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A. Kairzhan and D.E. Pelinovsky

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Nonlinear instability of half-solitons on star graphs

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**Abstract:**

We consider a half-soliton stationary state of the nonlinear Schrodinger equation
with the power nonlinearity on a star graph consisting of N edges and a single vertex. For
the subcritical power nonlinearity, the half-soliton state is a degenerate critical point of the
action functional under the mass constraint such that the second variation is nonnegative. By
using normal forms, we prove that the degenerate critical point is a nonlinear saddle point,
for which the small perturbations to the half-soliton state grow slowly in time resulting in the
nonlinear instability of the half-soliton state. The result holds for any N and arbitrary
subcritical power nonlinearity. It gives a precise dynamical characterization of the previous
result of Adami et al., where the half-soliton state was shown to be a saddle point of the
action functional under the mass constraint for N = 3 and for cubic nonlinearity.

**Keywords**:

Nonlinear Schrodinger equation; star graphs, normal forms, stability of stationary states