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: ( in press )
Received: 6 February 2017
Accepted in revised form: 4 May 2017
Mathematics Subject Classification (2010): 35R02, 35Q55, 81Q35, 49J40.
Keywords: Minimization, metric graphs, critical growth, nonlinear Schrödinger Equation.
Author address:
[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.

R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Fast solitons on star graphs, Rev. Math. Phys. 23 (2011), no. 4, 409-451. MR2804557
R. Adami, C. Cacciapuoti, D. Finco and D. Noja, On the structure of critical energy levels for the cubic focusing NLS on star graphs, J. Phys. A 45 (2012), no. 19, 192001. MR2924493
R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations 257 (2014), no. 10, 3738-3777. MR3260240
R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Constrained energy minimization and orbital stability for the NLS equation on a star graph, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 6, 1289-1310. MR3280068
R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differential Equations 260 (2016), no. 10, 7397-7415. MR3473445
R. Adami, F. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys. 127 (2007), no. 6, 1193-1220. MR2331036
R. Adami, E. Serra and P. Tilli, Lack of ground state for NLSE on bridge-type graphs, Springer Proc. Math. and Stat., 128, Springer, Cham, 2015, 1-11. MR3375151.
R. Adami, E. Serra and P. Tilli, NLS ground states on graphs, Calc. Var. Partial Differential Equations 54 (2015), 743-761. MR3385179
R. Adami, E. Serra and P. Tilli, Threshold phenomena and existence results for NLS ground states on metric graphs, J. Funct. Anal. 271 (2016), 201-223. MR3494248
R. Adami, E. Serra and P. Tilli, Negative energy ground states for the \(L^2\)-critical NLSE on metric graphs, Comm. Math. Phys. 352 (2017), no. 1, 387-406. MR3623262
F. Ali~Mehmeti, Nonlinear waves in networks, Mathematical Research, 80, Akademie-Verlag, Berlin, 1994. MR1287844
Z. Ammari, M. Falconi and B. Pawilowski, On the rate of convergence for the mean field approximation of bosonic many-body quantum dynamics, Commun. Math. Sci. 14 (2016), no. 5, 1417-1442. MR3506807
Z. Ammari and F. Nier, Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states, J. Math. Pures Appl. 95 (2011), no. 9, 585-626. MR2802894
C. Bardos, F. Golse and N. J. Mauser, Weak coupling limit of the $N$-particle Schrödinger equation, Methods Appl. Anal. 7 (2000), no. 2, 275-293. MR1869286
N. Benedikter, G. de Oliveira and B. Schlein, Quantitative derivation of the Gross-Pitaevskii equation, Comm. Pure App. Math. 68 (2015), no. 8, 1399-1482. MR3366749
N. Benedikter, M. Porta and B. Schlein, Effective evolution equations from quantum dynamics, SpringerBriefs Math. Phys., 7, Springer, Cham, 2016. MR3382225
G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, Math. Surveys Monogr., 186, AMS, Providence, RI, 2013. MR3013208
J. Bolte and J. Kerner, Many-particle quantum graphs and Bose-Einstein condensation, J. Math. Phys. 55 (2014), no. 6, 061901. MR3390648
J. Bona and R.C. Cascaval, Nonlinear dispersive waves on trees, Can. Appl. Math. Q. 16 (2008), no. 1, 1-18. MR2500096
S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik 26 (1924), 178-181. zbMAT
S. Buchholz, C. Saffirio and B. Schlein, Multivariate central limit theorem in quantum dynamics, J. Stat. Phys. 154 (2014), 113-152. MR3162535
C. Cacciapuoti, D. Finco and D. Noja, Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph, Phys. Rev. E (3) 91 (2015), no. 1, 013206. MR3416696
V. Caudrelier, On the inverse scattering method for integrable PDEs on a star graph, Commun. Math. Phys. 338 (2015), no. 2, 893-917. MR3351062
T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys. 85 (1982), no. 4, 549-561. MR0677997
X. Chen and J. Holmer, Focusing quantum many-body dynamics: the rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation, Arch. Ration. Mech. Anal. 221 (2016), no. 2, 631-676. MR3488534
K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein Condensation in a Gas of Sodium Atoms, Phys. Rev. Lett. 75 (1995), no. 22, 3969-3973. DOI: https://doi.org/10.1103/PhysRevLett.75.3969
E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell and C. E. Wieman, Dynamics of collapsing and exploding Bose-Einstein condensates, Nature 412 (2001), 295-299. DOI: 10.1038/35085500
A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften (1925), 3-25.
