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Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 957–978 | Cite as

On Reducibility of 1d Wave Equation with Quasiperiodic in Time Potentials

  • Zhenguo Liang
  • Xuting Wang
Article
  • 114 Downloads

Abstract

In this paper we use a KAM theorem of Grébert and Thomann (Commun Math Phys 307:383–427, 2011) to prove the reducibility of the 1d wave equation with Dirichlet boundery conditions on \([0,\pi ]\) with a quasi-periodic in time potential under some symmetry assumptions. From Mathieu–Hill operator’s known results (Eastham in The spectral theory of periodic differential operators, Hafner, New York, 1974; Magnus and Winkler in Hill’s equation, Wiley-Interscience, London, 1969) and Bourgain’s techniques (Commun Math Phys 204:207–247, 1999), we prove that for any \(\epsilon \) small enough, there exist a \(0<m_{\epsilon }\le 1\) and one solution \(u_{\epsilon }(t,x)\) with
$$\begin{aligned} \Vert u_{\epsilon }(t_n,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_n|\rightarrow \infty , \end{aligned}$$
where \(u_{\epsilon }(t,x)\) satisfies 1d wave equation
$$\begin{aligned} u_{tt}-u_{xx}+m_{\epsilon }u-\epsilon \cos 2t u=0, \end{aligned}$$
with Dirichlet boundery conditions on \([0,\pi ]\).

Notes

Acknowledgements

We thank the anonymous referee(s) for invaluable comments and suggestions. During the preparation of this work we benefit of many suggestions and discussions with Professor Geng Jiansheng.

