Title: Spectral stability of shifted states on star graphs,
Abstract: 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.