Stationary states of NLS on star graphs

We consider a generalized nonlinear Schrödinger equation (NLS) with a power nonlinearity | ψ | 2 μ ψ of focusing type describing propagation on the ramified structure given by N edges connected at a vertex (a star graph). To model the interaction at the junction, it is there imposed a boundary condition analogous to the δ potential of strength α on the line, including as a special case ( α = 0) the free propagation. We show that nonlinear stationary states describing solitons sitting at the vertex exist both for attractive ( α < 0, representing a potential well) and repulsive ( α > 0, a potential barrier) interaction. In the case of sufficiently strong attractive interaction at the vertex and power nonlinearity μ < 2, including the standard cubic case, we characterize the ground state as minimizer of a constrained action and we discuss its orbital stability. Finally we show that in the free case, for even N only, the stationary states can be used to construct traveling waves on the graph.