Kairzhan, Adilbek; Pelinovsky, Dmitry E. Nonlinear instability of half-solitons on star graphs. (English) Zbl 1392.35289 J. Differ. Equations 264, No. 12, 7357-7383 (2018). Summary: We consider a half-soliton stationary state of the nonlinear Schrödinger 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 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 R. Adami et al. [J. Phys. A, Math. 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