Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the …