We study the nonlinear Schrödinger (NLS) equation on star graphs with the Neumann-Kirchhoff (NK) boundary conditions at the vertex. We analyze the stability of standing wave solutions of the NLS equation.

For a half-soliton state of the NLS equation, we use reduction to finite-dimensional space to prove that the half-soliton state is nonlinearly unstable due to small perturbations growing slowly in time. Moreover, under certain constraints on parameters of the generalized NK conditions, we show the existence of a family of shifted states, which are parametrized by a translational parameter. For the spectrally stable shifted states, we show that the momentum of the NLS equation is not conserved which results in the irreversible drift of the family of shifted states towards the vertex of the star graph. As a result, the spectrally stable shifted states are nonlinearly unstable.

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