Bifurcating standing waves for effective equations in gapped honeycomb structures
https://doi.org/10.17586/2220-8054-2021-12-1-5-14
Abstract
In this paper, we deal with two-dimensional cubic Dirac equations, appearing as an effective model in gapped honeycomb structures. We give a formal derivation starting from cubic Schrodinger equations and prove the existence of standing waves bifurcating from one band-edge of the linear spectrum.
About the Authors
W. Borrelli
Centro De Giorgi, Scuola Normale Superiore
Italy
Italy
Piazza dei Cavalieri 3, I-56100, Pisa
R. Carlone
Universita “Federico II” di Napoli, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”
Italy
Italy
MSA, via Cinthia, I-80126, Napoli
References
1. Fefferman C.L., Weinstein M.I. Honeycomb lattice potentials and dirac points. J. Amer. Math. Soc., 2012, 25, P. 1169-1220.
2. Fefferman C.L., Weinstein M.I. Wave packets in honeycomb structures and two-dimensional Dirac equations. Comm. Math. Phys., 2014, 326, P. 251-286.
3. Erdös L., Schlein B., Yau H.-T. Derivation of the cubic non-linear Schrodinger equation from quantum dynamics of many-body systems. Invent. Math., 2007, 167, P. 515-614.
4. Moloney J., Newell A. Nonlinear optics. Westview Press, Advanced Book Program, Boulder, CO, 2004.
5. Pitaevskii L., Stringari S. Bose-Einstein condensation. International Series of Monographs on Physics, The Clarendon Press, Oxford University Press, Oxford, 2003,116.
6. Fefferman C.L., Weinstein M.I. Waves in honeycomb structures. Journ 'ees 'equations aux d 'eriv 'eespartielles, 2012, 12, 12 p.
7. Arbunich J., Sparber C. Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures. J. Math. Phys., 2018, 59,011509.
8. Borrelli W. Weakly localized states for nonlinear Dirac equations. Calc. Var. Partial Differential Equations, 2018, 57, P. 57-155.
9. Borrelli W., Frank R.L. Sharp decay estimates for critical Dirac equations. Trans. Amer. Math. Soc., 2020, 373, P. 2045-2070.
10. Borrelli W. Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity. J. Differential Equations, 2017, 263, P. 7941-7964.
11. Borrelli W. Symmetric solutions for a 2D critical Dirac equation. ArXiv e-prints, 2020, arXiv:2010.04630.
12. Ammann B., Grosjean J.-F., Humbert E., Morel B. A spinorial analogue of Aubin's inequality. Math. Z., 2008, 260, P. 127-151.
13. Grosse N. On a conformal invariant of the Dirac operator on noncompact manifolds. Ann. Global Anal. Geom., 2006, 30, P. 407-416.
14. Isobe T. Nonlinear Dirac equations with critical nonlinearities on compact Spin manifolds. J. Funct. Anal., 2011, 260, P. 253-307.
15. Maalaoui A. Infinitely many solutions for the spinorial Yamabe problem on the round sphere. Nonlinear Differential Equations Appl. NoDEA, 2016, 23, 25.
16. Borrelli W., Maalaoui A. Some properties of Dirac-Einstein bubbles. J. Geometric Analysis, 2020, https://doi.org/10.1007/s12220-020-00503-1.
17. Maalaoui A., Martino V. Characterization of the Palais-Smale sequences for the conformal Dirac-Einstein problem and applications. J. Differential Equations, 2019, 266, P. 2493-2541.
18. Ounaies H. Perturbation method for a class of nonlinear Dirac equations. Differential Integral Equations, 2000, 13, P. 707-720.
19. Borrelli W., Carlone R., Tentarelli L. On the nonlinear Dirac equation on noncompact metric graphs. Journal of Differential Equations, 2021, 278, P. 326-357.
20. Cazenave T., V'azquez L. Existence of localized solutions for a classical nonlinear Dirac field, Comm. Math. Phys., 1986,105, P. 35-47.
21. Cuevas-Maraver J., Kevrekidis P.G., et al. Stability of solitary waves and vortices in a 2D nonlinear Dirac model. Phys. Rev. Lett., 2016, 116, 214101.
22. Esteban M.J., S 'er'e E. Stationary states of the nonlinear Dirac equation: a variational approach. Comm. Math. Phys., 1995,171, P. 323-350.
23. Thaller B. The Dirac equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992.
24. Borrelli W. L' 'equation de Dirac en physique du solide et en optique non lin'eaire. PhD thesis, Universit'e Paris-Dauphine PSL, (2018).
25. Peleg O., Bartal G., et al. Conical diffraction and gap solitons in honeycomb photonic lattices. Phys. Rev. Lett., 2007, 98, 103901.
26. Reed M., Simon B. Methods of modern mathematical physics. IV Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.
27. Ilan B., Weinstein M.I. Band-edge solitons, nonlinear Schrodinger/Gross-Pitaevskii equations, and effective media. Multiscale Model. Simul., 2010, 8, P. 1055-1101.
28. Cazenave T. Semilinear Schrodinger equations. Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003, 10,
29. Chang S.-M., Gustafson S., Nakanishi K., Tsai T.-P. Spectra of linearized operators for NLS solitary waves. SIAM J. Math. Anal., 2007-2008, 39, P. 1070-1111.
30. Brezis H. Functional analysis, Sobolev spaces and partial differential equations. Universitat, Springer, New York, 2011.
Review
For citations:
Borrelli W., Carlone R. Bifurcating standing waves for effective equations in gapped honeycomb structures. Nanosystems: Physics, Chemistry, Mathematics. 2021;12(1):5-14. https://doi.org/10.17586/2220-8054-2021-12-1-5-14
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