Edge-localized states on quantum graphs in the limit of large mass

Abstract: In this work, we construct and quantify asymptotically in the limit of large mass a variety of edge-localized stationary states of the focusing nonlinear Schr\"odinger equation on a quantum graph. The method is applicable to general bounded and unbounded graphs. The solutions are constructed by matching a localized large amplitude elliptic function on a single edge with an exponentially smaller remainder on the rest of the graph. This is done by studying the intersections of Dirichlet-to-Neumann manifolds (nonlinear analogues of Dirichlet-to-Neumann maps) corresponding to the two parts of the graph. For the quantum graph with a given set of pendant, looping, and internal edges, we find the edge on which the state of smallest energy at fixed mass is localized. Numerical studies of several examples are used to illustrate the analytical results.