Skip to main content
SpringerLink
Log in

Quasi-periodic Solutions for a Class of Higher Dimensional Beam Equation with Quasi-periodic Forcing

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

This work focuses on higher-dimensional quasi-periodically forced nonlinear beam equation. This means studying

$$\begin{aligned} u_{tt} + ( -\Delta +M_\xi )^2u +\varepsilon \phi (t) ( u+u^3 ) =0, \quad x\in \mathbf {R}^d, t\in \mathbf {R} \end{aligned}$$

with periodic boundary conditions, where \(\varepsilon \) is a small positive parameter, \(\phi (t)\) is a real analytic quasi-periodic function in t with frequency vector \(\omega =(\omega _1,\omega _2,\ldots ,\omega _m).\) It is proved that there are many quasi-periodic solutions for the above equation via KAM theory.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bambusi, M., Graffi, S.: Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods. Commun. Math. Phys. 219, 465–480 (2001)

    Article  MATH  Google Scholar 

  2. Berti, M., Bolle, P.: Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity 25, 2579–2613 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berti, M., Bolle, P.: Quasi-periodic solutions with Sobolev regularity of NLS on \(\mathbb{T}^d\) with a multiplicative potential. Eur. J. Math. 15, 229–286 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bourgain, J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. Int. Math. Res. Not. 11, 475–497 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bourgain, J.: Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom. Funct. Anal. 5, 629–639 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. 148, 363–439 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bourgain, J.: Nonlinear Schrödinger Equations, Park City Series, vol. 5. American Mathematical Society, Providence (1999)

    Google Scholar 

  8. Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies, vol. 158. Princeton University Press, Princeton (2005)

    Google Scholar 

  9. Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations. Commun. Pure Appl. Math. 46, 1409–1498 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Eliasson, L.H., Kuksin, S.B.: KAM for the nonlinear Schrödinger equation. Ann. Math. 172, 371–435 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Geng, J., Xu, X., You, J.: An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation. Adv. Math. 226, 5361–5402 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Geng, J., Yi, Y.: Quasi-periodic solutions in a nonlinear Schrödinger equation. J. Differ. Equ. 233, 512–542 (2007)

    Article  MATH  Google Scholar 

  13. Geng, J., You, J.: A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions. J. Differ. Equ. 209, 1–56 (2005)

    Article  MATH  Google Scholar 

  14. Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun. Math. Phys. 262, 343–372 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Geng, J., You, J.: KAM tori for higher dimensional beam equations with constant potentials. Nonlinearity 19, 2405–2423 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kuksin, S.B.: Nearly Integrable Infinite-Dimensional Hamiltonian Systems. Lecture Notes in Mathematics, vol. 1556. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  17. Kuksin, S.B., Pöschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 143, 149–179 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Liang, Z., You, J.: Quasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity. SIAM J. Math. Anal. 36, 1965–1990 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Pöschel, J.: A KAM-theorem for some nonlinear partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23, 119–148 (1996)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Shi, Y., Xu, J., Xu, X.: On quasi-periodic solutions for a generalized Boussinesq equation. Nonlinear Anal. 105, 50–61 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  22. Shi, Y., Xu, J., Xu, X.: Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete Contin. Dyn. Syst. B 22, 2501–2519 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  23. Shi, Y., Lu, X., Xu, X.: Quasi-periodic solutions for Schrödinger equation with derivative nonlinearity. Dyn. Syst. 30, 158–188 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  24. Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127, 479–528 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  25. Xu, J., You, J.: Persistence of lower-dimensional tori under the first Melnikov’s non-resnonce condition. J. Math. Pures Appl. 80, 1045–1067 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Yuan, X.: Quasi-periodic solutions of completely resonant nonlinear wave equations. J. Differ. Equ. 230, 213–274 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Zhang, M., Si, J.: Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing. Physica D 238, 2185–2215 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Zhang, M.: Quasi-periodic solutions of two dimensional Schrödinger equations with quasi-periodic forcing. Nonlinear Anal. 135, 1–34 (2016)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their invaluable comments and suggestions which help to improve the presentation of this paper.

Author information

Authors and Affiliations

  1. College of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, 224051, People’s Republic of China

    Yanling Shi

  2. Department of Mathematics, Southeast University, Nanjing, 211189, People’s Republic of China

    Junxiang Xu & Xindong Xu

Authors
  1. Yanling Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junxiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xindong Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanling Shi.

Additional information

The first author is partially supported by NSFJS Grant (BK 20170472). The second author is supported by the NSFC Grant (11371090). The third author is supported by NSFC Grant (11771077).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shi, Y., Xu, J. & Xu, X. Quasi-periodic Solutions for a Class of Higher Dimensional Beam Equation with Quasi-periodic Forcing. J Dyn Diff Equat 31, 745–763 (2019). https://doi.org/10.1007/s10884-018-9657-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-018-9657-z

Keywords

Mathematics Subject Classification

Search

Navigation