Skip to main content
SpringerLink
Log in

Invariant tori for reversible nonlinear Schrödinger equations under quasi-periodic forcing

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, by infinite-dimensional reversible KAM (Kolmogorov–Arnold–Moser) theory, we prove the existence of invariant tori (thus quasi-periodic solutions) for a class of quasi-periodically forced reversible derivative nonlinear Schrödinger equations under periodic and Dirichlet boundary conditions. In the proof, we also use Birkhoff normal form techniques.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Baldi, P., Berti, M., Montalto, R.: KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation. Math. Ann. 359(1–2), 471–536 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baldi, P., Berti, M., Montalto, R.: KAM for quasi-linear KdV. C. R. Math. Acad. Sci. Paris 352(7–8), 603–607 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berti, M., Biasco, L., Procesi, M.: KAM theory for the Hamiltonian derivative wave equation. Ann. Sci. Éc. Norm. Supér. (4) 46(2), 301–373 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berti, M., Biasco, L., Procesi, M.: KAM for reversible derivative wave equations. Arch. Ration. Mech. Anal. 212(3), 905–955 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bogoljubov, N.N., Mitropoliskii, J.A., Samoĭlenko, A.M.: Methods of accelerated convergence in nonlinear mechanics. Hindustan Publishing Corp, Delhi (1976). (Translated from the Russian by Kumar, V. and edited by Sneddon, I. N.)

    Book  Google Scholar 

  6. Feola, R., Procesi, M.: Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations. J. Differ. Equ. 259(7), 3389–3447 (2015)

    Article  MATH  Google Scholar 

  7. Geng, J., Wu, J.: Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations. J. Math. Phys. 53(10), 102702,27 (2012)

    Article  Google Scholar 

  8. Kappeler, T.P., Pöschel, J.: KdV & KAM, volume 45 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics (Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics). Springer, Berlin (2003)

    Google Scholar 

  9. Kuksin, S.: On small-denominators equations with large variable coefficients. Z. Angew. Math. Phys. 48(2), 262–271 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kuksin, S.: A KAM-theorem for equations of the Korteweg-de Vries type. Rev. Math. Math. Phys 10(3), 1–64 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Liu, J., Si, J.: Invariant tori for a derivative nonlinear Schrödinger equation with quasi-periodic forcing. J. Math. Phys. 56(3), 032702, 25 (2015)

    MATH  Google Scholar 

  12. Liu, J., Si, J.: Invariant tori of a nonlinear Schrödinger equation with quasi-periodically unbounded perturbations. Commun. Pure Appl. Anal. 16(1), 25–68 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Liu, J., Yuan, X.: A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations. Commun. Math. Phys. 307(3), 629–673 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, J., Yuan, X.: KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions. J. Differ. Equ. 256(4), 1627–1652 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lou, Z., Si, J.: Quasi-periodic solutions for the reversible derivative nonlinear Schrödinger equations with periodic boundary conditions. J. Dyn. Differ. Equ. (2015). doi:10.1007/s10884-015-9481-7

    Google Scholar 

  16. Mi, L., Zhang, K.: Invariant tori for Benjamin-Ono equation with unbounded quasi-periodically forced perturbation. Discrete Contin. Dyn. Syst. 34(2), 689–707 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pöschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helv. 71(2), 269–296 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pöschel, J.: A lecture on the classical KAM theorem. In: Smooth ergodic theory and its applications (Seattle, WA, 1999), volume 69 of Proceedings of Symposia in Pure Mathematics, pp. 707–732. American Mathematical Society, Providence (2001)

  19. Sevryuk, M.: Reversible Systems. Lecture Notes in Mathematics, vol. 1211. Springer, Berlin (1986)

    MATH  Google Scholar 

  20. Sevryuk, M.: The finite-dimensional reversible KAM theory. Physica D 112(1–2):132–147 (1998). Time-reversal symmetry in dynamical systems (Coventry, 1996)

  21. Zhang, J., Gao, M., Yuan, X.: KAM tori for reversible partial differential equations. Nonlinearity 24(4), 1189–1228 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, People’s Republic of China

    Zhaowei Lou & Jianguo Si

Authors
  1. Zhaowei Lou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianguo Si
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhaowei Lou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lou, Z., Si, J. Invariant tori for reversible nonlinear Schrödinger equations under quasi-periodic forcing. Z. Angew. Math. Phys. 68, 101 (2017). https://doi.org/10.1007/s00033-017-0849-x

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-017-0849-x

Mathematics Subject Classification

Keywords

Search

Navigation