Symmetry-breaking bifurcation in nonlinear schrödinger/gross- pitaevskii equations

We consider a class of nonlinear Schrödinger/Gross-Pitaeveskii (NLS-GP) equations, i.e., NLS with a linear potential. NLS-GP plays an important role in the mathematical modeling of nonlinear optical as well as macroscopic quantum phenomena (BEC). We obtain conditions for a symmetry-breaking bifurcation in a symmetric family of states as M, the squared L ² norm (particle number, optical power), is increased. The bifurcating asymmetric state is a "mixed mode" which, near the bifurcation point, is approximately a superposition of symmetric and antisymmetric modes. In the special case where the linear potential is a double well with well-separation L, we estimate N _cr(L), the symmetry breaking threshold. Along the "lowest energy" symmetric branch, there is an exchange of stability from the symmetric to the asymmetric branch as N is increased beyond N _cr.