Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2015, Article ID 601657, 8 pages
http://dx.doi.org/10.1155/2015/601657
Research Article

Derivation of Asymptotic Dynamical Systems with Partial Lie Symmetry Groups

General Education Department, Aichi Institute of Technology, 1247 Yachigusa Yakusacho, Toyota 470-0392, Japan

Received 16 July 2015; Revised 12 September 2015; Accepted 14 September 2015

Academic Editor: Wan-Tong Li

Copyright © 2015 Masatomo Iwasa. 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. N. N. Bogoliubov and C. M. Place, Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon and Break, 1961.
  2. G. Hori, “Theory of general perturbation with unspecified canonical variable,” Publications of the Astronomical Society of Japan, vol. 18, p. 287, 1966. View at Google Scholar
  3. T. Taniuchi, “Reductive perturbation method and far fields of wave equations,” Progress of Theoretical Physics Supplement, vol. 55, pp. 1–35, 1974. View at Publisher · View at Google Scholar
  4. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields, Springer, New York, NY, USA, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. H. Nayfeh, Method of Normal Forms, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  6. T. Kunihiro, “A geometrical formulation of the renormalization group method for global analysis,” Progress of Theoretical Physics, vol. 94, no. 4, pp. 503–514, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  7. L.-Y. Chen, N. Goldenfeld, Y. Oono, and G. Paquette, “Selection, stability and renormalization,” Physica A: Statistical Mechanics and its Applications, vol. 204, no. 1–4, pp. 111–133, 1994. View at Publisher · View at Google Scholar · View at Scopus
  8. L. Y. Chen, N. Goldenfeld, and Y. Oono, “Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory,” Physical Review E, vol. 54, no. 1, pp. 376–394, 1996. View at Publisher · View at Google Scholar
  9. S.-I. Goto, Y. Masutomi, and K. Nozaki, “Lie-group approach to perturbative renormalization group method,” Progress of Theoretical Physics, vol. 102, no. 3, pp. 471–497, 1999. View at Publisher · View at Google Scholar · View at Scopus
  10. R. E. L. DeVille, A. Harkin, M. Holzer, K. Josić, and T. J. Kaper, “Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations,” Physica D, vol. 237, no. 8, pp. 1029–1052, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. H. Chiba, “C1 approximation of vector fields based on the renormalization group method,” SIAM Journal on Applied Dynamical Systems, vol. 7, no. 3, pp. 895–932, 2008. View at Publisher · View at Google Scholar
  12. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, NY, USA, 1986.
  13. V. A. Baĭkov, R. K. Gazizov, and N. H. Ibragimov, “Approximate symmetries,” Mathematics of the USSR-Sbornik, vol. 64, no. 2, p. 427, 1989. View at Publisher · View at Google Scholar
  14. W. I. Fushchich and W. M. Shtelen, “On approximate symmetry and approximate solutions of the nonlinear wave equation with a small parameter,” Journal of Physics A: Mathematical and General, vol. 22, no. 18, pp. L887–L890, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. N. Eular, M. W. Shulga, and W.-H. Steeb, “Approximate symmetries and approximate solutions for a multidimensional Landau-Ginzburg equation,” Journal of Physics A: Mathematical and General, vol. 25, no. 18, Article ID L1095, 1992. View at Publisher · View at Google Scholar
  16. Y. Y. Bagderina and R. K. Gazizov, “Equivalence of ordinary differential equations y=Rx,yy2+2Qx,yy+P(x,y),” Differential Equations, vol. 43, no. 5, pp. 595–604, 2007. View at Publisher · View at Google Scholar
  17. V. N. Grebenev and M. Oberlack, “Approximate Lie symmetries of the Navier-Stokes equations,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 2, pp. 157–163, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  18. R. K. Gazizov and N. K. Ibragimov, “Approximate symmetries and solutions of the Kompaneets equation,” Journal of Applied Mechanics and Technical Physics, vol. 55, no. 2, pp. 220–224, 2014. View at Publisher · View at Google Scholar
  19. R. K. Gazizov and C. M. Khalique, “Approximate transformations for van der Pol-type equations,” Mathematical Problems in Engineering, vol. 2006, Article ID 68753, 11 pages, 2006. View at Publisher · View at Google Scholar
