Skip to main content
SpringerLink
Log in

Qualitative analysis and approximate damped oscillatory solutions for a kind of nonlinear dispersive-dissipative equation

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

This paper makes qualitative analysis to the bounded traveling wave solutions for a kind of nonlinear dispersive-dissipative equation, and considers its solving problem. The relation is investigated between behavior of its solution and the dissipation coefficient. Further, all approximate damped oscillatory solutions when dissipation coefficient is small are presented by utilizing the method of undetermined coefficients according to the theory of rotated vector field in planar dynamical systems. Finally, error estimate is given by establishing the integral equation which reflects the relation between approximate and exact damped oscillatory solutions applying the idea of homogenization principle.

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. Bona, J.L., Schonbek, M.E. Travelling wave solutions to the Korteweg-de Vries-Burgers equation. Proc. R. Soc. Edin., 101A: 207–226 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cherkas, L.A. Estimation of the number of limit cycles of autonomous systems. Diff. Eqs., 13: 529–547 (1977)

    MATH  Google Scholar 

  3. Isidore, N. Exact solutions of a nonlinear dispersive-dissipative equation. J. Phys. A: Math. Gen., 29: 3679–3682 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jeffery, A., Kakutani, T. Stability of the Burgers shock wave and the Korteweg-de Vries soliton. Indiana Univ. Math. Journal, 20: 463–468 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  5. Korteweg, D.J., de Vries, G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag., 39: 422–443 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kakutani, T., Kawahara, T. Weak ion-acoustic shock waves. J. Phys. Soc. Jpn., 29: 1068–1073 (1970)

    Article  Google Scholar 

  7. Malfliet, M. The tanh method in nonlinear wave theory. Habilitation Thesis, University of Antwerp, Antwerp, Belgium, 1994

    Google Scholar 

  8. Nemytskii, V., Stepanov, V. Qualitative Theory of Differential Equations. Dover, New York, 1989

    MATH  Google Scholar 

  9. Octavio, C.P. Traveling wave solutions for some factorized nonlinear PDEs. J. Phys. A: Math. Theor., 42: 035204 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, D.S., Li, H. Single and multi-solitary wave solutions to a class of nonlinear evolution equations. J. Math. Anal. Appl., 343: 273–298 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhang, W., Chang, Q., Jiang, B. Explicit exact solitary-wave solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order. Chaos. Soliton. Fract., 13: 311–319 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhang, Z.F., Ding, T.R., Huang, W.Z., Dong, Z.X. Qualitative Theory of Differential Equations. Translations of Mathematical Monographs, Volume 101, American Mathematical Society, Providence, 1992

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Business School, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Shao-wei Li & Zheng-ming Li

  2. School of Mathematics and Information Engineering, Taizhou University, Taizhou, 317000, China

    Shao-wei Li

  3. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Wei-guo Zhang, Yan Zhao & Xiang Li

Authors
  1. Shao-wei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei-guo Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yan Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zheng-ming Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei-guo Zhang.

Additional information

Supported by the National Natural Science Foundation of China (No.11471215).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Sw., Zhang, Wg., Zhao, Y. et al. Qualitative analysis and approximate damped oscillatory solutions for a kind of nonlinear dispersive-dissipative equation. Acta Math. Appl. Sin. Engl. Ser. 33, 1–24 (2017). https://doi.org/10.1007/s10255-017-0632-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-017-0632-y

Keywords

2000 MR Subject Classification

Search

Navigation