SpringerLink
Log in

Breather solutions in conservative and dissipative nonlinear Klein–Gordon lattices

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

We study time-periodic and spatially localised solutions (breathers) in general infinite conservative and dissipative nonlinear Klein–Gordon lattices. First, in the time-reversible (conservative) case, we give a concise proof of the existence of breathers not using the concept of the anticontinuous limit. The existence problem is converted into an operator equation for time-reversal initial conditions generating breather solutions. A nontrivial solution of this operator equation is established facilitating Schauder’s fixed point theorem. Afterwards, we prove the existence and uniqueness of breather solutions in damped and forced infinite nonlinear Klein–Gordon lattice systems utilising the contraction map** principle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Availability of data and materials

Not applicable.

References

  1. Flach, S., Gorbach, A.: Discrete breathers: advances in theory and applications. Phys. Rep. 467, 1 (2008)

    Article  Google Scholar 

  2. Cuevas, J., English, L.Q., Kevrekidis, P.G., Anderson, M.: Discrete breathers in a forced-damped array of coupled pendula: modeling, computation, and experiment. Phys. Rev. Lett. 102, 224101 (2009)

    Article  Google Scholar 

  3. Palmero, F., English, L.Q., Cuevas-Maraver, J., Carretero-González, R., Kevrekidis, P.G.: Nonlinear edge modes in a honeycomb electrical lattice near the Dirac points. Phys. Rev. E 82, 026605 (2011)

    Article  Google Scholar 

  4. English, L.Q., Palmero, F., Stormes, J.F., Cuevas-Maraver, J., Carretero-González, R., Kevrekidis, P.G.: Nonlinear localized modes in two-dimensional electrical lattices. Phys. Rev. E 88, 022912 (2013)

    Article  Google Scholar 

  5. James, G., Kevrekidis, P.G., Cuevas, J.: Breathers in oscillator chains with Hertzian interactions. Physica D 251, 39 (2013)

    Article  MathSciNet  Google Scholar 

  6. James, G., Sánchez-Rey, B.: Breathers in inhomogeneous lattices: an analysis via centre manifold reduction. J. Cuevas, Rev. Math. Phys. 21, 1 (2009)

    Article  Google Scholar 

  7. Koukouloyannis, V., MacKay, R.S.: Existence and stability of 3-site breathers in a triangular lattice. J. Phys. A: Math. Gen. 38, 1021 (2005)

    Article  MathSciNet  Google Scholar 

  8. Koukouloyannis, V., Kevrekidis, P.G., Cuevas, J., Rothos, V.: Multibreathers in Klein-Gordon chains with interactions beyond nearest neighbors. Physica D 242, 16 (2013)

    Article  Google Scholar 

  9. Cuevas-Maraver, J., Kevrekidis, P.G., Pelinovsky, D.E.: Stability of discrete breathers. Stud. Appl. Math. 137, 214 (2016)

    MathSciNet  Google Scholar 

  10. Kevrekidis, P.G., Cuevas-Maraver, J., Pelinovsky, D.E.: Energy criterion for the spectral stability of discrete breathers. Phys. Rev. Lett. 119, 094101 (2016)

    Article  Google Scholar 

  11. Xu, H., Cuevas-Maraver, J., Kevrekidis, P.G., Vainchtein, A.: An energy-based stability criterion for solitary traveling waves in Hamiltonian lattices. Phil. Trans. Roy. Soc. A 376, 2017092 (2018)

    Google Scholar 

  12. Kevrekidis, P.G., Pelinovsky, D.E., Saxena, A.: When linear stability does not exclude nonlinear instability. Phys. Rev. Lett. 114, 214101 (2015)

    Article  Google Scholar 

  13. Kastner, M., Sepulchre, J.-A.: Effective Hamiltonian for traveling discrete breathers in the FPU chain. Discret. Contin. Dyn. Syst. Ser. B 5, 719 (2005)

    MathSciNet  Google Scholar 

  14. Pelinovsky, D.E., Sakovich, A.: Nonlinearity multi-site breathers in Klein-Gordon lattices: stability, resonances and bifurcations. Nonlinearity 25, 3423 (2012)

