17-48 Riccardo Adami, Enrico Serra, Paolo Tilli

nonlinear dynamics on branched structures and networks (157K, LaTex) May 1, 17

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Abstract. Nonlinear dynamics on graphs has rapidly become a topical issue with many physical applications, ranging fromnonlinear optics to Bose-Einstein condensation. Whenever in aphysical experiment a ramified structure is involved (e.g. in the propagation of signals, in a circuit of quantum wires or in trapping a boson gas), it can prove useful to approximate such a structure by a metric graph, or network. For the Schroedinger equation it turns out that the sixth power in the nonlinear term of the energy (corresponding to the quinticnonlinearity in the evolution equation) is critical in the sense that below that power the constrained energy is lower bounded irrespectively of the value of the mass (subcritical case}. On the other hand, if the nonlinearity power equals six, then the lower boundedness depends on the value of the mass: below a critical mass, the constrained energy is lower bounded, beyond it, it is not. For powers larger than six the constrained energy functional is never lower bounded, so that it is meaningless to speak about ground states (supercritical case). These results are the same as in the case of the nonlinear Schroedinger equation on the real line. In fact, as regards the existence of ground states, the results for systems on graphs differ, in general, from the ones for systems on the line even in the subcritical case: in the latter case, whenever the constrained energy is lower bounded there always exist ground states (the solitons, whose shape is explicitly known), whereas for graphs the existence of a ground state is not guaranteed. More precisely, we show that the existence of such constrained ground states is strongly conditioned by the topology and the matric of the graph.

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