Soliton dynamics for CNLS systems with potentials

The semiclassical limit of a weakly coupled nonlinear focusing Schrodinger system in presence of a nonconstant potential is studied. The initial data is of the form (u(1), u(2)) with u(i) = r(i)(x-(x) over tilde/epsilon)e((i/epsilon)x.xi), where (r(1), r(2)) is a real ground state solution, belonging to a suitable class, of an associated autonomous elliptic system. For epsilon sufficiently small, the solution (phi(1), phi(2)) will been shown to have, locally in time, the form (r(1)(x-x(t)/epsilon)e((i/epsilon)x.xi(t)),r(2)(x-x(t)/epsilon)e((i/epsilon)x.xi(t))), where (x(t), xi(t)) is the solution of the Hamiltonian system (x) over dot(t) = xi(t), (xi) over dot(t) = -del V(x(t)) with x(0) = (x) over tilde and xi(0) = (xi) over tilde.