Multi-soliton dynamics in the nonlinear Schr\"{o}dinger equation

In this paper, we study the Cauchy problem of the nonlinear Schr\"{o}dinger equation with a nontrival potential $V_\varepsilon(x)$. In particular, we consider the case where the initial data is close to a superposition of $k$ solitons with prescribed phase and location, and investigate the evolution of the Schr\"{o}dinger system. We prove that over a large time interval with the maximum time tending to infinity, all $k$ solitons will maintain the shape, and the solitons dynamics can be regarded as an approximation of $k$ particles moving in $\mathbb{R}^N$ with their accelerations dominated by $\nabla V_\varepsilon$, provided the barycenters of these solitons do not coincide.