The introduction of the nonlinear Schrödinger equation (NLS) on graphs is motivated by several physical reasons, e.g., the study of propagation of waves in ramified structures in nonlinear media, or the analysis of the dynamics of the Bose-Einstein condensates in the presence of impurities. The related mathematical problem can be formulated as a system of as many NLS as edges in the graph, coupled by suitable self-adjoint matching conditions at the vertices. The analysis is only at its beginning and mainly concerns the structure of the family of ground states of the NLS on star graphs; however, some results show the occurence of phenomena that are unexpected and far from the corresponding results for the standard NLS. Among them, the appearance of symmetry-breaking bifurcations, and the absence of ground states in the case of free (i.e. Kirchhoff’s) infinite graphs, that show the need for an adaptation of the rearrangement method and of techniques related with concentration-compactness theory. We give examples of such phenomena and related adaptations in the case of star graphs with delta and delta-prime vertex conditions.
This is part of a research project with Claudio Cacciapuoti (Bonn), Domenico Finco (Rome), and Diego Noja (Milan).