Soliton dynamics for CNLS systems with potentials

摘要

The semiclassical limit of a weakly coupled nonlinear focusing Schrodinger system in presence of a nonconstantrnpotential is studied. The initial data is of the form (u_1, u_2) with u_i = r_i((x-x)/ε)e~((i/ε)x·ξ), where (r_1,r_2) is a real ground state solution, belonging to a suitable class, of an associated autonomous elliptic system. For e sufficiently small, the solution (φ_1, φ_2) will been shown to have, locally in time, the form (r_1((x-x(t))/ε)e~((i/ε)x·ξ(t)),r_2((x-x(t))/ε)e~((i/ε)x·ξ(t))), where (x(t),ξ(t)) is the solution of the Hamiltonian system x(t) = ξ(t), ξ(t) = -▽V(x(t)) with x(0) = x and ξ(0) = ξ.