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Bounded Non-response Solutions with Liouvillean Forced Frequencies for Nonlinear Wave Equations

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Abstract

In this paper, one dimensional nonlinear wave equations

$$\begin{aligned} u_{tt}-u_{xx}+M_{\xi _{2}}u +\varepsilon f(\omega _{1} t, x, u)=0 \end{aligned}$$

with Dirichlet boundery conditions are considered, where \(M_{\xi _{2}}\) is a real Fourier multiplier, f is a real analytic function with \(f(\omega _{1} t, -x, -u)=-f(\omega _{1} t, x, u)\) and the forced frequencies \(\omega _1=(1,\alpha )\) are Liouvillean. We obtain a family of \(C^{\infty }\) smooth, bounded non-response solutions with Liouvillean forced frequencies. This is based on an infinite dimensional KAM theorem with angle-dependent normal form.

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Notes

  1. In the case \(b=2\), if \(\displaystyle \mathcal {B}(\omega )<\infty \), then \(\beta (\alpha )= 0\)

References

  1. Avila, A., Fayad, B., Krikorian, R.: A KAM scheme for \({\rm SL}(2,\mathbb{R})\) cocycles with Liouvillean frequencies. Geom. Funct. Anal. 21(5), 1001–1019 (2011)

    Article  MathSciNet  Google Scholar 

  2. Baldi, P., Berti, M., Montalto, R.: KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation. Math. Ann. 359(1–2), 471–536 (2014)

    Article  MathSciNet  Google Scholar 

  3. Baldi, P., Berti, M., Montalto, R.: KAM for quasi-linear KdV. C. R. Math. Acad. Sci. Paris 352(7–8), 603–607 (2014)

    Article  MathSciNet  Google Scholar 

  4. Bambusi, D.: Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations. I. Trans. Am. Math. Soc. 370(3), 1823–1865 (2018)

    Article  MathSciNet  Google Scholar 

  5. Bambusi, D.: Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations. II. Commun. Math. Phys. 353(1), 353–378 (2017)

    Article  MathSciNet  Google Scholar 

  6. Bambusi, D., Berti, M., Magistrelli, E.: Degenerate KAM theory for partial differential equations. J. Differ. Equ. 250(8), 3379–3397 (2011)

    Article  MathSciNet  Google Scholar 

  7. Berti, M., Biasco, L., Procesi, M.: KAM theory for the Hamiltonian derivative wave equation. Ann. Sci. Éc. Norm. Supér. (4) 46(2), 301–373 (2013)

    Article  MathSciNet  Google Scholar 

  8. Berti, M., Biasco, L., Procesi, M.: KAM for reversible derivative wave equations. Arch. Ration. Mech. Anal. 212(3), 905–955 (2014)

    Article  MathSciNet  Google Scholar 

  9. Berti, M., Montalto, R.: Quasi-periodic water waves. J. Fixed Point Theory Appl. 19(1), 129–156 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications, Volume 158 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (2005)

    Google Scholar 

  12. Chang, N., Geng, J., Lou, Z.: Response solutions of forced reversible wave equations with Liouvillean frequencies, preprint (2019)

  13. Chierchia, L., You, J.: KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Commun. Math. Phys. 211(2), 497–525 (2000)

    Article  MathSciNet  Google Scholar 

  14. Craig, W., Wayne, C.: Newton’s method and periodic solutions of nonlinear wave equations. Commun. Pure Appl. Math 46(11), 1409–1498 (1993)

    Article  MathSciNet  Google Scholar 

  15. Eliasson, L., Kuksin, S.: KAM for the nonlinear Schrödinger equation. Ann. Math. (2) 172(1), 371–435 (2010)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  18. Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun. Math. Phys. 262(2), 343–372 (2006)

    Article  MathSciNet  Google Scholar 

  19. Hou, X., You, J.: Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. Math. 190(1), 209–260 (2012)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  21. Kuksin, S.: Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum. Funktsional. Anal. i Prilozhen. 21(3), 22–37 (1987)

    Article  MathSciNet  Google Scholar 

  22. Kuksin, S.: A KAM-theorem for equations of the Korteweg–de Vries type. Rev. Math. Math. Phys. 10(3), ii+64 (1998)

    MathSciNet  MATH  Google Scholar 

  23. Krikorian, R., Wang, J., You, J., Zhou, Q.: Linearization of quasiperiodically forced circle flow beyond Brjuno condition. Commun. Math. Phys 358(1), 81–100 (2016)

    Article  MathSciNet  Google Scholar 

  24. Lou, Z., Geng, J.: Quasi-periodic response solutions in forced reversible systems with Liouvillean frequencies. J. Differ. Equ. 263(7), 3894–3927 (2017)

    Article  MathSciNet  Google Scholar 

  25. Pöschel, J.: On elliptic lower-dimensional tori in Hamiltonian systems. Math. Z. 202(4), 559–608 (1989)

    Article  MathSciNet  Google Scholar 

  26. Pöschel, J.: A KAM-theorem for some nonlinear partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(1), 119–148 (1996)

    MathSciNet  MATH  Google Scholar 

  27. Wang, J., You, J.: Boundedness of solutions for non-linear quasi-periodic differential equations with Liouvillean frequency. J. Differ. Equ. 261, 1068–1098 (2016)

    Article  MathSciNet  Google Scholar 

  28. Wang, J., You, J., Zhou, Q.: Response solutions for quasi-periodically forced harmonic oscillators. Trans. Am. Math. Soc. 369(6), 4251–4274 (2017)

    Article  MathSciNet  Google Scholar 

  29. Wayne, C.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127(3), 479–528 (1990)

    Article  MathSciNet  Google Scholar 

  30. Xu, X., You, J., Zhou, Q.: Quasi-Periodic Solutions of NLS with Liouvillean Frequency (2017). arXiv:1707.04048

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Ningning Chang & Jiansheng Geng

  2. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, People’s Republic of China

    Zhaowei Lou

Authors
  1. Ningning Chang
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  2. Jiansheng Geng
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  3. Zhaowei Lou
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Correspondence to Jiansheng Geng.

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The research was supported by NNSFC Grants 11971012, 11901291 and the Natural Science Foundation of Jiangsu Province of China BK20190395.

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Chang, N., Geng, J. & Lou, Z. Bounded Non-response Solutions with Liouvillean Forced Frequencies for Nonlinear Wave Equations. J Dyn Diff Equat 33, 2009–2046 (2021). https://doi.org/10.1007/s10884-020-09882-z

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  • DOI: https://doi.org/10.1007/s10884-020-09882-z

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