Quasi-Periodic Solutions of Derivative Beam Equation on Flat Tori

• Yingte Sun ^[1]
  1. [1] Yangzhou University

     Yangzhou University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • In this paper, we consider a class of nonlinear beam equations on flat tori Td L, utt + 2u = (−)α f (ωt, x, (−)βu), 0 < α + β ≤ 1 and prove that the equation admits many quasi-periodic solutions with the non-resonant frequencies ω. The main proof is based on an abstract Nash-Moser type implicit function theorem developed in Berti and Bolle (Nonlinearity 25(9):2579–2613, 2012;

    J Eur Math Soc 15(1):229–286, 2013).

• Referencias bibliográficas
  • 1. Bambusi, D., Grébert, B., Maspero, A., Robert, D.: Reducibility of the quantum harmonic oscillator in d-dimensions with polynomial time-dependent...
  • 2. Bambusi, D., Langella, B., Montalto, R.: Reducibility of non-resonant transport equation on Td with unbounded perturbations. Ann. Henri...
  • 3. Berti, M., Bolle, P.: Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity...
  • 4. Berti, M., Bolle, P.: Quasi-periodic solutions with Sobolev regularity of NLS onTd with a multiplicative potential. J. Eur. Math. Soc....
  • 5. Berti, M., Corsi, L., Procesi, M.: An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and...
  • 6. Berti, M., Maspero, A.: Long time dynamics of Schrödinger and wave equations on flat tori. J. Differ. Equ. 267(2), 1167–1200 (2019)
  • 7. Bhatia, R.: Matrix Analysis. Springer-Verlag, New York (1997)
  • 8. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Shrödinger equation. Ann. Math. 148, 363–439 (1998)
  • 9. Bourgain, J.: Greens function estimates for lattice Schrödinger operators and applications. Annals of Mathematics Studies vol. 158, Princeton...
  • 10. Chen, B., Gao, Y., Jiang, S., Li, Y.: Quasi-periodic solutions to nonlinear beam equations on compact Lie groups with a multiplicative...
  • 11. Corsi, L., Montalto, R.: Quasi-periodic solutions for the forced Kirchhoff equation on Td . Nonlinearity 31(11), 5075–5109 (2018)
  • 12. Craig, W., Wayne, C.E.: Newtons method and periodic solutions of nonlinear wave equations. Commun. Pure Appl. Math. 46(11), 1409–1498...
  • 13. Eliasson, H., Grébert, B., Kuksin, S.: KAM for the nonlinear beam equation. Geom. Funct. Anal. 26(6), 1588–1715 (2016)
  • 14. Eliasson, H., Kuksin, S.: KAM for the nonlinear Schrödinger equation. Ann. Math. 172(1), 371–435 (2010)
  • 15. Feola, R., Grebért, B.: Reducibility of Schrödinger equation on the Sphere. Int. Math. Res. Not. 2021(19), 15082–15120 (2021)
  • 16. Feola, R., Grebért, B., Nguyen, T.: Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential. J. Math. Phys. 61(7),...
  • 17. Feola, R., Giuliani, F., Montalto, R., Procesi, M.: Reducibility of first order linear operators on tori via Mosers theorem. J. Funct....
  • 18. Ge, C., Geng, J., Lou, Z.: KAM theory for the reversible perturbations of 2D linear beam equations. Math. Z. 297, 1693–1731 (2021)
  • 19. Ge, C., Geng, J., Lou, Z.: KAM Tori for Completely Resonant Hamiltonian Derivative Beam Equations on T2. J. Dyn. Differ. Equ. 33(1), 525–547...
  • 20. Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun. Math. Phys. 262(2),...
  • 21. Geng, J., Xu, X., You, J.: An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation....
  • 22. Grébert, B., Paturel, E.: KAM for the Klein Gordon equation on Sd . Boll. Unione Mat. Ital. 9, 237–288 (2016)
  • 23. Grébert, B., Paturel, E.: On reducibility of quantum harmonic oscillator on Rd with quasiperiodic in time potential. Ann. Fac. Sci. Toulouse...
  • 24. Mi, L., Cong, H.: Quasi-periodic solutions for d-dimensional beam equation with derivative nonlinear perturbation. J. Math. Phys. 56(7),...
  • 25. Montalto, R.: A reducibility result for a class of linear wave equations on Td . Int. Math. Res. Not. 2019(6), 1788–1862 (2019)
  • 26. Procesi, C., Procesi, M.: A KAM algorithm for the resonant non-linear Schrödinger equation. Adv. Math. 272, 399–470 (2015)
  • 27. Procesi, M., Procesi, C.: Reducible quasi-periodic solutions for the non linear Schrödinger equation. Boll. Unione Mat. Ital. 9, 189–236...
  • 28. Shi, Y., Xu, J.: Quasi-periodic solutions for two dimensional modified Boussinesq equation. J. Dyn. Differ. Equ. 33(2), 741–766 (2021)
  • 29. Shi, Y.: On the existence of Sobolev quasi-periodic solutions of multidimensional nonlinear beam equation. J. Math. Phys. 57(10), 102701...
  • 30. Yuan, X.: KAM theorem with normal frequencies of finite limit points for some shallow water equations. Comm. Pure Appl. Math. 74(6), 1193–1281...