Quantum graphs and dimensional crossover: the honeycomb

Adami, Riccardo; Dovetta, Simone; Ruighi, Alice

doi:10.2478/caim-2019-0016

Abstract

We summarize features and results on the problem of the existence of Ground States for the Nonlinear Schrödinger Equation on doubly-periodic metric graphs. We extend the results known for the two–dimensional square grid graph to the honeycomb, made of infinitely-many identical hexagons. Specifically, we show how the coexistence between one–dimensional and two–dimensional scales in the graph structure leads to the emergence of threshold phenomena known as dimensional crossover.

References

1. K. Ruedenberg and C. W. Scherr, Free-electron network model for conjugated systems. I. Theory, The Journal of Chemical Physics, vol. 21, no. 9, pp. 1565–1581, 1953.10.1063/1.1699299
Search in Google Scholar Back to article
2. F. A. Mehmeti, Nonlinear Waves in Networks. Mathematical Research Series, Akademie Verlag, 1994.
Search in Google Scholar Back to article
3. R. Adami, C. Cacciapuoti, D. Finco, and D. Noja, Fast solitons on star graphs, Reviews in Mathematical Physics, vol. 23, no. 04, pp. 409–451, 2011.10.1142/S0129055X11004345
Search in Google Scholar Back to article
4. M. Keel and T. Tao, Endpoint Strichartz estimates, American Journal of Mathematics, vol. 120, no. 5, pp. 955–980, 1998.10.1353/ajm.1998.0039
Search in Google Scholar Back to article
5. C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing non-linear Schrödinger equation in the radial case, Inventiones mathematicae, vol. 166, no. 3, pp. 645– 675, 2006.10.1007/s00222-006-0011-4
Search in Google Scholar Back to article
6. S. N. Bose, Plancks gesetz und lichtquantenhypothese, Zeitschrift für Physik, vol. 26, pp. 178–181, Dec 1924.10.1007/BF01327326
Search in Google Scholar Back to article
7. A. Einstein, Quantentheorie des einatomigen idealen Gases. No. 2 in Sitzungsberichte der Preussischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse, Verlag d. Akad. d. Wiss., 1924.
Search in Google Scholar Back to article
8. T. Cazenave, Semilinear Schrödinger Equations. Courant lecture notes in mathematics, American Mathematical Society, 2003.10.1090/cln/010
Search in Google Scholar Back to article
9. T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Communications in Mathematical Physics, vol. 85, no. 4, pp. 549–561, 1982.10.1007/BF01403504
Search in Google Scholar Back to article
10. V. E. Zakharov and B. Shabat, Exact theory of two–dimensional self–focusing and one–dimensional self–modulation of waves in nonlinear media, Journal of Experimental and Theoretical Physics, vol. 34, no. 1, pp. 62–71, 1972.
Search in Google Scholar Back to article
11. R. Adami, C. Cacciapuoti, D. Finco, and D. Noja, On the structure of critical energy levels for the cubic focusing NLS on star graphs, Journal of Physics A: Mathematical and Theoretical, vol. 45, p. 192001, apr 2012.10.1088/1751-8113/45/19/192001
Search in Google Scholar Back to article
12. R. Adami, E. Serra, and P. Tilli, NLS ground states on graphs, Calculus of Variations and Partial Differential Equations, vol. 54, pp. 743–761, Sep 2015.10.1007/s00526-014-0804-z
Search in Google Scholar Back to article
13. S. Dovetta and L. Tentarelli, Ground states of the L²-critical NLS equation with localized nonlinearity on a tadpole graph, Operator Theory: Advances and Applications. to appear.
Search in Google Scholar Back to article
14. J. L. Marzuola and D. E. Pelinovsky, Ground state on the dumbbell graph, Applied Mathematics Research eXpress, vol. 2016, no. 1, pp. 98–145, 2016.10.1093/amrx/abv011
Search in Google Scholar Back to article
15. D. Noja, D. Pelinovsky, and G. Shaikhova, Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph, Nonlinearity, vol. 28, no. 7, pp. 2343–2378, 2015.10.1088/0951-7715/28/7/2343
Search in Google Scholar Back to article
