Riv. Mat. Univ. Parma, Vol. 8, No. 1, 2017

Riccardo Adami,[a] Enrico Serra,[b] and Paolo Tilli[c]

Nonlinear dynamics on branched structures and networks

Pages: 109-159
Accepted in revised form: 4 May 2017
Mathematics Subject Classification (2010): 35R02, 35Q55, 81Q35, 49J40.
Keywords: Minimization, metric graphs, critical growth, nonlinear Schrödinger Equation.
[a],[b],[c]: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, 10129, Italy

Abstract: In these lectures we review on a recently developed line of research, concerning the existence of ground states with prescribed mass (i.e. $$L^2$$-norm) for the focusing nonlinear Schrödinger equation with a power nonlinearity, on noncompact quantum graphs. Nonlinear dynamics on graphs has rapidly become a topical issue with many physical applications, ranging from nonlinear optics to Bose-Einstein condensation. Whenever in a physical 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 Schrödinger equation it turns out that the sixth power in the nonlinear term of the energy (corresponding to the quintic nonlinearity 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 Schrödinger 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 of the graph. In particular, in the subcritical case we single out a topological hypothesis that prevents a graph from having ground states for every value of the mass. For the critical case, our results show a phenomenology much richer than the analogous on the line: if some topological assumptions are fulfilled, then there may exist a whole interval of masses for which a ground state exist. This behaviour is highly non-standard for $$L^2$$-critical nonlinearities.

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