Symmetry-breaking bifurcation in the Gross-Pitaevskii equation with a double-well potential

We classify bifurcations of the asymmetric states from a family of symmetric states in the focusing (attractive) Gross-Pitaevskii equation with a symmetric double-well potential. Depending on the shape of the potential, both supercritical and subcritical pitchfork bifurcations may occur. We also consider the limit of large energies and show that the asymmetric states always exist near a non-degenerate extremum of the symmetric potential. These states are stable (unstable) in the case of subcritical nonlinearity if the extremum is a minimum (a maximum). All states are unstable for large energy in the case of supercritical nonlinearity. This is a joint work with E. Kirr and P. Kevrekidis.