Bose–Einstein condensates in the presence of a magnetic trap and optical lattice: two-mode approximation

Abstract

In this paper we study the solutions to a nonlinear Schrödinger equation with harmonic and periodic potentials, motivated from the recent interest in these models as mean-field descriptions of Bose–Einstein condensates. We use a two-mode Galerkin approximation to study the dynamics of the full model. The phase plane and stability analysis of the reduced model yield very good agreement with the findings of the full partial differential equation. A particularly interesting finding of the stability analysis is a spontaneous symmetry breaking through a branching bifurcation, resulting in the stabilization of asymmetric states and the destabilization of symmetric or anti-symmetric ones. We also highlight the important differences between the cases of symmetric potentials and those of weakly asymmetric potentials.