G. Berkolaiko, J.L. Marzuola and D.E. Pelinovsky
Edge-localized states on quantum graphs in the limit of large mass
Annales de l'Institut Henri Poincaré C, Analyse non linéaire 38 (2021) 1295-1335
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.
Keywords:
Nonlinear Schrodinger equation; quantum graphs, stationary solutions,
ground state, Jacobi elliptic functions.