Skip to main content
SpringerLink
Log in

Painlevé analysis, auto-Bäcklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries (KdV) equation

  • Statistical and Nonlinear Physics
  • Published:
The European Physical Journal B - Condensed Matter and Complex Systems Aims and scope Submit manuscript

Abstract.

There has been considerable interest in the study on the variable-coefficient nonlinear evolution equations in recent years, since they can describe the real situations in many fields of physical and engineering sciences. In this paper, a generalized variable-coefficient KdV (GvcKdV) equation with the external-force and perturbed/dissipative terms is investigated, which can describe the various real situations, including large-amplitude internal waves, blood vessels, Bose-Einstein condensates, rods and positons. The Painlevé analysis leads to the explicit constraint on the variable coefficients for such a equation to pass the Painlevé test. An auto-Bäcklund transformation is provided by use of the truncated Painlevé expansion and symbolic computation. Via the given auto-Bäcklund transformation, three families of analytic solutions are obtained, including the solitonic and periodic solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge Univ. Press, Cambridge, 1991)

  • B. Tian, Y.T. Gao, Eur. Phys. J. B 22, 351 (2001); B. Tian, Y.T. Gao, Phys. Lett. A 340, 243 (2005); B. Tian, Y.T. Gao, Eur. Phys. J. D 33, 59 (2005); Y.T. Gao, B. Tian, Phys. Lett. A 349, 314 (2006)

    Article  ADS  Google Scholar 

  • G. Das, J. Sarma, Phys. Plasmas 5, 3918 (1998); G. Das, J. Sarma, Phys. Plasmas 6, 4394 (1999); H. Hong, J. Lee, Phys. Plasmas 6, 3422 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  • A. Ludu, J. Draayer, Phys. Rev. Lett. 80, 2125 (1998); L. Reatto, D. Galli, Int. J. Mod. Phys. B 13, 607 (1999)

    Article  ADS  Google Scholar 

  • A. Osborne, Chaos, Solitons & Fractals 5, 2623 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • L. Ostrovsky, Yu. A. Stepanyants, Rev. Geophys. 27, 293 (1989)

    ADS  Google Scholar 

  • C. Gardner, J. Greene, M. Kruskal, R. Miura, Comm. Pure Appl. Math. 27, 97 (1974)

    MATH  MathSciNet  Google Scholar 

  • J. Weiss, M. Tabor, G. Carnevale, J. Math. Phys. 24, 522 (1983)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • M. Lakshmanan, P. Kaliappan, J. Math. Phys. 24, 795 (1983)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • P. Clarkson, M. Kruskal, J. Math. Phys. 30, 2201 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • M. Coffey, Phys. Rev. B 54, 1279 (1996)

    Article  ADS  Google Scholar 

  • B. Tian, W.R. Shan, C.Y. Zhang, G.M. Wei, Y.T. Gao, Eur. Phys. J. B (Rapid Note) 47, 329 (2005); B. Tian, Y.T. Gao, Phys. Plasmas (Lett.) 12, 070703 (2005); Y.T. Gao, B. Tian, Phys. Plasmas (Lett.), in press 2006 (AIP ID: 009611PHP)

    Article  ADS  Google Scholar 

  • S. Turitsyn, A. Aceves, C. Jones, V. Zharnitsky, Phys. Rev. E 58, R48 (1998); B. Tian, Y.T. Gao, Phys. Lett. A 342, 228 (2005); Phys. Lett. A 359, 241 (2006)

    Google Scholar 

  • P. Holloway, E. Pelinovsky, T. Talipova, J. Geophys. Res. C 104, 18333 (1999)

    Article  ADS  Google Scholar 

  • E. Pelinovsky, T. Talipova, V. Ivanov, Nonlinear Processes Geophys. 2, 80 (1995)

    ADS  Google Scholar 

  • P. Holloway, E. Pelinovsky, T. Talipova, B. Barnes, J. Phys. Oceanogr. 27, 871 (1997)

    Article  ADS  Google Scholar 

  • T. Talipova, E. Pelinovsky, T. Kouts, Oceanology 38, 33 (1998)

    Google Scholar 

  • B. Tian, G.M. Wei, C.Y. Zhang, W.R. Shan, Y.T. Gao, Phys. Lett. A 356, 8 (2006)

    Article  ADS  MATH  Google Scholar 

  • H.A. Erbay, S. Erbay, S. Dost, Acta Mech. 95, 87 (1992); H. Demiray, Math. Computer Modelling 39, 151 (2004); I. Bakirtas, H. Demiray, Int. J. Non-Linear Mech. 40, 785 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • H. Demiray, Int. J. Engng. Sci. 42, 203 (2004)

    Article  MathSciNet  Google Scholar 

  • M. Olufsen, Stud. Health Technol. Inform. 71, 79 (2000); A. Quarteroni, M. Tuveri, A. Veneziani, Comp. Visual Sci. 2, 163 (2000); M. Zamir, The Physics of Pulsatile Flow (Springer-Verlag, New York, 2000)

    Google Scholar 

  • R.C. Cascaval, in Evolution Equations, edited by G.R. Goldstein, R. Nagel, S. Romanelli, Lecture Notes Pure Appl. Math., Vol. 234 (Marcel Dekker, 2003)

  • C.C. Bradley, C.A. Sackett, J.J. Tollett, R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269, 198 (1995); F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71, 463 (1999); A.J. Leggett, Rev. Mod. Phys. 73, 307 (2001)

