High frequency solutions for the singularly-perturbed one-dimensional nonlinear Schrodinger equation

Abstract

This article is devoted to the nonlinear Schrodinger equation epsilon(2)u'' - V(x)u + vertical bar u vertical bar(p-1)u = 0,

when the parameter epsilon approaches zero. All possible asymptotic behaviors of bounded solutions can be described by means of envelopes, or alternatively by adiabatic profiles. We prove that for every envelope, there exists a family of solutions reaching that asymptotic behavior, in the case of bounded intervals. We use a combination of the Nehari finite dimensional reduction together with degree theory. Our main contribution is to compute the degree of each cluster, which is a key piece of information in order to glue them.