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 $u_i=r_i\big( \fracx-\tilde x\vep\big)e^\frac\rm i\vep x\cdot \tilde\xi$, where $(r_1,r_2) $ is a real ground state solution, belonging to a suitable class, of an associated autonomous elliptic system. For $\vep$ sufficiently small, the solution $(\phi_1,\phi_2)$ will been shown to have, locally in time, the form $(r_1\big(\fracx- x(t)\vep\big) e^\frac\rm i\vep x\cdot \xi(t),r_2\big(\fracx- x(t)\vep\big) e^\frac\rm i\vep x\cdot \xi(t))$, where $(x(t),\xi(t))$ is the solution of the Hamiltonian system $\dot x(t)=\xi(t)$, $\dot \xi(t)=-\nabla V(x(t))$ with $x(0)=\tildex$ and $\xi(0)=\tilde\xi$