(\Bbb{R}^n) : |x|u\in L^2(\Bbb{R}^n)\}$, we obtain its scattering theory. Of course ,the global existence and the uniqueness of the solution for the Cauchy problem are studied.
Key words and Phrases: Nonlinear Davey-Stewartson equations, generalized Davey-Stewartson equation, Morawetz-type estimate, pseudo conformally invariant conservation law and scattering operator.