Variational properties and orbital stability of standing waves for NLS equation on a star graph

We study standing waves for a nonlinear Schrödinger equation on a star graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength $α\leqslant 0$. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form $i \frac{d}{dt}Ψ_t = H Ψ_t - | Ψ_t |^{2μ} Ψ_t$, where $H$ is the Hamiltonian operator which generates the linear Schrödinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every $ω> \frac{α^2}{N^2}$. Furthermore, we determine the ground states, as minimizers of the action on the Nehari manifold, and order the various families. Finally, we show that the ground states are orbitally stable for every allowed $ω$ if the nonlinearity is subcritical or critical, and for $ω<ω^\ast$ otherwise.

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