On the soliton dynamics under slowly varying medium for Nonlinear Schr\"odinger equations

We consider the problem of the soliton propagation, in a slowly varying medium, for a generalized, variable-coefficients nonlinear Schr"odinger equation. We prove existence and uniqueness of new soliton-like solutions for a large class of slowly varying media. Moreover, we describe for all time the behavior of this new generalized soliton solution.