Asymptotic behavior of the Schrödinger-Debye system with refractive index of square wave amplitude

Abstract

We obtain local well-posedness for the one-dimensional Schrödinger-Debye interactions in nonlinear optics in the spaces $L^2\times L^p,1\le p<\infty$. When $p=1$ we show that the local solutions extend globally. In the focusing regime, we consider a family of solutions $\{(u_{\tau },v_{\tau }){\}}_{\tau >0}$ in $H^1\times H^1$ associated to an initial data family $\{(u_{{\tau }_0},v_{{\tau }_0}){\}}_{\tau >0}$ uniformly bounded in $H^1\times L^2$, where $\tau$ is a small response time parameter. We prove prove that $(u_{\tau },v_{\tau })$ converges to $(u,-|u{|}^2)$ in the space $L_{[0,T]}^{\infty }{L}_x^2\times L_{[0,T]}^1{L}_x^2$ whenever $u_{{\tau }_0}$ converges to $u_0$ in $H^1$ as long as $\tau$ tends to 0, where $u$ is the solution of the one-dimensional cubic non-linear Schrödinger equation with initial data $u_0$. The convergence of $v_{\tau }$ for $-|u{|}^2$ in the space $L_{[0,T]}^{\infty }{L}_x^2$ is shown under compatibility conditions of the initial data. For non compatible data we prove convergence except for a corrector term which looks like an initial layer phenomenon.