Analytic smoothing effects of global small solutions to the elliptic-hyperbolic Davey-Stewartson system

Abstract

We study the elliptic-hyperbolic Davey-Stewartson system $$ \left\{ \begin{array}{l} i\partial_t u + (\partial_{x_1}^2 + \partial_{x_2}^2) u = c_1 |u|^2 u + c_2 u \partial_{x_1} {\varphi}, \qquad (t,x) \in {\bf R} \times {\bf R}^2, \\ (\partial_{x_1}^2 - \partial_{x_2}^2) {\varphi} = \partial_{x_1} |u|^2, \\ u(0,x) = u_0 (x), \qquad x \in {\bf R}^2, \end{array} \right. $$ where $c_1$, $c_2 \in {\bf R}$, $u$ is a complex valued function and ${\varphi}$ is a real valued function. The system can be translated to nonlocal nonlinear Schr{\"o}dinger equations after a rotation and a rescaling $$ {\mbox{(NLS)}} \qquad\qquad\qquad \left\{ \begin{array}{l} i\partial_t u + (\partial_{x_1}^2 + \partial_{x_2}^2) u \\ = c_0 |u|^2u + c_1 u \displaystyle{\int_{x_2}^\infty} \partial_{x_1} |u|^2 ({x_1},{x_2}')d{x_2}' + c_2 u \displaystyle{\int_{x_1}^\infty} \partial_{x_2} |u|^2 ({x_1}',{x_2})d{x_1}',\\ u(0,x) = u_0(x), \end{array} \right. $$ where $c_0,c_1,c_2 \in {\bf C}$. Our purpose in this paper is to prove that if the norm $ \| ( \prod_{j=1}^2 \cosh \theta x_j )u_0 \|_{3,0} + \| ( \prod_{j=1}^2 \cosh \theta x_j )u_0 \|_{0,3} $ for $\theta \neq 0$ is sufficiently small, then small solutions of (NLS) exist and become analytic with respect to $x$ for any $t \ne 0$ . Here we denote the weighted Sobolev space by ${\bf H}^{m,s} = \bigm\{ \phi \in {\bf L}^2 ; \| \phi \|_{m,s} = \bigm\| (1 + x_1^2 + x_2^2)^{s/2} (1 - \partial_{x_1}^2 - \partial_{x_2}^2)^{m/2} \phi \bigm\| < \infty \bigm\}$, $m,s \in {\bf R}^+$.