A system of coupled Schrödinger equations with time-oscillating nonlinearity

• Autor(a) principal: Panthee, Mahendra Prasad
• Data de Publicação: 2012
• Outros Autores: Carvajal, Xavier, Gamboa, Pedro
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/1822/21097
• topic: Schrödinger equation; Initial value problem; Strichartz estimate; Well-posedness

This paper is concerned with the initial value problem (IVP) associated to the coupled system of supercritical nonlinear Schrödinger equations \begin{equation} \begin{cases} iu_{t}+\Delta u+\theta_1(\omega t)(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1})u = 0, \\ iv_{t}+\Delta v+\theta_2(\omega t)(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1})v = 0, \end{cases} \end{equation} where $\theta_1$ and $\theta_2$ are periodic functions, which has applications in many physical problems, especially in nonlinear optics. We prove that, for given initial data $\varphi,\psi\in H^{1}(\mathbb{R}^{n})$, as $|\omega|\;\rightarrow\;\infty$, the solution $(u_{\omega},v_{\omega})$ converges to the solution $(U,V)$ of the IVP associated to \begin{equation}\label{eq-0.2} \begin{cases} iU_{t}+\Delta U+I(\theta_1)(|U|^{2p}+\beta|U|^{p-1}|V|^{p+1})U = 0, \\ iV_{t}+\Delta V+I(\theta_2)(|V|^{2p}+\beta|V|^{p-1}|U|^{p+1})V = 0, \end{cases} \end{equation} with the same initial data, where $I(g)$ is the average of the periodic function $g$. Moreover, if the solution $(U,V)$ is global and bounded, then we prove that the solution $(u_{\omega},v_{\omega})$ is also global provided $|\omega|\gg 1$.