Topology induced bifurcations for the NLS on the tadpole graph

In this paper we give the complete classification of solitons for a cubic NLS equation on the simplest network with a non-trivial topology: the tadpole graph, i.e. a ring with a half-line attached to it and free boundary conditions at the junction. The model, although simple, exhibits a surprisingly rich behavior and in particular we show that it admits: 1) a denumerable family of continuous branches of embedded solitons bifurcating from linear eigenstates and threshold resonances of the system; 2) a continuous branch of edge solitons displaying a pitchfork symmetry breaking bifurcation at the threshold of the continuous spectrum; 3) a finite family of continuous branches of solitons without linear analogue. 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. % and the analysis could be easily generalized to other simple and phenomenologically relevant topologies. The main interest of the example lies in the fact that the simplest equation (cubic NLS) in the simplest topologically nontrivial situation gives rise to unexpected and definite soliton patterns; in this sense the model appears as prototypical and its properties not bound to the explicit character of the presented example. Finally we highlight %, seemingly for the first time, the role of resonances in the linearization as a signature of the occurrence of bifurcations of solitons from the continuous spectrum.