Stationary solutions for multi-dimensional Gross-Pitaevskii equation with double-well potential

Abstract

In this paper we consider a non-linear Schroedinger equation with a cubic nonlinearity and a multi-dimensional double well potential. In the semiclassical limit the problem of the existence of stationary solutions simply reduces to the analysis of a finite dimensional Hamiltonian system which exhibits different behavior depending on the dimension. In particular, in dimension 1 the symmetric stationary solution shows a standard pitchfork bifurcation effect, while in dimension 2 and 3 new asymmetrical solutions associated to saddle points occur. These last solutions are localized on a single well and this fact is related to the phase transition effect observed in Bose-Einstein condensates in periodical lattices.