SYMMETRY-BREAKING BIFURCATION IN NONLINEAR SCHRODINGER/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 ${\cal N}$, 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 ${\cal N}_{cr}(L)$, the symmetry breaking threshold. Along the “lowest energy” symmetric branch, there is an exchange of stability from the symmetric to the asymmetric branch as ${\cal N}$ is increased beyond ${\cal N}_{cr}$