A. Elgart, L. Erdős, B. Schlein and H.-T. Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Ration. Mech. Anal. 179 (2006), no. 2, 265-283. MR2209131
L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 (2007), 515-614. MR2276262
L. Erdős, B. Schlein and H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc. 22 (2009), no. 4, 1099-1156. MR2525781
L. Erdős, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math. (2) 172 (2010), no. 1, 291-370. MR2680421
L. Erdős and H.-T. Yau, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys. 5 (2001), no. 6, 1169-1205. MR1926667
L. Friedlander, Extremal properties of eigenvalues for a metric graph, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 199-211. MR2141695
S. Gilg, D. Pelinovsky and G. Schneider, Validity of the NLS approximation for periodic quantum graphs, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no.~6, Art.~63. MR3571159
J. Ginibre and G. Velo, The classical field limit of scattering theory for nonrelativistic many-boson systems, I and II, Comm. Math. Phys. 66 (1979), no. 1, 37-76, and, Comm. Math. Phys. 68 (1979), no. 1, 45-68. MR0530915 MR0539736
S. Gnutzmann and D. Waltner, Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory, Phys. Rev. E 93 (2016), no. 3, 032204. DOI: https://doi.org/10.1103/PhysRevE.93.032204
M. Grillakis, M. Machedon and D. Margetis, Second-order corrections to mean field evolution of weakly interacting bosons, I, Commun. Math. Phys. 294 (2010), no. 1, 273-301. MR2575484
M. Grillakis, M. Machedon and D. Margetis, Second-order corrections to mean field evolution of weakly interacting bosons, II, Adv. Math. 228 (2011), no. 3, 1788-1815. MR2824569
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74 (1987), no. 1, 160-197. MR0901236
K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. 35 (1974), 265-277. MR0332046
W. Ketterle and N. J. van Druten, Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions, Phys. Rev. A 54 (1996), no. 1, 656-660. DOI: https://doi.org/10.1103/PhysRevA.54.656
A. Knowles and P. Pickl, Mean-field dynamics: singular potentials and rate of convergence, Comm. Math. Phys. 298 (2010), no. 1, 101-138. MR2657816
V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A 32 (1999), no. 4, 595-630. MR1671833
M. Lewin, P. T. Nam and B. Schlein, Fluctuations around Hartree states in the mean-field regime, Amer. J. Math. 137 (2015), no. 6, 1613-1650. MR3432269
E. H. Lieb and R. Seiringer, Proof of Bose-Einstein condensation for dilute trapped gases, Phys. Rev. Lett. 88 (2002), no. 17, 170409. DOI: https://doi.org/10.1103/PhysRevLett.88.170409
E. H. Lieb, R. Seiringer and J. Yngvason, Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A 61 (2000), no. 4, 043602. DOI: https://doi.org/10.1103/PhysRevA.61.043602
E. H. Lieb and J. Yngvason, Ground state energy of the low density Bose gas, Phys. Rev. Lett. 80 (1998), no. 12, 2504-2507. DOI: https://doi.org/10.1103/PhysRevLett.80.2504
M. Lorenzo, M. Lucci, V. Merlo, I. Ottaviani, M. Salvato, M. Cirillo, F. Müller, T. Weimann, M. G. Castellano, F. Chiarello and G.~Torrioli, On Bose-Einstein condensation in Josephson junctions star graph arrays, Phys. Lett. A 378 (7-8) (2014), 655-658. DOI: https://doi.org/10.1016/j.physleta.2013.12.032
J. L. Marzuola and D. Pelinovsky, Ground state on the dumbbell graph, Appl. Math. Res. Express. AMRX 2016 (2016), no. 1, 98-145. MR3483843
D. Noja, D. Pelinovsky and G. Shaikhova, Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph, Nonlinearity 28 (2015), no. 7, 2343-2378. MR3366647
D. Pelinovsky and G. Schneider, Bifurcations of standing localized waves on periodic graphs, Ann. Henri Poincaré 18 (2017), no. 4, 1185-1211. MR3626301
L. Pitaevskii and S. Stringari, Bose-Einstein condensation, Internat. Ser. Monogr. Phys., 116, Clarendon Press, Oxford, 2003. MR2012737
I. Rodnianski and B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Comm. Math. Phys. 291 (2009), no. 1, 31-61. MR2530155
K. Ruedenberg and C. W. Scherr, Free-electron network model for conjugated systems, I, Theory, J. Chem. Phys. 21 (1953), 1565-1581. DOI: http://dx.doi.org/10.1063/1.1699299
K. K. Sabirov, Z. A. Sobirov, D. Babajanov and D. U. Matrasulov, Stationary nonlinear Schrödinger equation on simplest graphs, Phys. Lett. A 377 (2013), no. 12, 860-865. MR3028326
R. Seiringer and J. Yin, The Lieb-Liniger model as a limit of dilute bosons in three dimensions, Comm. Math. Phys. 284 (2008), 459-479. MR2448137
E. Serra and L. Tentarelli, Bound states of the NLS equation on metric graphs with localized nonlinearities, J. Differential Equations 260 (2016), no.~7, 5627-5644. MR3456809
Z. Sobirov, D. Matrasulov, K. Sabirov, S. Sawada and K. Nakamura, Integrable nonlinear Schrödinger equation on simple networks: connection formula at vertices, Phys. Rev. E 81 (2010), no. 6, 066602. MR2736292
H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys. 52 (1980), no. 3, 569-615. MR0578142
L. Tentarelli, NLS ground states on metric graphs with localized nonlinearities, J. Math. Anal. Appl. 433 (2016), no. 1, 291-304. MR3388792
V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34 (1972), no. 1, 62-69. MR0406174

Home Riv.Mat.Univ.Parma