From nonlinear Schrödinger equation to interacting particle system

We study the limiting behavior of solutions to nonlinear Schrödinger equations$-{\epsilon }^2\Delta u_{\epsilon }+u_{\epsilon }=u_{\epsilon }^p,u_{\epsilon }>0\text{in}{R}^n,$ as $\epsilon \to 0$, where p is Sobolev subcritical. These solutions are assumed to have infinitely many peaks. We derive the interaction form between the limiting peak points. This is achieved by first describing the main order term of $u_{\epsilon }$ and providing a very precise estimate on the error by the reverse Lyapunov-Schmidt reduction method, and then extracting information from the reduction equation in a limiting way.