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KAM tori for higher dimensional beam equation with a fixed constant potential

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Abstract

In this paper, we consider the higher dimensional nonlinear beam equation: u tt + Δ2 u + σu + f(u) = 0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u) = u 3 + h.o.t near u = 0 and σ is a positive constant. It is proved that for any fixed σ > 0, the above equation admits a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.

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References

  1. Geng J, You J. KAM tori of Hamiltonian perturbations of 1D linear beam equations. J Math Anal Appl, 277: 104–121 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Xu J, Qiu Q, You J. A KAM theorem of degenerate infinite dimensional Hamiltonian systems (I, II). Sci China Ser A, 39: 372–383, 384-394 (1996)

    MATH  MathSciNet  Google Scholar 

  5. Whitney H. Analytical extensions of differentiable functions defined on closed set. Trans Amer Math Sci, 36: 63–89 (1934)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Chierchia L, You J. KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Commun Math Phys, 211: 498–525 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Institute of Mathematical Science, Nanjing University, Nanjing, 210093, China

    XinDong Xu & JianSheng Geng

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  1. XinDong Xu
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  2. JianSheng Geng
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Correspondence to JianSheng Geng.

Additional information

This work was supported by National Natural Science Foundation of China (Grant Nos. 10531050, 10771098), the Major State Basic Research Development of China and the Natural Science Foundation of Jiangsu Province (Grant No. BK2007134)

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Xu, X., Geng, J. KAM tori for higher dimensional beam equation with a fixed constant potential. Sci. China Ser. A-Math. 52, 2007–2018 (2009). https://doi.org/10.1007/s11425-008-0158-0

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  • DOI: https://doi.org/10.1007/s11425-008-0158-0

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