Skip to main content
SpringerLink
Log in

Painlevé Test, Generalized Symmetries, Bäcklund Transformations and Exact Solutions to the Third-Order Burgers’ Equations

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In this paper, the Painlevé analysis is performed on the physical form of the third-order Burgers’ equation, the Painlevé property and integrability (C-integrable) of the equation is verified. Then, the generalized symmetries of the equation are presented and the generalized symmetries of the other equation are given by the symmetry transformation method. The Bäcklund Transformations of the equations are constructed based on the symmetries, respectively. Furthermore, the exact explicit solutions to the equations are investigated in terms of the symmetries, Bäcklund transformations and transformations of the equations.

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

  1. Liu, H., Li, J., Zhang, Q.: Lie symmetry analysis and exact explicit solutions for general Burgers’ equation. J. Comput. Appl. Math. 228, 1–9 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Liu, H., Li, J., Liu, L.: The recursion operator method for generalized symmetries and Bäcklund transformations of the Burgers’ equations. J. Appl. Math. Comput. 42, 159–170 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Conte, R., Musette, M.: The Painlevé Handbook. Springer, Dordrecht (2008)

    MATH  Google Scholar 

  4. Weiss, J., Tabor, M., Carnevale, G.: The Painlevé property for partial differential equations. J. Math. Phys. 24, 522–526 (1983)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Newell, A., et al.: A unified approach to Painlevé expansions. Physics D 29, 1–68 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Cariello, F., Tabor, M.: The Painlevé expansions for nonintegrable evolution equations. Physics D 39, 77–94 (1989)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Clarkson, P.: Painlevé analysis and the complete integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation. IMA J. Appl. Math. 44, 27–53 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  8. Olver, P.: Applications of Lie groups to differential equations. Graduate Texts in Mathematics. Springer, New York (1993)

    Google Scholar 

  9. Bluman, G., Anco, S.: Symmetry and integration methods for differential equations. Applied Mathematical Sciences. Applied Mathematical Sciences, New York (2002)

    Google Scholar 

  10. Chou, T.: New strong symmetries, symmetries and Lie algebra for Burgers’ equation. Sci. China Ser. A 10, 1009–1018 (1987). (in Chinese)

    Google Scholar 

  11. Cheng, Y., et al.: Connections among symmetries, Bäcklund transformation and the Painlevé property for Burgers’ hierarchies. Acta Math. Appl. Sin. 14, 180–184 (1991). (in Chinese)

    MATH  Google Scholar 

  12. Liu, H., Li, J.: Lie symmetry analysis and exact solutions for the short pulse equation. Nonlinear Anal. TMA 71, 2126–2133 (2009)

    Article  MATH  Google Scholar 

  13. Liu, H., Li, J., Liu, L.: Painlevé analysis. Lie symmetries, and exact solutions for the time-dependent coefficients Gardner equations. Nonlinear Dyn. 59, 497–502 (2010)

    Article  MATH  Google Scholar 

  14. Liu, H., Li, J., Liu, L.: Symmetry and conservation law classification and exact solutions to the generalized KdV types of equations. Int. J. Bifur. Chaos 22(1250188), 1–11 (2012)

    MATH  Google Scholar 

  15. Liu, H., Li, J., Liu, L.: Lie symmetry analysis, Bäcklund transformations and exact solutions to (2 + 1)-dimensional Burgers’ types of equations. Commun. Theor. Phys. 57, 737–742 (2012)

    Article  ADS  MATH  Google Scholar 

  16. Liu, H., Geng, Y.: Symmetry reductions and exact solutions to the systems of carbon nanotubes conveying fluid. J. Differ. Equ. 254, 2289–2303 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Fokas, A., Fuchssteiner, B.: On the structure of sympletic operators and hereditary symmetries. Lett. Nuovo Cimento 28, 299–303 (1980)

    Article  MathSciNet  Google Scholar 

  18. Tian, C.: Transformation of equations and transformation of symmetries. Acta Math. Appl. Sin. 12, 238–249 (1989). (in Chinese)

    MATH  Google Scholar 

  19. Calogero, F., Eckhaus, W.: Nonlinear evolution equations, rescalings, model PDEs and their integrability: I. Inverse Probl. 3, 229–262 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Wang, Z., Guo, D.: Introduction to special functions. The Series of Advanced Physics of Peking University. Peking University Press, Beijing (2000). (in Chinese)

    Google Scholar 

  21. Polyanin, A., Zaitsev, V.: Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. Chapman & Hall/CRC, Boca Raton (2003)

    MATH  Google Scholar 

Download references

Acknowledgments

We are grateful to the Editor and anonymous Reviewers comments on our paper. The author would like to thank Professor Wen Zhang for her invitation to visit Oakland University for hosting his research in 2014. This work is supported by the National Natural Science Foundation of China under Grant Nos. 11171041, 11401274, and the doctorial foundation of Liaocheng University under Grant No. 31805.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Liaocheng University, Liaocheng, 252059, Shandong, China

    Hanze Liu

  2. Department of Mathematics, Binzhou University, Binzhou, 256603, Shandong, China

    Hanze Liu

Authors
  1. Hanze Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hanze Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H. Painlevé Test, Generalized Symmetries, Bäcklund Transformations and Exact Solutions to the Third-Order Burgers’ Equations. J Stat Phys 158, 433–446 (2015). https://doi.org/10.1007/s10955-014-1130-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-014-1130-8

Keywords

Mathematics Subject Classification

Search

Navigation