Global existence and asymptotic behavior in time of small solutions to the elliptic-hyperbolic Davey-Stewrtson system

Abstract

We study the initial value problem for the Davey-Stewartson systems $$\cases i\partial_t u+c_0\partial_{x_1}^2u+\partial_{x_2}^2 u =c_1|u|^2u+c_2u\partial_{x_1}\varphi, \quad (x,t)\in{\bold R}^3, \partial_{x_1}^2\varphi+c_3\partial_{x_2}^2\varphi = \partial_{x_1}|u|^2, u(x,0) = \phi(x),\endcases$$where $c_0,c_3\in{\bold R}$, $c_1,c_2\in{\bold C}$,$u$ is a complex valued function and $\varphi$ is a real valued function. The initial data $\phi$ is $\bold C$-valued function on $\bold R^n$, and usually it belongs to some kind of Sobolev type spaces.