Spike vector solutions for some coupled nonlinear Schrödinger equations

We consider spike vector solutions for the nonlinearSchrödinger system\begin{equation*}\left\{\begin{array}{ll}-\varepsilon^{2}\Delta u+P(x)u=\mu u^{3}+\beta v^2u \ \hbox{in}\ \mathbb{R}^3,\\-\varepsilon^{2}\Delta v+Q(x)v=\nu v^{3} +\beta u^2v \ \ \hbox{in}\ \mathbb{R}^3,\\u, v >0 \,\ \hbox{in}\ \mathbb{R}^3,\end{array}\right.\end{equation*}where $\varepsilon > 0$ is a small parameter, $P(x)$ and $Q(x)$ arepositive potentials, $\mu>0, \nu>0$ are positive constants and$\beta\neq 0$ is a coupling constant. We investigate the effect ofpotentials and the nonlinear coupling on the solution structure. Forany positive integer $k\ge 2$, we construct $k$interacting spikes concentrating near the local maximum point$x_{0}$ of $P(x)$ and $Q(x)$ when $P(x_{0})=Q(x_{0})$ in theattractive case. In contrast, for any two positive integers $k\ge 2$ and $m\ge 2$, we construct $k$ interacting spikes for $u$near the local maximum point $x_{0}$ of $P(x)$ and $m$ interacting spikes for $v$near the local maximum point $\bar{x}_{0}$ of$Q(x)$ respectively when $x_{0}\neq \bar{x}_{0}$, moreover,spikes of $u$ and $v$ repel each other. Meanwhile, we prove theattractive phenomenon for $\beta 0$.