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 $$\{\begin{array}{l}i\end{array}{\partial }_{t}u+c_0{\partial }_{x_1}^{2}u+{\partial }_{x_2}^{2}u=c_1|u{|}^{2}u+c_{2}u{\partial }_{x_1}\phi ,(x,t)\in {\mathbf{R}}^3,{\partial }_{x_1}^2\phi +c_3{\partial }_{x_2}^2\phi ={\partial }_{x_1}|u{|}^2,u(x,0)=\varphi (x),\text{endcases}$$where $c_0,c_3\in \mathbf{R}$, $c_1,c_2\in \mathbf{C}$,$u$ is a complex valued function and $\phi$ is a real valued function. The initial data $\varphi$ is $\mathbf{C}$-valued function on ${\mathbf{R}}^{\mathbf{n}}$, and usually it belongs to some kind of Sobolev type spaces.