Juan C. Cordero | Searcharxiv

arXiv2020

Uniform Adiabatic Limit of Benney type Systems

In this paper we show that solutions of the cubic nonlinear Schrödinger equation are asymptotic limit of solutions to the Benney system. Due to the special characteristic of the one-dimensional transport equation same result is obtained for solutions of the one-dimensional Zakharov and 1d-Zakharov-Rubenchik systems. Convergence is reached in the topology $L^2(\mathbb{R})\times L^2(\mathbb{R})$ and with an approximation in the energy space $H^1(\mathbb{R})\times L^2(\mathbb{R})$. In the case of the Zakharov system this is achieved without the condition $\partial_t n(x,0) \in \dot H^{-1}(\mathbb{R})$ for the wave component, improving previous results.

math.AP↗

arXiv2017

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

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 $\{(u_τ, v_τ)\}_{τ>0}$ in $ H^1\times H^1$ associated to an initial data family $\{(u_{τ_0},v_{τ_0})\}_{τ>0}$ uniformly bounded in $H^1\times L^2$, where $τ$ is a small response time parameter. We prove prove that $(u_τ, v_τ)$ converges to $(u, -|u|^2)$ in the space $L^{\infty}_{[0, T]}L^2_x\times L^1_{[0, T]}L^2_x$ whenever $u_{τ_0}$ converges to $u_0$ in $H^1$ as long as $τ$ 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 $v_τ$ 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.

math-ph↗