Table of Contents
Physics Research International
Volume 2014, Article ID 786965, 4 pages
http://dx.doi.org/10.1155/2014/786965
Research Article

Full Symmetry Groups and Exact Solutions to BKP and GKP Equations

1Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China
2Department of Physics, Shanghai Jiao Tong University, Shanghai 200040, China

Received 1 July 2014; Accepted 9 September 2014; Published 18 September 2014

Academic Editor: Juan Jose Hernandez-Rey

Copyright © 2014 Bo Ren and Jian-Yong Wang. 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. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  2. R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  3. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Physical Review Letters, vol. 19, no. 19, pp. 1095–1097, 1967. View at Publisher · View at Google Scholar · View at Scopus
  4. J. Lin, B. Ren, H. Li, and Y. Li, “Soliton solutions for two nonlinear partial differential equations using a Darboux transformation of the Lax pairs,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 77, no. 3, Article ID 036605, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. R. Bo and L. Ji, “Painlevé properties and exact solutions for the high-dimensional Schwartz Boussinesq equation,” Chinese Physics B, vol. 18, no. 3, pp. 1161–1167, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. P. J. Olver, Application of Lie Group to Differential Equation, Springer, Berlin, Germany, 1986.
  7. G. W. Bluman and J. D. Cole, “The general similarity solution of the heat equation,” Journal of Applied Mathematics and Mechanics, vol. 18, pp. 1025–1042, 1969. View at Google Scholar · View at MathSciNet
  8. P. A. Clarkson and M. D. Kruskal, “New similarity reductions of the Boussinesq equation,” Journal of Mathematical Physics, vol. 30, no. 10, pp. 2201–2213, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. S. Y. Lou and H. C. Ma, “Non-Lie symmetry groups of (2+1)-dimensional nonlinear systems obtained from a simple direct method,” Journal of Physics A: Mathematical and General, vol. 38, no. 7, pp. L129–L137, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. B. Ren, X. J. Xu, and J. Lin, “Symmetry group and exact solutions for the 2+1 dimensional Ablowitz-Kaup-Newell-Segur equation,” Journal of Mathematical Physics, vol. 50, Article ID 123505, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  11. B. Li, W. Ye, and Y. Chen, “Symmetry, full symmetry groups, and some exact solutions to a generalized Davey-Stewartson system,” Journal of Mathematical Physics, vol. 49, no. 10, Article ID 103503, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. H. Ma, A. Deng, and Y. Wang, “Exact solution of a KdV equation with variable coefficients,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2278–2280, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. L. Wang and Z. Dong, “Symmetry reduction and exact solutions of the (3+1)-dimensional breaking soliton equation,” Communications in Theoretical Physics, vol. 50, no. 4, pp. 832–840, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. Z.-Z. Dong, C. Yong, and Y.-H. Lang, “Symmetry reduction and exact solutions of the (3+1)-dimensional Zakharov-Kuznetsov equation,” Chinese Physics B, vol. 19, Article ID 090205, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. A. Wazwaz, “Multiple-front solutions for the Burgers-Kadomtsev-Petviashvili equation,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 437–443, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. A. M. Wazwaz, “Solitons and singular solitons for the Gardner-KP equation,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 162–169, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. N. Taghizadeh, M. Mirzazadeh, and F. Farahrooz, “Exact soliton solutions of the modified KdV-KP equation and the Burgers-{KP} equation by using the first integral method,” Applied Mathematical Modelling, vol. 35, no. 8, pp. 3991–3997, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. B. Xu and X. Liu, “Classification, reduction, group invariant solutions and conservation laws of the Gardner-KP equation,” Applied Mathematics and Computation, vol. 215, no. 3, pp. 1244–1250, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. B. Ren and J. Lin, “A new (2+1)-dimensional integrable equation,” Communications in Theoretical Physics, vol. 51, no. 1, pp. 13–16, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. S. Y. Lou and X. B. Hu, “Infinitely many Lax pairs and symmetry constraints of the KP equation,” Journal of Mathematical Physics, vol. 38, no. 12, pp. 6401–6427, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. A. Maccari, “The investigation into new integrable systems of equations in 2+1-dimensions,” Journal of Mathematical Physics, vol. 44, no. 1, pp. 242–250, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus