Edgelocalized 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 edgelocalized 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 DirichlettoNeumann manifolds (nonlinear analogues of DirichlettoNeumann 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.
 Publication:

Annales de L'Institut Henri Poincare Section (C) Non Linear Analysis
 Pub Date:
 September 2021
 DOI:
 10.1016/j.anihpc.2020.11.003
 arXiv:
 arXiv:1910.03449
 Bibcode:
 2021AIHPC..38.1295B
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Nonlinear Sciences  Pattern Formation and Solitons;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 43 pages