Nonlinear dynamics on branched structures and networks

Abstract

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, 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 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 Schrodinger 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. For the critical case, our results show a phenomenology much richer than the analogous on the line.

47 pages, 44 figure. Lecture notes for a course given at the Summer School "MMKT 2016, Methods and Models of Kinetic Theory, Porto Ercole, June 5-11, 2016. To be published in Riv. Mat. Univ. Parma

References in corpus (2)

• Bifurcations of standing localized waves on periodic graphs
• Topology induced bifurcations for the NLS on the tadpole graph

Cited by in corpus (6)

• On the nonlinear Dirac equation on noncompact metric graphs
• Ground states of the $L^2$-critical NLS equation with localized nonlinearity on a tadpole graph
• Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover
• NLS Bifurcations on the bowtie combinatorial graph and the dumbbell metric graph
• Existence of infinite stationary solutions of the $L^2$-subcritical and critical NLSE on compact metric graphs
• Variational and stability properties of constant solutions to the NLS equation on compact metric graphs