References

  1. 1.
    Bambusi, D.: Birkhoff normal form for some nonlinear PDEs. Commun. Math. Phys. 234, 253–283 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bambusi, D., Graffi, S.: Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods. Commun. Math. Phys. 219, 465–480 (2001)CrossRefMATHGoogle Scholar
  3. 3.
    Bambusi, D., Grébert, B.: Birkhoff normal form for PDEs with tame modulus. Duke Math. J. 135, 507–567 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Berti, M., Biasco, L., Procesi, M.: KAM for reversible derivative wave equations. Arch. Ration. Mech. Anal. 212, 905–955 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bogoliubov, N., Mitropolsky, Yu., Samoilenko, A.: The Method of Rapid Convergence in Nonlinear Mechanics. Naukova Dumka, Kiev (1969). (Russian), English translation: Springer Verlag, 1976Google Scholar
  6. 6.
    Bourgain, J.: Growth of Sobolev norms in linear Schrödinger equation with quasi-periodic potential. Commun. Math. Phys. 204, 207–247 (1999)CrossRefMATHGoogle Scholar
  7. 7.
    Bourgain, J.: On growth of Sobolev norms in linear Schrödinger equation with time dependent potential. J. Anal. Math. 77, 315–348 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chavaudret, C.: Reducibility of quasiperiodic cocycles in linear Lie groups. Ergod. Theory Dyn. Syst. 31(3), 741–769 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chavaudret, C.: Strong almost reducibility for analytic and Gevrey quasi-periodic cocyles. Bull. Soc. Math. Fr. 141(1), 47–106 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chierchia, L., You, J.: KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Commun. Math. Phys. 211, 498–525 (2000)Google Scholar
  11. 11.
    Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations. Commun. Pure. Appl. Math. 46, 1409–1498 (1993)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Eastham, M.: The Spectral Theory of Periodic Differential Operators. Hafner, New York (1974)Google Scholar
  13. 13.
    Eliasson, L.H.: Floquent solutions for the one-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)CrossRefMATHGoogle Scholar
  14. 14.
    Eliasson, L.H.: Almost reducibility of linear quasi-periodic systems. In: Proceedings of Symposia in Pure Mathematics, vol. 69, pp. 679–705 (2001)Google Scholar
  15. 15.
    Eliasson, L.H.: Reducibility for Linear Quasi-periodic Differential Equations. Winter School, St Etienne de Tinée (2011)Google Scholar
  16. 16.
    Eliasson, L.H., Grébert, B., Kuksin, S.B.: Almost reducibility of linear quasi-periodic wave equation. KAM Theory and Dispersive PDEs, Roman (2014)Google Scholar
  17. 17.
    Eliasson, L.H., Kuksin, S.B.: On reducibility of Schrödinger equations with quasiperiodic potentials. Commun. Math. Phys. 286, 125–135 (2009)CrossRefMATHGoogle Scholar
  18. 18.
    Eliasson, L.H., Kuksin, S.B.: KAM for the nonlinear Schrödinger equation. Ann. Math. 172, 371–435 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Geng, J., Ren, X.: Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation. J. Differ. Equ. 249, 2796–2821 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun. Math. Phys. 262, 343–372 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Grébert, B., Kappeler, T., Pöschel, J.: The defocusing NLS equation and its normal form. To appear in EMS Series of Lectures in Mathematics. EMS Publishing HouseGoogle Scholar
  22. 22.
    Grébert, B., Thomann, L.: KAM for the quantum harmonic oscillator. Commun. Math. Phys. 307, 383–427 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hou, X., You, J.: Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. Math. 190, 209–260 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kappeler, T., Liang, Z.: A KAM theorem for the defocusing NLS equation with periodic boundary conditions. J. Differ. Equ. 252, 4068–4113 (2012)CrossRefMATHGoogle Scholar
  25. 25.
    Kappeler, T., Pöschel, J.: KDV & KAM. Springer, Berlin (2003)CrossRefMATHGoogle Scholar
  26. 26.
    Kirac, A.: On the instability intervals of the Mathieu–Hill operator. Lett Math Phys 83, 149–161 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Krikorian, R.: Rédutibilité presque partout des systèmes quasi périodiques analytiques dans le cas SO(3). C. R. Acad. Sci. Paris Sr. I Math. 321, 1039–1044 (1995)MathSciNetMATHGoogle Scholar
  28. 28.
    Krikorian, R.: Rédutibilité des systèmes produits-croisés à valeurs dans des groupes compacts. Astérisque 259, 1–216 (1999)Google Scholar
  29. 29.
    Kuksin, S.B.: Nearly integrable infinite-dimensional Hamiltonian systems. In: Lecture Notes in Mathematics, vol. 1556. Springer, Berlin (1993)Google Scholar
  30. 30.
    Kuksin, S.B.: Analysis of Hamiltonina PDEs. Oxford University Press, Oxford (2000)Google Scholar
  31. 31.
    Kuksin, S.B.: A KAM theorem for equations of the Korteweg–de Vries type. Rev. Math. Math. Phys. 10(3), 1–64 (1998)MathSciNetMATHGoogle Scholar
  32. 32.
    Liang, Z.: Quasi-periodic solutions for 1D Schrödinger equations with the nonlinearity \(|u|^{2p}u\). J. Differ. Equ. 244, 2185–2225 (2008)CrossRefMATHGoogle Scholar
  33. 33.
    Liang, Z., You, J.: Quasi-periodic solutions for 1D Schrödinger equations with higher order nonlinearity. SIAM J. Math. Anal. 36, 1965–1990 (2005)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Liu, J., Yuan, X.: Spectrum for quantum duffing oscillator and small divisor equation with large variable coefficient. Commun. Pure Appl. Math. 63, 1145–1172 (2010)MathSciNetMATHGoogle Scholar
  35. 35.
    Magnus, W., Winkler, S.: Hill’s Equation. Wiley-Interscience, London (1969)MATHGoogle Scholar
  36. 36.
    Moser, J.: A rapidly convergent iteration method and non-linear partial differential equations I and II. Ann. Sc. Norm. Super. Pisa 20(265–315), 499–535 (1966)MATHGoogle Scholar
  37. 37.
    Moser, J.: On the construction of almost periodic solutions for ordinary differential equations. In: Proceedings of the International Conference on Functional Analysis and Related Topics, Tokyo, pp. 60–67 (1969)Google Scholar
  38. 38.
    Pöschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helv. 71, 269–296 (1996)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Pöschel, J.: A KAM theorem for some nonlinear partial differential equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 23, 119–148 (1996)MathSciNetMATHGoogle Scholar
  40. 40.
    Wang, W.-M.: Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations. J. Funct. Anal. 254, 2926–2946 (2008)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Wang, W.-M.: Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations. Commun. Math. Phys. 277, 459–496 (2008)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Wang, W.-M.: Logarithmic bounds on Sobolev norms for time dependent linear Schrödinger equations. Commun. Partial Differ. Equ. 33, 2164–2179 (2008)CrossRefMATHGoogle Scholar
  43. 43.
    Wang, Z., Liang, Z.: 1D quantum harmonic oscillator perturbed by a potential with logarithmic decay (preprint). arXiv:1605.05480
  44. 44.
    Yuan, X.: Quasi-periodic solutions of completely resonant nonlinear wave equations. J. Differ. Equ. 230, 213–274 (2006)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematical Sciences and Key Lab of Mathematics for Nonlinear ScienceFudan UniversityShanghaiChina

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