We consider the focusing (attractive) nonlinear Schr\"odinger (NLS) equationwith an external, symmetric potential which vanishes at infinity and supports alinear bound state. We prove that the symmetric, nonlinear ground states mustundergo a symmetry breaking bifurcation if the potential has a non-degeneratelocal maxima at zero. Under a generic assumption we show that the bifurcationis either subcritical or supercritical pitchfork. In the particular case ofdouble-well potentials with large separation, the power of nonlinearitydetermines the subcritical or supercritical character of the bifurcation. Theresults are obtained from a careful analysis of the spectral properties of theground states at both small and large values for the corresponding eigenvalueparameter. We employ a novel technique combining concentration--compactness andspectral properties of linearized Schr\"odinger type operators to show that thesymmetric ground states can either be uniquely continued for the entireinterval of the eigenvalue parameter or they undergo a symmetry--breakingpitchfork bifurcation due to the second eigenvalue of the linearized operatorcrossing zero. In addition we prove the appropriate scaling for the stationarystates in the limit of large values of the eigenvalue parameter. The scalingand our novel technique imply that all ground states at large eigenvalues mustbe localized near a critical point of the potential and bifurcate from thesoliton of the focusing NLS equation without potential localized at the samepoint. The theoretical results are illustrated numerically for a double-wellpotential obtained after the splitting of a single-well potential. We comparethe cases before and after the splitting, and numerically investigatebifurcation and stability properties of the ground states which are beyond thereach of our theoretical tools.
展开▼
