Title: Nonlinear Instability of Half-Solitons on Star Graphs,
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 $N \geq 3$ and arbitrary subcritical power nonlinearity. It gives
a precise dynamical characterization of the previous result of Adami {\em 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.