A. Kairzhan and D.E. Pelinovsky

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 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