16. C. Cacciapuoti, S. Dovetta, and E. Serra, Variational and stability properties of constant solutions to the NLS equation on compact metric graphs, Milan Journal of Mathematics, vol. 86, no. 2, pp. 305– 327, 2018.10.1007/s00032-018-0288-y
Search in Google Scholar Back to article
17. S. Dovetta, Existence of infinitely many stationary solutions of the L²–subcritical and critical NLSE on compact metric graphs, Journal of Differential Equations, vol. 264, no. 7, pp. 4806–4821, 2018.10.1016/j.jde.2017.12.025
Search in Google Scholar Back to article
18. S. Dovetta and L. Tentarelli, L²–critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features, Calculus of Variations and Partial Differential Equations, 2019. to appear.10.1007/s00526-019-1565-5
Search in Google Scholar Back to article
19. E. Serra and L. Tentarelli, Bound states of the NLS equation on metric graphs with localized nonlinearities, Journal of Differential Equations, vol. 260, no. 7, pp. 5627–5644, 2016.10.1016/j.jde.2015.12.030
Search in Google Scholar Back to article
20. E. Serra and L. Tentarelli, On the lack of bound states for certain NLS equations on metric graphs, Nonlinear Analysis: Theory, Methods and Applications, vol. 145, pp. 68–82, 2016.10.1016/j.na.2016.07.008
Search in Google Scholar Back to article
21. L. Tentarelli, NLS ground states on metric graphs with localized nonlinearities, Journal of Mathematical Analysis and Applications, vol. 433, no. 1, pp. 291–304, 2016.10.1016/j.jmaa.2015.07.065
Search in Google Scholar Back to article
22. R. Adami, E. Serra, and P. Tilli, Negative energy ground states for the L² -critical NLSE on metric graphs, Communications in Mathematical Physics, vol. 352, pp. 387–406, May 2017.10.1007/s00220-016-2797-2
Search in Google Scholar Back to article
23. W. Borrelli, R. Carlone, and L. Tentarelli, Nonlinear Dirac equation on graphs with localized non- linearities: Bound states and nonrelativistic limit, SIAM Journal on Mathematical Analysis, vol. 51, pp. 1046–1081, 01 2019.10.1137/18M1211714
Search in Google Scholar Back to article
24. W. Borrelli, R. Carlone, and L. Tentarelli, An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity, Symmetry, vol. 11, no. 2, 2019.10.3390/sym11020169
Search in Google Scholar Back to article
25. S. Gilg, D. Pelinovsky, and G. Schneider, Validity of the NLS approximation for periodic quantum graphs, Nonlinear Differential Equations and Applications NoDEA, vol. 63, no. 6, p. 30, 2016.10.1007/s00030-016-0417-7
Search in Google Scholar Back to article
26. A. Pankov, Nonlinear Schrödinger equations on periodic metric graphs, Discrete e Continuous Dynamical Systems, vol. 38, no. 2, pp. 697–714, 2018.10.3934/dcds.2018030
Search in Google Scholar Back to article
27. D. Pelinovsky and G. Schneider, Bifurcations of standing localized waves on periodic graphs, Annales Henri Poincaré, vol. 18, no. 4, pp. 1185–1211, 2017.10.1007/s00023-016-0536-z
Search in Google Scholar Back to article
28. S. Dovetta, Mass-constrained ground states of the stationary NLSE on periodic metric graphs, Non- linear Differential Equations and Applications, 2019. to appear.10.1007/s00030-019-0576-4
Search in Google Scholar Back to article
29. R. Adami, S. Dovetta, E. Serra, and P. Tilli, Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs, Analysis & PDE, vol. 12, pp. 1597–1612, 02 2019.10.2140/apde.2019.12.1597
Search in Google Scholar Back to article
30. R. Adami and S. Dovetta, One-dimensional versions of three-dimensional system: Ground states for the NLS on the spatial grid, Rendiconti di Matematica e delle sue Applicazioni, vol. 39, pp. 181–194, 2018.
Search in Google Scholar Back to article
31. R. Adami, E. Serra, and P. Tilli, Threshold phenomena and existence results for NLS ground states on metric graphs, Journal of Functional Analysis, vol. 271, no. 1, pp. 201 – 223, 2016.10.1016/j.jfa.2016.04.004
Search in Google Scholar Back to article

Quantum graphs and dimensional crossover: the honeycomb

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