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

  • Zhaowei Lou
  • Jianguo Si
Article

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.

Keywords

Invariant tori Quasi-periodic solutions Quasi-periodically forced Schrödinger equations Birkhoff normal form Reversible systems KAM theorem 

Mathematics Subject Classification

37K55 35Q41 35B15 

References

  1. 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)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Baldi, P., Berti, M., Montalto, R.: KAM for quasi-linear KdV. C. R. Math. Acad. Sci. Paris 352(7–8), 603–607 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Berti, M., Biasco, L., Procesi, M.: KAM for reversible derivative wave equations. Arch. Ration. Mech. Anal. 212(3), 905–955 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 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.)CrossRefGoogle Scholar
  6. 6.
    Feola, R., Procesi, M.: Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations. J. Differ. Equ. 259(7), 3389–3447 (2015)CrossRefMATHGoogle Scholar
  7. 7.
    Geng, J., Wu, J.: Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations. J. Math. Phys. 53(10), 102702,27 (2012)CrossRefGoogle Scholar
  8. 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. 9.
    Kuksin, S.: On small-denominators equations with large variable coefficients. Z. Angew. Math. Phys. 48(2), 262–271 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kuksin, S.: A KAM-theorem for equations of the Korteweg-de Vries type. Rev. Math. Math. Phys 10(3), 1–64 (1998)MathSciNetMATHGoogle Scholar
  11. 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)MATHGoogle Scholar
  12. 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)MathSciNetMATHGoogle Scholar
  13. 13.
    Liu, J., Yuan, X.: A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations. Commun. Math. Phys. 307(3), 629–673 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Liu, J., Yuan, X.: KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions. J. Differ. Equ. 256(4), 1627–1652 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 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. 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)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pöschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helv. 71(2), 269–296 (1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 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)Google Scholar
  19. 19.
    Sevryuk, M.: Reversible Systems. Lecture Notes in Mathematics, vol. 1211. Springer, Berlin (1986)MATHGoogle Scholar
  20. 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)Google Scholar
  21. 21.
    Zhang, J., Gao, M., Yuan, X.: KAM tori for reversible partial differential equations. Nonlinearity 24(4), 1189–1228 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanPeople’s Republic of China

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