Stationary States of NLS on Star Graphs

We consider a generalized nonlinear Schrödinger equation (NLS) with a power nonlinearity |ψ|^2μψ, 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 δpotential of strength αon 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.

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