    Article  MathSciNet  Google Scholar 

  15. MacKay, R.S., Aubry, S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623 (1994)

    Article  MathSciNet  Google Scholar 

  16. Aubry, S.: Breathers in nonlinear lattices: existence, linear stability and quantization. Physica D 103, 201 (1998)

    Article  MathSciNet  Google Scholar 

  17. Aubry, S., Kopidakis, G., Kadelburg, V.: Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems. Discret. Contin. Dyn. Syst. B 1, 271 (2001)

    MathSciNet  Google Scholar 

  18. James, G.: Centre manifold reduction for quasilinear discrete systems. J. Nonlinear Sci. 13, 27 (2003)

    Article  MathSciNet  Google Scholar 

  19. James, G., Sire, Y.: Travelling breathers with exponentially small tails in a chain of nonlinear oscillators. Commun. Math. Phys. 257, 51 (2005)

    Article  MathSciNet  Google Scholar 

  20. James, G., Pelinovsky, D.: Breather continuation from infinity in nonlinear oscillator chains. Discrete Contin. Dyn. Syst. A 32, 1775 (2012)

    Article  MathSciNet  Google Scholar 

  21. James, G.: Existence of breathers on FPU lattices. Acad. Sci. Paris 332, 581 (2001)

    Article  MathSciNet  Google Scholar 

  22. Sepulchre, J.A., MacKay, R.S.: Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators. Nonlinearity 10, 679 (1997)

    Article  MathSciNet  Google Scholar 

  23. MacKay, R.S.: Discrete breathers: classical and quantum. Phys. A 288, 174 (2000)

    Article  MathSciNet  Google Scholar 

  24. Hennig, D., Karachalios, N.I.: Existence of exponentially localized solutions for lattices of nonlinearly coupled particles: Schauder’s fixed point theorem approach. J. Math. Phys. 62, 123506 (2021)

    Article  MathSciNet  Google Scholar 

  25. Hennig, D., Karachalios, N.I.: Existence of exponentially and superexponentially spatially locaized breather solutions for nonlinear Klein-Gordon latticed in \({\mathbb{Z} }^d\), \(d\ge 1\). Edinburgh Math. Soc. 65, 480 (2022)

    Article  Google Scholar 

  26. Bambusi, D.: Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators. Nonlinearity 9, 433 (1996)

    Article  MathSciNet  Google Scholar 

  27. Hennig, D.: Existence of breathing patterns in globally coupled finite-size nonlinear lattices. Appl. Anal. 98, 2511 (2019)

    Article  MathSciNet  Google Scholar 

  28. Schauder, E.: Der Fixpunktsatz in Funktionsräumen. Stud. Math. 2, 171 (1930)

    Article  Google Scholar 

  29. Smart, D.R.: Fixed Point Theorems (Cambridge Univ. Press, (1973))

  30. DeVito, C.: Functional analysis and linear operator theory. Addison-Wesley, Reading, MA (1990)

    Google Scholar 

  31. Bocarra, N.: Functional analysis. Academic Press, New York (1990)

    Google Scholar 

  32. Zeidler, E.: Applied functional analysis. Springer-Verlag, New York (1995)

    Book  Google Scholar 

  33. Deimling, K.: Nonlinear functional analysis. Dover Publications, Mineola NY (2010)

    Google Scholar 

  34. Teman, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, Berlin (1997)

    Book  Google Scholar 

Download references

Acknowledgements

I am very grateful to the referee.

Funding

There are no fundings.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Thessaly, GR35100, Lamia, Greece

    Dirk Hennig

Authors
  1. Dirk Hennig
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

As I am the sole author of the manuscript, the entire manuscript was written by me.

Corresponding author

Correspondence to Dirk Hennig.

Ethics declarations

Conflict of interest

There are no conflicts of interest.

Ethical approval and consent to participate

I approve and consent.

Consent for publication

I consent.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hennig, D. Breather solutions in conservative and dissipative nonlinear Klein–Gordon lattices. J. Fixed Point Theory Appl. 26, 18 (2024). https://doi.org/10.1007/s11784-024-01106-x

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11784-024-01106-x

Mathematics Subject Classification

Search

Navigation