Stationary States of NLS on Star Graphs

We consider a nonlinear Schr\"odinger equation (NLS) with a power focusing nonlinearity on a star graph with $N$ edges and a vertex with boundary conditions of $\delta$ type, including the special case of a Kirchhoff vertex. We show that nonlinear stationary states exist both for attractive and repulsive $\delta$ interaction and we give explicitly their expression. In the case of attractive interaction at the vertex and subcritical nonlinearity we characterize the ground state as suitably constrained action minimum and we rigorously discuss its orbital stability. Finally we show that in the Kirchhoff case, for even $N$ only, the stationary states can be used to construct traveling waves on the graph.