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Quasi-Periodic Solution of Nonlinear Beam Equation on \({\mathbb T}^d\) with Forced Frequencies

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Abstract

In this paper, we study the higher dimensional nonlinear beam equation under periodic boundary condition:

$$\begin{aligned} u_{tt} + \Delta ^2 _x u + M_{\xi }u + f({\bar{\omega }} t,u)=0,\quad u=u(t,x),t\in {{\mathbb {R}}},x\in {\mathbb T}^d,d\ge 2 \end{aligned}$$

where \(M_{\xi }\) is a real Fourier multiplier, \(f=f({\bar{\theta }},u)\) is a real analytic function with respect to \(({\bar{\theta }},u)\), and \(f({\bar{\theta }},u)=O(|u|^2)\). This equation can be viewed as an infinite dimensional nearly integrable Hamiltonian system. We establish an infinite dimensional KAM theorem, and apply it to this equation to prove that there exist a class of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori.

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References

  1. Berti, M., Bolle, P.: Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity 25, 2579–2613 (2012)

    Article  MathSciNet  Google Scholar 

  2. 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)

    Article  MathSciNet  Google Scholar 

  3. Berti, M., Bolle, P.: A Nash-Moser approach to KAM theory. Hamiltonian partial differential equations and applications, 255-284, Fields Inst. Commun., 75, Fields Inst. Res. Math. Sci., Toronto, ON (2015)

  4. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equation. Ann. Math. 148, 363–439 (1998)

    Article  MathSciNet  Google Scholar 

  5. Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Ann. of Math.Stud. vol.158, Princeton University Press, Princeton, NJ (2005)

  6. Bourgain, J., Wang, W.M.: Quasi-periodic solutions of nonlinear random Schrödinger equations. J. Eur. Math. Soc. 10(1), 1–45 (2008)

    Article  MathSciNet  Google Scholar 

  7. 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 

  8. Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations. Commun. Pure Appl. Math. 46, 1409–1498 (1993)

    Article  MathSciNet  Google Scholar 

  9. Eliasson, L.H., Grebert, B., Kuksin, S.B.: KAM for the nonlinear beam equation. Geom. Funct. Anal. 26(6), 1588–1715 (2016)

    Article  MathSciNet  Google Scholar 

  10. Eliasson, L.H., Kuksin, S.B.: KAM for nonlinear Schrödinger equation. Ann. Math. 172, 371–435 (2010)

    Article  MathSciNet  Google Scholar 

  11. Eliasson, L.H., Kuksin, S.B.: Infinite Töeplitz–Lipschitz matrices and operators. Z. Angew. Math. Phys. 59, 24–50 (2008)

    Article  MathSciNet  Google Scholar 

  12. Eliasson, L.H., Kuksin, S.B.: On reducilibity of Schrödinger equations with quasiperiodic in time potentials. Commun. Math. Phys. 286, 125–135 (2009)

    Article  Google Scholar 

  13. 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)

    Article  MathSciNet  Google Scholar 

  14. Geng, J., Xue, S.: A KAM theorem for higher dimensional forced nonlinear Schrödinger equations. J. Dyn. Differ. Equ. 30(3), 979–1010 (2018)

    Article  MathSciNet  Google Scholar 

  15. Geng, J., Yi, Y.: Quasi-periodic solutions in a nonlinear Schrödinger equation. J. Differ. Equ. 233, 512–542 (2007)

    Article  Google Scholar 

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

    Article  Google Scholar 

  17. Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun. Math. Phys. 262, 343–372 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Grebert, B., Paturel, E.: KAM for the Klein–Gordon equation on \(S^d\). Boll. Unione. Mat. Ital. 9, 237–288 (2016)

    Article  MathSciNet  Google Scholar 

  20. Grebert, B., Thomann, L.: KAM for the quantum harmonic oscillator. Commun. Math. Phys. 307, 383–427 (2011)

    Article  MathSciNet  Google Scholar 

  21. Kappeler, T., Pöschel, J.: KdV & KAM. Springer, Berlin (2003)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Kuksin, S.B.: Nearly Intergrable Infinite Dimensional Hamiltonian Systems. Lecture Notes in Math, vol. 1556. Springer, Berlin (1992)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Kuksin, S.B.: Analysis of Hamiltonian PDEs. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  26. Pöschel, J.: On elliptic lower dimensional tori in Hamiltonian systems. Math. Z. 202, 559–608 (1989)

    Article  MathSciNet  Google Scholar 

  27. Pöschel, J.: A KAM theorem for some nonlinear partial differential equations. Ann. Sc Norm. Sup. Pisa Cl. Sci. 23, 119–148 (1996)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Procesi, C., Procesi, M.: A normal form for the Schrödinger equation with analytic non-linearities. Commun. Math. Phys. 312, 501–557 (2012)

    Article  Google Scholar 

  30. Procesi, C., Procesi, M.: A KAM algorithm for the resonant non-linear Schrödinger equation. Adv. Math. 272, 399–470 (2015)

    Article  MathSciNet  Google Scholar 

  31. Procesi, C., Procesi, M.: Reducible quasi-periodic solutions for the nonlinear Schrödinger equation. Boll. Unione. Mat. Ital. 9, 189–236 (2016)

    Article  MathSciNet  Google Scholar 

  32. Procesi, M.: A normal form for beam and non-local nonlinear Schrödinger equations. J. Phys. A. 43, 434028 (2010). 13 pp

    Article  MathSciNet  Google Scholar 

  33. Wang, W.M.: Energy supercritical nonlinear Schrödinger equations: quasiperiodic solutions. Duke Math. J. 165, 1129–1192 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Wayne, C.E.: Periodic and quasi-periodic solutions for nonlinear wave equations via KAM theory. Commun. Math. Phys. 127, 479–528 (1990)

    Article  MathSciNet  Google Scholar 

  35. Yuan, X.: Quasi-periodic solutions of completely resonant nonlinear wave equations. J. Differ. Equ. 230, 213–274 (2006)

    Article  MathSciNet  Google Scholar 

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  1. School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, Guangdong, People’s Republic of China

    Shidi Zhou

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Zhou, S. Quasi-Periodic Solution of Nonlinear Beam Equation on \({\mathbb T}^d\) with Forced Frequencies. Qual. Theory Dyn. Syst. 19, 89 (2020). https://doi.org/10.1007/s12346-020-00425-x

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