Asymptotic behavior of the Schr\"odinger-Debye system with refractive index of square wave amplitude

We obtain local well-posedness for the one-dimensional Schr\"odinger-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 $L^{\infty}_{[0, T]}L^2_x\times L^1_{[0, T]}L^2_x$ 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 Schr\"odinger equation with initial data $u_0$. The convergence of $v_{\tau}$ for $-|u|^2$ in the space $L^{\infty}_{[0, T]}L^2_x$ 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.