# 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 ##IMG## [http://ej.iop.org/images/1751-8121/51/9/095203/aaaa89fieqn001.gif]

{$N \geqslant 4$} are saddle points of the action functional which are spe...