Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2014, Article ID 712183, 14 pages
http://dx.doi.org/10.1155/2014/712183
Research Article

Uniform Attractor for the Fractional Nonautonomous Long-Short Wave Equations

School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

Received 14 March 2014; Accepted 3 September 2014; Published 20 October 2014

Academic Editor: Prasanta K. Panigrahi

Copyright © 2014 Huanmin Ge and Jie Xin. 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. B. Guo, “The global solution for one class of the system of LS nonlinear wave interaction,” Journal of Mathematical Research and Exposition, vol. 7, no. 1, pp. 69–76, 1987. View at Google Scholar · View at MathSciNet
  2. B. Guo, “The periodic initial value problems and initial value problems for one class of generalized long-short type equations,” Journal of Engineering Mathematics, vol. 8, pp. 47–53, 1991. View at Google Scholar
  3. X. Du and B. Guo, “The global attractor for LS type equation in R1,” Acta Mathematicae Applicatae Sinica, vol. 28, pp. 723–734, 2005. View at Google Scholar
  4. Y. Li, “Long time behavior for the weakly damped driven long-wave—short-wave resonance equations,” Journal of Differential Equations, vol. 223, no. 2, pp. 261–289, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. R.-F. Zhang, “Existence of global attractor for LS type equations,” Journal of Mathematical Research and Exposition, vol. 26, no. 4, pp. 708–714, 2006. View at Google Scholar · View at MathSciNet
  6. H. Cui, J. Xin, and A. Li, “Weakly compact uniform attractor for the nonautonomous long-short wave equations,” Abstract and Applied Analysis, vol. 2013, Article ID 601325, 13 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, NY, USA, 1965.
  8. N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Physics Letters A, vol. 268, no. 4–6, pp. 298–305, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. N. Laskin, “Fractional quantum mechanics,” Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, vol. 62, no. 3, pp. 3135–3145, 2000. View at Publisher · View at Google Scholar · View at Scopus
  10. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, 2012.
  11. X. Guo and M. Xu, “Some physical applications of fractional Schrödinger equation,” Journal of Mathematical Physics, vol. 47, no. 8, Article ID 082104, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  12. D. Baleanu and A. K. Golmankhaneh, “Solving of the fractional non-linear and linear Schrödinger equations by homotopy perturbation method,” Romanian Journal of Physics, vol. 54, no. 10, pp. 823–832, 2009. View at Google Scholar · View at MathSciNet · View at Scopus
  13. R. Eid, S. I. Muslih, D. Baleanu, and E. Rabei, “On fractional Schrödinger equation in α-dimensional fractional space,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1299–1304, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. B. Guo, Y. Han, and J. Xin, “Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 468–477, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. O. Goubet, “Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in R2,” Advances in Differential Equations, vol. 3, no. 3, pp. 337–360, 1998. View at Google Scholar · View at MathSciNet · View at Scopus
  16. V. V. Chepyzhov and M. I. Vishik, “Attractors of nonautonomous dynamical systems and their dimension,” Journal de Mathématiques Pures et Appliquées, vol. 73, no. 3, pp. 279–333, 1994. View at Google Scholar · View at MathSciNet
  17. J. Bergh and J. Laofstraom, Interpolation Spaces, Springer, Berlin, Germany, 1976.
  18. V. Chepyzhov and M. Vishik, “Non-autonomous evolutionary equations with translation-compact symbols and their attractors,” Comptes Rendus de l'Académie des Sciences Série I: Mathématique, vol. 321, no. 2, pp. 153–158, 1995. View at Google Scholar · View at MathSciNet
  19. J. M. Ball, “Global attractors for damped semilinear wave equations,” Discrete and Continuous Dynamical Systems Series A, vol. 10, no. 1-2, pp. 31–52, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. B. Guo, Infinite Dimensional Dynamical Systems, National Defense Industry Press, Beijing, China, 1st edition, 2000.
  21. J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, Germany, 1976. View at MathSciNet
  22. J. Xin, B. Guo, Y. Han, and D. Huang, “The global solution of the (2+1)-dimensional long wave-short wave resonance interaction equation,” Journal of Mathematical Physics, vol. 49, no. 7, Article ID 073504, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. B. Guo and B. Wang, “The global solution and its long time behavior for a class of generalized LS type equations,” Progress in Natural Science, vol. 6, no. 5, pp. 533–546, 1996. View at Google Scholar · View at MathSciNet
  24. R. Zhang and B. Guo, “Global solution and its long time behavior for the generalized long-short wave equations,” Journal of Partial Differential Equations, vol. 18, no. 3, pp. 206–218, 2005. View at Google Scholar · View at MathSciNet