Multi-soliton dynamics in the nonlinear Schrödinger equation

Abstract

In this paper, we study the Cauchy problem of the nonlinear Schrödinger equation with a nontrival potential $V_{\epsilon }(x)$. 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 Schrödinger 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 $\nabla V_{\epsilon }$, provided the barycenters of these solitons do not coincide.

27 pages

References in corpus (1)

• Solitary Wave Dynamics in an External Potential