Publicação
A system of coupled Schrödinger equations with time-oscillating nonlinearity
| Resumo: | 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 $|\omega|\gg 1$. |
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| Autores principais: | Panthee, Mahendra Prasad |
| Outros Autores: | Carvajal, Xavier; Gamboa, Pedro |
| Assunto: | Schrödinger equation Initial value problem Strichartz estimate Well-posedness |
| Ano: | 2012 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade do Minho |
| Idioma: | inglês |
| Origem: | RepositóriUM - Universidade do Minho |
| Resumo: | 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 $|\omega|\gg 1$. |
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