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In this paper we give the complete classification of solitons for a cubic NLSequation on the simplest network with a non-trivial topology: the tadpolegraph, i.e. a ring with a half-line attached to it and free boundary conditionsat the junction. The model, although simple, exhibits a surprisingly richbehavior and in particular we show that it admits: 1) a denumerable family ofcontinuous branches of embedded solitons bifurcating from linear eigenstatesand threshold resonances of the system; 2) a continuous branch of edge solitonsdisplaying a pitchfork symmetry breaking bifurcation at the threshold of thecontinuous spectrum; 3) a finite family of continuous branches of solitonswithout linear analogue. All the solutions are explicitly constructed in termsof elliptic Jacobian functions. Moreover we show that families of nonlinearbound states of the above kind continue to exist in the presence of a uniformmagnetic field orthogonal to the plane of the ring when a well definite fluxquantization condition holds true. Finally we highlight the role of resonancesin the linearization as a signature of the occurrence of bifurcations ofsolitons from the continuous spectrum.

Topology induced bifurcations for the NLS on the tadpole graph