Topological Resonances and Nonlinear Waves in Metric Graphs

We consider wave scattering from a complex system. Our model is a metric graph with a nonlinear Schrödinger (NLS) operator -- a simple theoretical model either for a BEC in a quasi-1D trap with non-trivial topology or for an optical fibre network. In the low intensity limit the NLS operator becomes linear (a quantum graph). For scattering with very low incoming intensities, one may expect that the nonlinearity is either irrelevant or may be treated perturbatively. However, at resonances this expectation often breaks down as the intensity inside the network may be amplified by some orders of magnitude (constructive interference on network cycles). In certain networks narrow resonances with very high amplification are far more frequent than in most other complex scattering models. We identify the origin of these resonances which is of topological nature and derive power laws for the intensity amplification inside the network