Advertisement

Annales Henri Poincaré

, Volume 18, Issue 4, pp 1185–1211 | Cite as

Bifurcations of Standing Localized Waves on Periodic Graphs

  • Dmitry PelinovskyEmail author
  • Guido Schneider
Article

Abstract

The nonlinear Schrödinger (NLS) equation is considered on a periodic graph subject to the Kirchhoff boundary conditions. Bifurcations of standing localized waves for frequencies lying below the bottom of the linear spectrum of the associated stationary Schrödinger equation are considered by using analysis of two-dimensional discrete maps near hyperbolic fixed points. We prove the existence of two distinct families of small-amplitude standing localized waves, which are symmetric about the two symmetry points of the periodic graph. We also prove properties of the two families, in particular, positivity and exponential decay. The asymptotic reduction of the two-dimensional discrete map to the stationary NLS equation on an infinite line is discussed in the context of the homogenization of the NLS equation on the periodic graph.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ablowitz, M.J., Curtis, C.W., Zhu, Y.: On tight-binding approximations in optical lattices. Stud. Appl. Math. 129, 362–388 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: Constrained energy minimization and orbital stability for the NLS equation on a star graph. Ann. Inst. H. Poincaré AN 31, 1289–1310 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: Variational properties and orbital stability of standing waves for NLS equation on a star graph. J. Differ. Equ. 257, 3738–3777 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Adami, R., Serra, E., Tilli, P.: NLS ground states on graphs. Calc. Var. PDEs 54, 743–761 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Adami, R., Serra, E., Tilli, P.: Threshold phenomena and existence results for NLS ground state on graphs. J. Funct. Anal. 271, 201–223 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Banica, V., Ignat, L.I.: Dispersion for the Schrödinger equation on the line with multiple Dirac delta potentials and on delta trees. Anal. PDE 7, 903–927 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Mathematical Surveys and Monographs, vol. 186. AMS, Providence (2013)Google Scholar
  8. 8.
    Busch, K., Schneider, G., Tkeshelashvili, L., Uecker, H.: Justification of the nonlinear Schrödinger equation in spatially periodic media. Z. Angew. Math. Phys. 57, 905–939 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cacciapuoti, C., Finco, D., Noja, D.: Topology induced bifurcations for the NLS on the tadpole graph. Phys. Rev. E 91, 013206 (2015)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dohnal, T., Pelinovsky, D., Schneider, G.: Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential. J. Nonlinear Sci. 19, 95–131 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gilg, S., Pelinovsky, D., Schneider, G.: Validity of the NLS approximation for periodic quantum graphs. Nonlinear Differ. Equ. Appl. 23, 63 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gnutzmann, S., Smilansky, U.: Quantum graphs: applications to quantum chaos and universal spectral statistics. Adv. Phys. 55, 527–625 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    Gnutzmann, S., Waltner, D.: Stationary waves on nonlinear quantum graphs: general framework and canonical perturbation theory. Phys. Rev. E 93, 032204 (2016)ADSCrossRefGoogle Scholar
  14. 14.
    Ilan, B., Weinstein, M.: Band-edge solitons, nonlinear Schrödinger (Gross–Pitaevskii) equations and effective media. SIAM J. Multiscale Model. Simul. 8, 1055–1101 (2010)CrossRefzbMATHGoogle Scholar
  15. 15.
    James, G., Sánchez-Rey, B., Cuevas, J.: Breathers in inhomogeneous nonlinear lattices: an analysis via center manifold reduction. Rev. Math. Phys. 21, 1–59 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Korotyaev, E., Lobanov, I.: Schrödinger operators on zigzag nanotubes. Ann. Henri Poincaré 8, 1151–1176 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kuchment, P., Post, O.: On the spectra of carbon nano-structures. Commun. Math. Phys. 275, 805–826 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lekner, J.: Reflectionless eigenstates of the \({\rm sech}^2\) potential. Am. J. Phys. 75, 1151–1157 (2007)ADSCrossRefGoogle Scholar
  19. 19.
    Marzuola, J.L., Pelinovsky, D.E.: Ground state on the dumbbell graph. Appl. Mat. Res. Express 2016, 98–145 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Niikuni, H.: Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph. Opusc. Math. 35, 199–234 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Noja, D.: Nonlinear Schrödinger equation on graphs: recent results and open problems. Phil. Trans. R. Soc. A 372, 20130002 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Noja, D., Pelinovsky, D., Shaikhova, G.: Bifurcations and stability of standing waves on tadpole graphs. Nonlinearity 28, 2343–2378 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pelinovsky, D.E.: Localization in Periodic Potentials: from Schrödinger Operators to the Gross–Pitaevskii equation. LMS Lecture Note Series, vol. 390. Cambridge University Press, Cambridge (2011)Google Scholar
  24. 24.
    Pelinovsky, D., Schneider, G., MacKay, R.: Justification of the lattice equation for a nonlinear problem with a periodic potential. Comm. Math. Phys. 284, 803–831 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pelinovsky, D., Schneider, G.: Bounds on the tight-binding approximation for the Gross–Pitaevskii equation with a periodic potentia. J. Diff. Eqs. 248, 837–849 (2010)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Qin, W.-X., Xiao, X.: Homoclinic orbits and localized solutions in nonlinear Schrödinger lattices. Nonlinearity 20, 2305–2317 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsMcMaster UniversityHamiltonCanada
  2. 2.Institut für Analysis, Dynamik und ModellierungUniversität StuttgartStuttgartGermany

Personalised recommendations