  20. G. Gaeta, “Asymptotic symmetries and asymptotically symmetric solutions of partial differential equations,” Journal of Physics A: Mathematical and General, vol. 27, no. 2, pp. 437–451, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  21. G. Gaeta, “Asymptotic symmetries in an optical lattice,” Physical Review A, vol. 72, no. 3, Article ID 033419, 2005. View at Publisher · View at Google Scholar
  22. G. Gaeta, D. Levi, and R. Mancinelli, “Asymptotic symmetries of difference equations on a lattice,” Journal of Nonlinear Mathematical Physics, vol. 12, supplement 2, pp. 137–146, 2005. View at Publisher · View at Google Scholar · View at Scopus
  23. G. Gaeta and R. Mancinelli, “Asymptotic scaling symmetries for nonlinear PDEs,” International Journal of Geometric Methods in Modern Physics, vol. 2, no. 6, pp. 1081–1114, 2005. View at Publisher · View at Google Scholar
  24. G. Gaeta and R. Mancinelli, “Asymptotic scaling in a model class of anomalous reaction-diffusion equations,” Journal of Nonlinear Mathematical Physics, vol. 12, no. 4, pp. 550–566, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. D. Levi and M. A. Rodríguez, “Asymptotic symmetries and integrability: the KdV case,” Europhysics Letters, vol. 80, no. 6, Article ID 60005, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  26. V. F. Kovalev, V. V. Pustovalov, and D. V. Shirkov, “Group analysis and renormgroup symmetries,” Journal of Mathematical Physics, vol. 39, no. 2, pp. 1170–1188, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. V. F. Kovalev, “Renormalization group analysis for singularities in the wave beam self-focusing problem,” Theoretical and Mathematical Physics, vol. 119, no. 3, pp. 719–730, 1999. View at Publisher · View at Google Scholar
  28. V. F. Kovalev and D. V. Shirkov, “Functional self-similarity and renormalization group symmetry in mathematical physics,” Theoretical and Mathematical Physics, vol. 121, no. 1, pp. 1315–1332, 1999. View at Publisher · View at Google Scholar
  29. D. V. Shirkov and V. F. Kovalev, “The Bogoliubov renormalization group and solution symmetry in mathematical physics,” Physics Report, vol. 352, no. 4–6, pp. 219–249, 2001. View at Publisher · View at Google Scholar · View at Scopus
  30. V. F. Kovalev and D. V. Shirkov, “The renormalization group symmetry for solution of integral equations,” Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 50, no. 2, pp. 850–861, 2004. View at Google Scholar
  31. M. Iwasa and K. Nozaki, “A method to construct asymptotic solutions invariant under the renormalization group,” Progress of Theoretical Physics, vol. 116, no. 4, pp. 605–613, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  32. M. Iwasa and K. Nozaki, “Renormalization group in difference systems,” Journal of Physics A: Mathematical and Theoretical, vol. 41, no. 8, Article ID 085204, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  33. H. Chiba and M. Iwasa, “Lie equations for asymptotic solutions of perturbation problems of ordinary differential equations,” Journal of Mathematical Physics, vol. 50, no. 4, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  34. Y. Y. Yamaguchi and Y. Nambu, “Renormalization group equations and integrability in hamiltonian systems,” Progress of Theoretical Physics, vol. 100, no. 1, pp. 199–204, 1998. View at Publisher · View at Google Scholar
  35. N. D. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, 1992.
  36. J. Bricmont, A. Kupiainen, and G. Lin, “Renormalization group and asymptotics of solutions of nonlinear parabolic equations,” Communications on Pure and Applied Mathematics, vol. 47, no. 6, pp. 893–922, 1994. View at Publisher · View at Google Scholar
  37. N. V. Antonov and J. Honkonen, “Field-theoretic renormalization group for a nonlinear diffusion equation,” Physical Review E, vol. 66, no. 4, Article ID 046105, 7 pages, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. S. Yoshida and T. Fukui, “Exact renormalization group approach to a nonlinear diffusion equation,” Physical Review E, vol. 72, no. 4, Article ID 046136, 2005. View at Publisher · View at Google Scholar · View at MathSciNet