title: Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph.
abstract: In this paper we give the complete classification of solitons for a cubic nonlinear Schrödinger equation on the simplest network with a nontrivial topology: the tadpole graph, i.e., a ring with a half line attached to it and free boundary conditions at the junction. This is a step toward the modelization of condensate propagation and confinement in quasi-one-dimensional traps. The model, although simple, exhibits a surprisingly rich behavior and in particular we show that it admits: (i) a denumerable family of continuous branches of embedded solitons vanishing on the half line and bifurcating from linear eigenstates and threshold resonances of the system; (ii) a continuous branch of edge solitons bifurcating from the previous families at the threshold of the continuous spectrum with a pitchfork bifurcation; and (iii) a finite family of continuous branches of solitons without linear analog. All the solutions are explicitly constructed in terms of elliptic Jacobian functions. Moreover we show that families of nonlinear bound states of the above kind continue to exist in the presence of a uniform magnetic field orthogonal to the plane of the ring when a well definite flux quantization condition holds true. In this sense the magnetic field acts as a control parameter. Finally we highlight the role of resonances in the linearization as a signature of the occurrence of bifurcations of solitons from the continuous spectrum.
authors: Dipartimento di Scienza e Alta Tecnologia, Università dell'Insubria, Via Valleggio 11, 22100 Como, Italy.
authors: Claudio
authors: C
authors: Cacciapuoti
authors: Facoltà di Ingegneria, Università Telematica Internazionale Uninettuno, Corso Vittorio Emanuele II 39, 00186 Roma, Italy.
authors: Domenico
authors: D
authors: Finco
authors: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 55, 20125 Milano, Italy.
authors: Diego
authors: D
authors: Noja