Spectral stability of shifted states on star graphs

We consider the nonlinear Schrödinger (NLS) equation with the subcritical power nonlinearity on a star graph consisting of $N$ edges and a single vertex under generalized Kirchhoff boundary conditions. The stationary NLS equation may admit a family of solitary waves parameterized by a translational parameter, which we call the shifted states. The two main examples include (i) the star graph with even $N$ under the classical Kirchhoff boundary conditions and (ii) the star graph with one incoming edge and $N-1$ outgoing edges under a single constraint on coefficients of the generalized Kirchhoff boundary conditions. We obtain the general counting results on the Morse index of the shifted states and apply them to the two examples. In the case of (i), we prove that the shifted states with even $N \geq 4$ are saddle points of the action functional which are spectrally unstable under the NLS flow. In the case of (ii), we prove that the shifted states with the monotone profiles in the $N-1$ outgoing edges are spectrally stable, whereas the shifted states with non-monotone profiles in the $N-1$ outgoing edges are spectrally unstable, the two families intersect at the half-soliton states which are spectrally stable but nonlinearly unstable. Since the NLS equation on a star graph with shifted states can be reduced to the homogeneous NLS equation on a line, the spectral instability of shifted states is due to the perturbations breaking this reduction. We give a simple argument suggesting that the spectrally stable shifted states are nonlinear unstable under the NLS flow due to the perturbations breaking the reduction to the NLS equation on a line.