Soliton dynamics for CNLS systems with potentials

The semiclassical limit of a weakly coupled nonlinear focusing Schrödinger system in presence of a nonconstant potential is studied. The initial data is of the form (u₁, u₂) with u_i=r_i ((x−x˜)/ε)e^{(i/ε)x·ξ˜}, where (r₁, r₂) is a real ground state solution, belonging to a suitable class, of an associated autonomous elliptic system. For ε sufficiently small, the solution (ϕ₁, ϕ₂) will been shown to have, locally in time, the form (r₁ ((x−x(t))/ε)e^{(i/ε)x·ξ(t)}, r₂ ((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˜.