Skip to main content
SpringerLink
Log in

A Nekhoroshev Type Theorem for the Nonlinear Wave Equation in Gevrey Space

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

In this paper, the authors prove a Nekhoroshev type theorem for the nonlinear wave equation

$${u_{tt}} = {u_{xx}} - mu - f(u),\;\;\;\;\;x \in [0,\pi ]$$

in Gevrey space.

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, D., On long time stability in Hamiltonian perturbations of non-resonant linear PDEs, Nonlinearity, 12, 1999, 823–850.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bambusi, D., Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations, Math. Z., 230, 1999, 345–387.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bambusi, D., Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phy., 234, 2003, 253–285.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bambusi, D., A Birkhoff normal form theorem for some semilinear PDEs, Hamiltonian dynamical systems and applications, 2008, 213–247.

  5. Bambusi, D., Berti, M. and Magistrelli, E., Degenerate KAM theory for partial differential equations, J. Diff. Eqs., 250, 2011, 3379–3397.

    Article  MathSciNet  MATH  Google Scholar 

  6. Bambusi, D. and Grébert, B., Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., 135, 2006, 507–567.

    Article  MathSciNet  MATH  Google Scholar 

  7. Bambusi, D. and Nekhoroshev, N. N., A property of exponential stability in nonlinear wave equations near the fundamental linear mode, Phys. D., 122, 1998, 73–104.

    Article  MathSciNet  MATH  Google Scholar 

  8. Benettin, G., Galgani, L. and Giorgilli, A., A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems, Celestial Mech., 37, 1985, 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  9. Bounemoura, A., Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians, J. Diff. Eqs., 249, 2000, 2905–2920.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bounemoura, A., Normal forms, stability and splitting of invariant manifolds I, Gevrey Hamiltonians, Regular and Chaotic Dynamics, 18, 2013, 237–260.

    Article  MathSciNet  MATH  Google Scholar 

  11. Bourgain, J., Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Gemo. Funct. Anal., 6, 1996, 201–230.

    Article  MathSciNet  MATH  Google Scholar 

  12. Bourgain, J., Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations, Ergodic Theory and Dynamical Systems, 24(5), 2004, 1331–1357.

    Article  MathSciNet  MATH  Google Scholar 

  13. Cong, H., Gao, M. and Liu, J., Long time stability of KAM tori for nonlinear wave equation, J. Diff. Eqs., 258, 2015, 2823–2846.

    Article  MathSciNet  MATH  Google Scholar 

  14. Cong, H., Liu, J. and Yuan, X., Stability of KAM tori for nonlinear Schrödinger equation, Mem. Amer. Math. Soc., 239(1134), 2016, 1–85.

    MATH  Google Scholar 

  15. Faou, E. and Grébert, B., A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus, Analysis and PDE., 6, 2013, 1243–1262.

    Article  MathSciNet  MATH  Google Scholar 

  16. Kuksin, S. B., Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21, 1987, 192–205.

    Article  MathSciNet  MATH  Google Scholar 

  17. Kuksin, S. B., Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Math. USSR Izv., 32, 1989, 39–62.

    Article  MathSciNet  MATH  Google Scholar 

  18. Kuksin, S. B., Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Springer-Verlag, Berlin, 1993.

    Book  MATH  Google Scholar 

  19. Marco, J. P. and Sauzin, D., Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publications Mathématiques de l′IHÉS, 96, 2003, 199–275.

    Article  MATH  Google Scholar 

  20. Mi, L., Liu, C., Shi, G. and Zhao, R., A Nekhoroshev type theorem for the nonlinear wave equation, Pure and Applied Mathematics Quarterly, preprint.

  21. Popov, G., KAM theorem for Gevrey Hamiltonians, Ergodic Theory Dynam. Systems, 24, 2004, 1753–1786.

    Article  MathSciNet  MATH  Google Scholar 

  22. Pöschel, J., Quasi-periodic solutions for nonlinear wave equation, Comm. Math. Helv., 71, 1996, 269–296.

    Article  MathSciNet  MATH  Google Scholar 

  23. Wayne, C. E., Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127, 1990, 479–528.

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu, J., You, J. and Qiu, Q., Invariant tori for nearly integrable Hamtonian systems with degenercy, Math. Z., 226, 1997, 375–387.

    Article  MathSciNet  MATH  Google Scholar 

  25. Yuan, X., Quasi-periodic solutions of nonlinear wave equations with a prescribed potential, Discrete Contin. Dyn. Syst., 16, 2006, 615–634.

    Article  MathSciNet  MATH  Google Scholar 

  26. Yuan, X., Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Diff. Eqs., 230, 2006, 213–274.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Hongzi Cong for his invaluable discussions and suggestions. The authors would also like to thank the anonymous referee for helpful comments and suggestions.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, Liaoning, China

    Chunyong Liu & Huayong Liu

  2. Shangdong Chemical Engineering & Vocational College, Weifang, 261108, Shandong, China

    Chunyong Liu

  3. Dalian Airforce Communication Non-Commissioned Officer Academy, Dalian, 116600, Liaoning, China

    Rong Zhao

Authors
  1. Chunyong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huayong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Chunyong Liu, Huayong Liu or Rong Zhao.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 11671066) and the Fundamental Research Funds for the Central Universities (No. DUT15LK05).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, C., Liu, H. & Zhao, R. A Nekhoroshev Type Theorem for the Nonlinear Wave Equation in Gevrey Space. Chin. Ann. Math. Ser. B 40, 389–410 (2019). https://doi.org/10.1007/s11401-019-0140-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-019-0140-x

Keywords

2000 MR Subject Classification

Search

Navigation