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A. Kairzhan and D.E. Pelinovsky

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Spectral stability of shifted states on star graphs

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**Abstract:**

We consider the nonlinear Schrodinger (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>=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 nonmonotone
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 under the NLS flow.
Since the NLS equation on a star graph with shifted states can be reduced to the homogeneous NLS
equation on an infinite 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
unstable under the NLS flow due to the perturbations breaking the reduction to the NLS equation on
an infinite line.

**Keywords**:

Nonlinear Schrodinger equation; star graphs, spectral stability of stationary states, Morse index, Sturm theory.