Stationary solutions to the multi-dimensional Gross–Pitaevskii equation with double-well potential

In this paper we consider a nonlinear Schr ̈odinger 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 behaviour depending on the dimension. In particular, in dimension 1 the symmetric stationary solution shows a standard pitchfork bifurcation effect, while in dimensions 2 and 3 new asymmetrical solutions associated with 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.