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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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J. Phys. A: Math. Theor. 51 (2018) 095203 (23pp)

**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 edges
are spectrally stable, whereas the shifted states with non-monotone profiles
in the N - 1 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
in the case of (ii) are nonlinearly unstable under the NLS flow due to the
perturbations breaking the reduction to the homogeneous NLS equation.

**Keywords**:

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