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

### Dmitry E. Pelinovsky

Department of Mathematics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada### Gregory Berkolaiko

Department of Mathematics, Texas A & M University, College Station, TX 77843-3368, USA### Jeremy L. Marzuola

Department of Mathematics, University of North Carolina - Chapel Hill, Chapel Hill, NC 27599, USA

## Abstract

We construct and quantify asymptotically in the limit of large mass a variety of edge-localized stationary states of the focusing nonlinear Schrödinger 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.

## Cite this article

Dmitry E. Pelinovsky, Gregory Berkolaiko, Jeremy L. Marzuola, Edge-localized states on quantum graphs in the limit of large mass. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 5, pp. 1295–1335

DOI 10.1016/J.ANIHPC.2020.11.003