Semiclassical limit of a class of Schrodinger equations with potential

This note is related to the paper by J. C. Bronski and R. L. Jerrard (2) on the soliton dynamics in a potential. By using a similar argument to that carried out in (2) and some ideas of the WKB method, we give a sharper description of the asymptotic behavior of the family of solutions to the nonlinear focusing Schrodinger equation (NLS) with a potential term: iepsilonpartial derivative(t)u(epsilon) + epsilon(2)/2 Deltau(epsilon) - V(x)u(epsilon) + \u(epsilon)\(2sigma) u(epsilon) = 0, t epsilon R, x epsilon R-N, with initial data u(epsilon)(0, x) = R(x-x(0)/epsilon)epsilon(i(x,v0/epsilon)), where R is the ground state of the associated unscaled elliptic problem. This gives, in particular, a complete description to the dynamics of the t-dependent Wigner measure associated to the family ((1/epsilon(N/2))u(epsilon)(t)).