KAM Tori for Higher Dimensional Quintic Beam Equation

  • Chuanfang Ge
  • Jiansheng GengEmail author


In this paper, we consider higher dimensional quintic beam equation
$$\begin{aligned} u_{tt} +\triangle ^2 u+u+u^5 =0, \end{aligned}$$
with periodic boundary conditions, it is proved that the above equation admits a family of small-amplitude linearly stable quasi-periodic solutions.


Beam equation KAM Tori Birkhoff normal form 

Mathematics Subject Classification

Primary 37K60 37K55 


  1. 1.
    Bambusi, D.: On long time stability in Hamiltonian perturbations of non-resonant linear PDEs. Nonlinearity 12, 823–850 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berti, M., Bolle, P.: Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity 25, 2579–2613 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berti, M., Bolle, P.: Quasi-periodic solutions with Sobolev regularity of NLS on \(\mathbb{T}^{d}\) and a multiplicative potential. J. Eur. Math. Soc. 15, 229–286 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourgain, J.: Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. 148, 363–439 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bourgain, J.: Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom. Funct. Anal. 5, 629–639 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourgain, J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. Int. Math. Res. Not. 1994, 475–497 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bourgain, J.: Green’s function estimates for lattice Schrödinger operators and applications. In: Annals of Mathematics Studies, vol. 158. Princeton University Press, Princeton (2005)Google Scholar
  8. 8.
    Bourgain, J.: Nonlinear Schrödinger Equations, Park City Series 5. American Mathematical Society, Providence (1999)Google Scholar
  9. 9.
    Bourgain, J.: On diffusion in high-dimensional Hamiltonian systems and PDE. J. Anal. Math. 80, 1–35 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bourgain, J., Wang, W.-M.: Quasi-periodic solutions of nonlinear random Schrödinger equations. J. Eur. Math. Soc. 10, 1–45 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chierchia, L., You, J.: KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Commun. Math. Phys. 211, 498–525 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations. Commun. Pure Appl. Math. 46, 1409–1498 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eliasson, L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Ann. Sc. Norm. Super. Pisa 15, 115–147 (1988)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Eliasson, L.H., Kuksin, S.B.: KAM for the nonlinear Schrödinger equation. Ann. Math. 172, 371–435 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Eliasson, L., Grebert, B., Kuksin, S.B.: KAM for the nonlinear beam equation. Geom. Funct. Anal. 26, 1588–1715 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Geng, J., You, J.: KAM tori of Hamiltonian perturbations of 1D linear beam equations. J. Math. Anal. Appl. 277, 104–121 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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
  19. 19.
    Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun. Math. Phys. 262, 343–372 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Geng, J., You, J.: KAM tori for higher dimensional beam equations with constant petentials. Nonlinearity 19, 2405–2523 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Geng, J., Yi, Y.: Quasi-periodic solutions in a nonlinear Schrödinger equation. J. Differ. Equ. 233, 512–542 (2007)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kappeler, T., Pöschel, J.: KdV & KAM. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  23. 23.
    Kuksin, S.B.: Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funct. Anal. Appl. 21, 192–205 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kuksin, S.B.: Nearly Integrable Infinite Dimensional Hamiltonian Systems. Lecture Notes in Mathematics, vol. 1556. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  25. 25.
    Kuksin, S.B., Pöschel, J.: Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 143, 149–179 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pöschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helv. 71, 269–296 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pöschel, J.: A KAM theorem for some nonlinear partial differential equations. Ann. Sc. Norm. Super. Pisa CI. Sci. 23, 119–148 (1996)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Pöschel, J.: On the construction of almost periodic solutions for a nonlinear Schrödinger equations. Ergod. Theory Dyn. Syst. 22, 1537–1549 (2002)CrossRefzbMATHGoogle Scholar
  29. 29.
    Procesi, M., Procesi, C.: A normal form for the Schrödinger equation with analytic nonlinearities. Commun. Math. Phys. 312, 501–557 (2012)CrossRefzbMATHGoogle Scholar
  30. 30.
    Procesi, C., Procesi, M.: A KAM algorithm for the resonant nonlinear Schrödinger equation. Adv. Math. 272, 399–470 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wang, W.-M.: Energy supercritical nonlinear Schrödinger equations: quasiperiodic solutions. Duke Math. J. 165, 1129–1192 (2016)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Wayne, C.E.: Periodic and quasi-periodic solutions for nonlinear wave equations via KAM theory. Commun. Math. Phys. 127, 479–528 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Whitney, H.: Analytical extensions of differentiable functions defined on closed set. Trans. Am. Math. Soc. 36, 63–89 (1934)CrossRefzbMATHGoogle Scholar
  34. 34.
    Xu, J., Qiu, Q., You, J.: A KAM theorem of degenerate infinite dimensional Hamiltonian systems (I). Sci. China Ser. A 39, 372–383 (1996)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Xu, J., Qiu, Q., You, J.: A KAM theorem of degenerate infinite dimensional Hamiltonian systems (II). Sci. China Ser. A 39, 384–394 (1996)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Xu, X., Geng, J.: KAM tori for higher dimensional beam equation with a fixed constand potential. Sci. China Ser. A 52, 2007–2018 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Yuan, X.: Quasi-periodic solutions of completely resonant nonlinear wave equations. J. Differ. Equ. 230, 213–274 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China

Personalised recommendations