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

  • Yanling ShiEmail author
  • Junxiang Xu
  • Xindong Xu


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.


Beam equation Quasi-periodic solution Infinite dimensional KAM theory 

Mathematics Subject Classification

37K50 58E05 



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


  1. 1.
    Bambusi, M., Graffi, S.: Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods. Commun. Math. Phys. 219, 465–480 (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Berti, M., Bolle, P.: Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity 25, 2579–2613 (2012)CrossRefzbMATHGoogle Scholar
  3. 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)CrossRefzbMATHGoogle Scholar
  4. 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)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bourgain, J.: Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom. Funct. Anal. 5, 629–639 (1995)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. 148, 363–439 (1998)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bourgain, J.: Nonlinear Schrödinger Equations, Park City Series, vol. 5. American Mathematical Society, Providence (1999)Google Scholar
  8. 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. 9.
    Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations. Commun. Pure Appl. Math. 46, 1409–1498 (1993)CrossRefzbMATHGoogle Scholar
  10. 10.
    Eliasson, L.H., Kuksin, S.B.: KAM for the nonlinear Schrödinger equation. Ann. Math. 172, 371–435 (2010)CrossRefzbMATHGoogle Scholar
  11. 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)CrossRefzbMATHGoogle Scholar
  12. 12.
    Geng, J., Yi, Y.: Quasi-periodic solutions in a nonlinear Schrödinger equation. J. Differ. Equ. 233, 512–542 (2007)CrossRefzbMATHGoogle Scholar
  13. 13.
    Geng, J., You, J.: A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions. J. Differ. Equ. 209, 1–56 (2005)CrossRefzbMATHGoogle Scholar
  14. 14.
    Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun. Math. Phys. 262, 343–372 (2006)CrossRefzbMATHGoogle Scholar
  15. 15.
    Geng, J., You, J.: KAM tori for higher dimensional beam equations with constant potentials. Nonlinearity 19, 2405–2423 (2006)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kuksin, S.B.: Nearly Integrable Infinite-Dimensional Hamiltonian Systems. Lecture Notes in Mathematics, vol. 1556. Springer, Berlin (1993)zbMATHGoogle Scholar
  17. 17.
    Kuksin, S.B., Pöschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 143, 149–179 (1996)CrossRefzbMATHGoogle Scholar
  18. 18.
    Liang, Z., You, J.: Quasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity. SIAM J. Math. Anal. 36, 1965–1990 (2005)CrossRefzbMATHGoogle Scholar
  19. 19.
    Pöschel, J.: A KAM-theorem for some nonlinear partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23, 119–148 (1996)zbMATHGoogle Scholar
  20. 20.
    Pöschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helv. 71, 269–296 (1996)CrossRefzbMATHGoogle Scholar
  21. 21.
    Shi, Y., Xu, J., Xu, X.: On quasi-periodic solutions for a generalized Boussinesq equation. Nonlinear Anal. 105, 50–61 (2014)CrossRefzbMATHGoogle Scholar
  22. 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)CrossRefzbMATHGoogle Scholar
  23. 23.
    Shi, Y., Lu, X., Xu, X.: Quasi-periodic solutions for Schrödinger equation with derivative nonlinearity. Dyn. Syst. 30, 158–188 (2015)CrossRefzbMATHGoogle Scholar
  24. 24.
    Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127, 479–528 (1990)CrossRefzbMATHGoogle Scholar
  25. 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)CrossRefzbMATHGoogle Scholar
  26. 26.
    Yuan, X.: Quasi-periodic solutions of completely resonant nonlinear wave equations. J. Differ. Equ. 230, 213–274 (2006)CrossRefzbMATHGoogle Scholar
  27. 27.
    Zhang, M., Si, J.: Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing. Physica D 238, 2185–2215 (2009)CrossRefzbMATHGoogle Scholar
  28. 28.
    Zhang, M.: Quasi-periodic solutions of two dimensional Schrödinger equations with quasi-periodic forcing. Nonlinear Anal. 135, 1–34 (2016)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and PhysicsYancheng Institute of TechnologyYanchengPeople’s Republic of China
  2. 2.Department of MathematicsSoutheast UniversityNanjingPeople’s Republic of China

Personalised recommendations