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Abstract and Applied Analysis
Volume 2014, Article ID 649270, 15 pages
http://dx.doi.org/10.1155/2014/649270
Research Article

Quasiperiodic Solutions of Completely Resonant Wave Equations with Quasiperiodic Forced Terms

1School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China
2Key Laboratory of Symbolic Computation and Knowledge Engineering, Ministry of Education, Jilin University, Changchun, Jilin 130012, China
3Fundamental Department, Aviation University of Air Force, Changchun 130023, China

Received 28 February 2014; Revised 25 April 2014; Accepted 28 April 2014; Published 18 May 2014

Academic Editor: Yongli Song

Copyright © 2014 Yixian Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Bourgain, “Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations,” Annals of Mathematics, vol. 148, no. 2, pp. 363–439, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. J. Bourgain, “Periodic solutions of nonlinear wave equations,” in Harmonic Analysis and Partial Differential Equations, Chicago Lectures in Mathematics, pp. 69–97, University of Chicago Press, Chicago, Ill, USA, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, vol. 158 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 2005. View at MathSciNet
  4. L. Chierchia and J. You, “KAM tori for 1D nonlinear wave equations with periodic boundary conditions,” Communications in Mathematical Physics, vol. 211, no. 2, pp. 497–525, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. W. Craig and C. E. Wayne, “Newton's method and periodic solutions of nonlinear wave equations,” Communications on Pure and Applied Mathematics, vol. 46, no. 11, pp. 1409–1498, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. Gao, Y. Li, and J. Zhang, “Invariant tori of nonlinear Schrödinger equation,” Journal of Differential Equations, vol. 246, no. 8, pp. 3296–3331, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. J. Geng and J. You, “A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces,” Communications in Mathematical Physics, vol. 262, no. 2, pp. 343–372, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. J. Geng and X. Ren, “Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation,” Journal of Differential Equations, vol. 249, no. 11, pp. 2796–2821, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. J. Geng, “Almost periodic solutions for a class of higher dimensional Schrödinger equations,” Frontiers of Mathematics in China, vol. 4, no. 3, pp. 463–482, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. J. Pöschel, “Quasi-periodic solutions for a nonlinear wave equation,” Commentarii Mathematici Helvetici, vol. 71, no. 2, pp. 269–296, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. X. Yuan, “A KAM theorem with applications to partial differential equations of higher dimensions,” Communications in Mathematical Physics, vol. 275, no. 1, pp. 97–137, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. C. E. Wayne, “Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory,” Communications in Mathematical Physics, vol. 127, no. 3, pp. 479–528, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. V. Benci and P. Rabinowitz, “Critical point theorems for indefinite functionals,” Inventiones Mathematicae, vol. 52, no. 3, pp. 241–273, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. H. Brézis and L. Nirenberg, “Forced vibrations for a nonlinear wave equation,” Communications on Pure and Applied Mathematics, vol. 31, no. 1, pp. 1–30, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. H. Brézis, J.-M. Coron, and L. Nirenberg, “Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz,” Communications on Pure and Applied Mathematics, vol. 33, no. 5, pp. 667–684, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. E. R. Fadell and P. H. Rabinowitz, “Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems,” Inventiones Mathematicae, vol. 45, no. 2, pp. 139–174, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. P. H. Rabinowitz, “Periodic solutions of nonlinear hyperbolic partial differential equations,” Communications on Pure and Applied Mathematics, vol. 20, no. 1, pp. 145–205, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. P. H. Rabinowitz, “Free vibrations for a semilinear wave equation,” Communications on Pure and Applied Mathematics, vol. 31, no. 1, pp. 31–68, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986. View at MathSciNet
  20. D. Bambusi and M. Berti, “A Birkhoff-Lewis-type theorem for some Hamiltonian PDEs,” SIAM Journal on Mathematical Analysis, vol. 37, no. 1, pp. 83–102, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. D. Bambusi and S. Paleari, “Families of periodic solutions of resonant PDEs,” Journal of Nonlinear Science, vol. 11, no. 1, pp. 69–87, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. M. Berti and P. Bolle, “Periodic solutions of nonlinear wave equations with general nonlinearities,” Communications in Mathematical Physics, vol. 243, no. 2, pp. 315–328, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. M. Berti and P. Bolle, “Multiplicity of periodic solutions of nonlinear wave equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 7, pp. 1011–1046, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. M. Berti and P. Bolle, “Cantor families of periodic solutions for completely resonant nonlinear wave equations,” Duke Mathematical Journal, vol. 134, no. 2, pp. 359–419, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. B. V. Lidskii and E. I. Shul'man, “Periodic solutions of the equation utt-uxx+u3=0,” Functional Analysis and Its Applications, vol. 22, no. 4, pp. 332–33, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  26. G. Gentile, V. Mastropietro, and M. Procesi, “Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions,” Communications in Mathematical Physics, vol. 256, no. 2, pp. 437–490, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. G. Gentile and M. Procesi, “Conservation of resonant periodic solutions for the one-dimensional nonlinear Schrödinger equation,” Communications in Mathematical Physics, vol. 262, no. 3, pp. 533–553, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  28. X. Yuan, “Quasi-periodic solutions of completely resonant nonlinear wave equations,” Journal of Differential Equations, vol. 230, no. 1, pp. 213–274, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  29. M. Berti and M. Procesi, “Quasi-periodic solutions of completely resonant forced wave equations,” Communications in Partial Differential Equations, vol. 31, no. 6, pp. 959–985, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  30. M. Procesi, “Quasi-periodic solutions for completely resonant non-linear wave equations in ID and 2D,” Discrete and Continuous Dynamical Systems, vol. 13, no. 3, pp. 541–552, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. P. Baldi, “Quasi-periodic solutions of the equation vtt-vxx+v3=f(v),” Discrete and Continuous Dynamical Systems, vol. 15, no. 3, pp. 883–903, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. Y. Ma and W. Lou, “Quasi-periodic solutions of completely resonant wave equations with quasi-periodically forced vibrations,” Acta Applicandae Mathematicae, vol. 112, no. 3, pp. 309–322, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  33. D. Bambusi, “Lyapunov center theorem for some nonlinear PDE's: a simple proof,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, vol. 29, no. 4, pp. 823–837, 2000. View at Google Scholar · View at MathSciNet
  34. A. Ambrosetti and M. Badiale, “Homoclinics: Poincaré-Melnikov type results via a variational approach,” Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 15, no. 2, pp. 233–252, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  35. W. M. Schmidt, Diophantine Approximation, vol. 785 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1980. View at MathSciNet