    Article  ADS  Google Scholar 

  • Y. Wu, X. Yang, C.P. Sun, X.J. Zhou, Y.Q. Wang, Phys. Rev. A 61, 043604 (2000); W.M. Liu, B. Wu, Q. Niu, Phys. Rev. Lett. 84, 2294 (2000); I. Bloch, M. Kohl, M. Greiner, T. W. Hansch, T. Esslinger, Phys. Rev. Lett. 87, 030401 (2001) N. Robins, C. Savage, E.A. Ostrovskaya, Phys. Rev. A 64, 043605 (2001); J.R. Abo-Shaeer, C. Raman, J.M. Vogels, W. Ketterle, Science 292, 476 (2001); Z. Dutton, M. Budde, C. Slowe, L.V. Hau, Science 293, 663 (2001)

    Article  ADS  Google Scholar 

  • G.X. Huang, J. Szeftel, S.H. Zhu, Phys. Rev. A 65, 053605 (2002)

    Article  ADS  Google Scholar 

  • D.J. Frantzeskakis, N.P. Proukakis, P.G. Kevrekidis, Phys. Rev. A 70, 015601 (2004)

    Article  ADS  Google Scholar 

  • A.M. Samsonov, E.V. Sokurinskaya, J. Appl. Math. Mech. 51, 376 (1988); H. Cohenn H.H. Dai, Acta Mech. 100, 223 (1993); A.M. Samsonov, in Nonlinear Waves in Solids, edited by A. Jeffrey, J. Engelbreght (Springer-Verlag, New York, 1994); A.M. Samsonov, G.V. Drcidcn, A.V. Proubov, I.V. Scmcnova, Phys. Rev. B 57, 5778 (1998)

    Article  MathSciNet  Google Scholar 

  • H.H. Dai, Y. Huo, Wave Motion 35, 55 (2002)

    Article  MATH  Google Scholar 

  • F. Capasso, C. Sirtori, J. Faist, D.L. Sivco, S.N.G. Cho, Nature 358, 565 (1992)

    Article  ADS  Google Scholar 

  • Y. Chen, B. Li, H.Q. Zhang, Int. J. Mod. Phys. C 14, 99 & 471 (2003); X.D. Zheng, Y. Chen, B. Li, H.Q. Zhang, Comm. Theor. Phys. 39, 647 (2003); Y. Chen, Z. Yu, Int. J. Mod. Phys. C 14, 601 (2003); Y.C. Xie, Chaos, Solitons & Fractal 20, 337 (2004)

    Article  MATH  ADS  Google Scholar 

  • Y.T. Gao, B. Tian, Int. J. Mod. Phys. C 12, 1431 (2001)

    Article  ADS  Google Scholar 

  • F. Calogero, A. Degasperis, Lett. Nuovo Cim. 23, 150 (1978)

    MathSciNet  Google Scholar 

  • R. Hirota, Phys. Lett. A 71, 393 (1979); M. Tajiri, S. Kawamoto, J. Phys. Soc. Jpn. 51, 1678 (1982); W.-H. Steeb, M. Kloke, B.M. Spieker, W. Oevel, J. Phys. A 16, L447 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  • B. Tian, Y.T. Gao, Phys. Lett. A 340, 449 (2005)

    Article  ADS  MATH  Google Scholar 

  • B. Tian, Y.T. Gao, Phys. Plasmas 12, 054701 (2005)

    Article  ADS  Google Scholar 

  • Z.T. Fu, S.D. Liu, S. K. Liu, Q. Zhao, Appl. Math. Mech. 25, 73 (2004)

    Article  MATH  Google Scholar 

  • V. Hlavaty, J. Phys. Soc. Jpn. 55, 1405 (1986); B. Baby, J. Phys. A 20, L555 (1987); P. Clarkson, IMA J. Appl. Math. 44, 27 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  • R. Grimshaw, Proc. R. Soc. Lond. A 368, 359 (1979); N. Joshi, Phys. Lett. A 125, 456 (1987); Z. Chen, B. Guo, L. Xiang, J. Math. Phys. 31, 2851 (1990); W. Hong, Y. Jung, Phys. Lett. A 257, 149 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • E.G. Fan, Phys. Lett. A 294, 26 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • N. Nirmala, M. Vedan, B. Baby, J. Math. Phys. 27, 2640 (1986); 2644 (1986)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • B. Tian, H. Li, Y.T. Gao, Z. Angew. Math. Phys. 56, 783 (2005); Y.T. Gao, B. Tian, C.Y. Zhang, Acta Mech. 182, 17 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • A. Ramani, B. Grammaticos, T. Bountis, Phys. Rep. 180, 159 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  • W.-H. Steeb, N. Eulur, Nonlinear Evolution Equations and Painlevé Test (World Scientific, Singapore, 1988)

  • E.L. Ince, Ordinary Differential Equations (Dover, New York, 1956)

  • E. Abellanas, A. Galindo, Phys. Lett. A 108, 123 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  • T. Brugarino, J. Math. Phys. 30, 1013 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • M. Musette, Painlevé Analysis for Nonlinear Partial Differential Equations in The Painlevé Property : One Century Later, edited by R. Conte (Springer-Verlag, New York, 1999)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and LMIB, Beijing University of Aeronautics and Astronautics, Beijing, 100083, China

    Guang-Mei Wei & Wei Hu

  2. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100083, China

    Guang-Mei Wei, Yi-Tian Gao & Chun-Yi Zhang

  3. CCAST (World Lab.), PO Box 8730, Beijing, 100080, China

    Yi-Tian Gao

  4. Meteorology Center of Air Force Command Post, Changchun, 130051, China

    Chun-Yi Zhang

Authors
  1. Guang-Mei Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi-Tian Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chun-Yi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guang-Mei Wei.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wei, GM., Gao, YT., Hu, W. et al. Painlevé analysis, auto-Bäcklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries (KdV) equation. Eur. Phys. J. B 53, 343–350 (2006). https://doi.org/10.1140/epjb/e2006-00378-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2006-00378-3

PACS.

Search

Navigation