主讲人：钟学秀（华南数学应用与交叉中心）
时 间：2019年9月19日下午 9:00-11:00
地 点：华南数学应用与交叉研究中心学术课室114
题 目：The solutions of Nonlinear Schr\"odinger system
摘要：
Consider the following time-dependent system
\begin{equation}\label{system-t}
\left\{\begin{array}{l}{-i \frac{\partial}{\partial t} \Phi_{1}=\Delta \Phi_{1}+\mu_{1}\left|\Phi_{1}\right|^{2} \Phi_{1}+\beta\left|\Phi_{2}\right|^{2} \Phi_{1}, \quad x \in \Omega, t>0} \\ {-i \frac{\partial}{\partial t} \Phi_{2}=\Delta \Phi_{2}+\mu_{2}\left|\Phi_{2}\right|^{2} \Phi_{2}+\beta\left|\Phi_{1}\right|^{2} \Phi_{2}, \quad x \in \Omega, t>0} \\ {\Phi_{j}=\Phi_{j}(x, t) \in \mathbb{C}, \quad j=1,2} \\ {\Phi_{j}(x, t)=0, \quad x \in \partial \Omega, t>0, j=1,2}\end{array}\right.
\end{equation}
where $\Omega=\R^N$ or $\Omega\subset \R^N$ is a bounded smooth domain. $i$ is the imaginary unit,$\mu_1,\mu_2>0$ are constants. The ansatz $\Phi_1(x,t)=e^{i\lambda_1 t}u(x)$ and $\Phi_2(x,t)=e^{i\lambda_2t}v(x)$ for solitary wave solutions leads to the elliptic system:
\begin{equation}\label{system}
\left\{\begin{array}{l}{-\Delta u+\lambda_{1} u=\mu_{1} u^{3}+\beta u v^{2}, \quad x \in \Omega} \\ {-\Delt