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

Berkolaiko, Gregory; Marzuola, Jeremy L.; Pelinovsky, Dmitry E.

doi:10.1016/j.anihpc.2020.11.003

Berkolaiko, Gregory ¹ ; Marzuola, Jeremy L.² ; Pelinovsky, Dmitry E.³

¹ a Department of Mathematics, Texas A & M University, College Station, TX 77843-3368, USA
² b Department of Mathematics, University of North Carolina - Chapel Hill, Chapel Hill, NC 27599, USA
³ c Department of Mathematics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1295-1335.

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Résumé

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.

Reçu le : 2020-03-24
Révisé le : 2020-09-28
Accepté le : 2020-11-09

DOI : 10.1016/j.anihpc.2020.11.003

Mots-clés : Standing waves on metric graphs, Existence of energy minimizers, Nonlinear Schrödinger equation, Large-mass asymptotic limit

Affiliations des auteurs :

Berkolaiko, Gregory ¹ ; Marzuola, Jeremy L. ² ; Pelinovsky, Dmitry E. ³

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     title = {Edge-localized states on quantum graphs in the limit of large mass},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1295--1335},
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Berkolaiko, Gregory; Marzuola, Jeremy L.; Pelinovsky, Dmitry E. Edge-localized states on quantum graphs in the limit of large mass. Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1295-1335. doi : 10.1016/j.anihpc.2020.11.003. http://www.numdam.org/articles/10.1016/j.anihpc.2020.11.003/

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