Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates



In this paper, we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates. We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation. We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions. Furthermore, we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation. We obtain two critical values of r, and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value, while it appears as a damped oscillatory wave if |r| is less than some critical value. By means of analysis and the undetermined coefficients method, we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation. Based on the above discussions and according to the evolution relations of orbits in the global phase portraits, we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method. Finally, using the homogenization principle, we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions. Moreover, we also give the error estimates for these approximate solutions.


compound KdV-Burgers equation qualitative analysis solitary wave solution damped oscillatory solution error estimate 

2000 MR Subject Classification

34C05 34C37 35Q51 37C29 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ablowitz, M.J., Segur, H. Solitons and the inverse scattering transform. SIAM, Philiadelphia, 1981CrossRefMATHGoogle Scholar
  2. [2]
    Aronson, D.G., Weiberger, H.F. Multidimentional nonlinear diffusion arising in population genetics. Adv. in Math., 30: 33–76 (1978)MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    Benjamin, T.B., Bona, J.L., Mahony, J.J. Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. London, Ser. A, 272: 47–78 (1972)MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Benney, D.J. Long waves on liquid films. J. Math. Phys., 45: 150–155 (1966)MathSciNetMATHGoogle Scholar
  5. [5]
    Bona, J.L., Dougalia, V.A. An initial-and boundary-value problem for a model equation for propagation of long waves. J. Math. Anal. Appl., 75: 513–522 (1980)CrossRefGoogle Scholar
  6. [6]
    Bona, J.L., Schonbek, M.E. Travelling wave solutions to the Korteweg-de Vries-Burgers equation. Proc. R. Soc. Edin., 101A: 207–226 (1985)MathSciNetCrossRefGoogle Scholar
  7. [7]
    Canosa, J., Gazdag, J. The Korteweg-de Vries-Burgers equation. J. Comput. Phys., 23: 393–403 (1977)MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Coffey, M.W. On series expansions giving closed-form solutions of Korteweg-de Vries-like equations. SIAM J. Appl. Math., 50: 1580–1592 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Dai, S.Q. Approximate analytical solutions for some strong nonlinear problems. Science in China Series A, 2: 43–52 (1990)Google Scholar
  10. [10]
    Dai, S.Q. Solitary waves at the interface of a two-layer fluid. Appl. Math. Mech., 3: 721–731 (1982)Google Scholar
  11. [11]
    Dey, B. Domain wall solutions of KdV like equations with higher order nonlinearity. J. Phys. A, 19: L9–L12 (1986)CrossRefMATHGoogle Scholar
  12. [12]
    Dood, R.K. Solitons and nonlinear wave equations. Academic Press Inc Ltd, London, 1982Google Scholar
  13. [13]
    Feng, Z.S. On explicit exact solutions to the compound Burgers-KdV equations, Phys. Lett. A, 293: 57–66 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Fife, P.C. Mathematical aspects of reacting and diffusing systems. Lect. notes in biomath., 28, Springer-Verlag, New York, 1979CrossRefMATHGoogle Scholar
  15. [15]
    Gao, G. A theory of interaction between dissipation and dispersion of turbulence. Science in China Series A, 28: 616–627 (1985)MATHGoogle Scholar
  16. [16]
    Grad, H., Hu, P.N. Unified shock profile in a plasma. Phys. Fluids, 10: 2596–2602 (1967)CrossRefGoogle Scholar
  17. [17]
    Guan, K.Y., Gao, G. Qualitative Analysis for the travelling wave solutions of Burgers-KdV mixed type equation. Science in China Series A, 30: 64–73 (1987)Google Scholar
  18. [18]
    Hu, P.N. Collisional theory of shock and nonlinear waves in a plasma. Phys. Fluids, 15: 854–864 (1972)CrossRefGoogle Scholar
  19. [19]
    Johnson, R.S. A modern introduction to the mathematical theory of water waves. Cambridge University Press, Cambridge, 1997CrossRefMATHGoogle Scholar
  20. [20]
    Johnson, R.S. A nonlinear incorporating damping and dispersion. J. Fluid Mech., 42: 49–60 (1970)MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    Johnson, R.S. Shallow water waves on a viscous fluid-the undular bore. Phys. Fluids, 15: 1693–1699 (1972)CrossRefMATHGoogle Scholar
  22. [22]
    Karahara, T. Weak nonlinear magneto-acoustic waves in a cold plasma in the presence of effective electronion collisions. J. Phys. Soc. Japan, 27: 1321–1329 (1970)CrossRefGoogle Scholar
  23. [23]
    Konno, K., Ichikawa, Y.H. A modified Korteweg-de Vries equation for ion acoustic waves. J. Phys. Soc. Japan, 37: 1631–1636 (1974)CrossRefGoogle Scholar
  24. [24]
    Korteweg, D.J., de Vries, G. On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. Mag., 39: 422–443 (1895)CrossRefMATHGoogle Scholar
  25. [25]
    Liu, S.D., Liu, S.K. KdV-Burgers equation modelling of turbulence. Science in China Series A, 35: 576–586 (1992)MATHGoogle Scholar
  26. [26]
    Liu, S.D., Liu, S.K. Solitons and Turbulent Flows. Shanghai Science and Education Press, Shanghai, 1994 (in Chinese)Google Scholar
  27. [27]
    Narayanamurti, V., Varma, C.M. Nonlinear propagation of heat pulses in solids. Phys. Rev. Lett., 25: 1105–1108 (1970)CrossRefGoogle Scholar
  28. [28]
    Nemytskii, V., Stepanov, V. Qualitative theory of differential equations. Dover, New York, 1989Google Scholar
  29. [29]
    Pan, X.D. Solitary wave and similarity solutions of the combined KdV equation. Appl. Math. Mech., 9: 281–285 (1988)Google Scholar
  30. [30]
    Raadu, M., Chanteur, G. Formation of double layers: shock like solutions of an mKdV-equation. Phys. Scripta, 33: 240–245 (1986)CrossRefGoogle Scholar
  31. [31]
    Tappert, F.D., Varma, C.M. Asymptotic theory of self-trapping of heat pulses in solids. Phys. Rev. Lett., 25: 1108–1111 (1970)MathSciNetCrossRefGoogle Scholar
  32. [32]
    Wadati, M. Wave propagation in nonlinear lattice, I, II. J. Phys. Soc. Japan, 38: 673–686 (1975)MathSciNetCrossRefGoogle Scholar
  33. [33]
    Wang, M.L. Exact solution for a compound KdV-Burgers equation. Phys. Lett. A, 213: 279–287 (1996)MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    Whitham, G.B. Linear and nonlinear wave. Springer-Verlag, New York, 1974Google Scholar
  35. [35]
    Wijngaarden, L.V. On the motion of gas bubbles in a perfect fluid. Ann. Rev. Fluid Mech., 4: 369–373 (1972)CrossRefGoogle Scholar
  36. [36]
    Xiong, S.L. An analytic solution of Burgers-KdV equation. Chinese Science Bulletin, 1: 26–29 (1989)Google Scholar
  37. [37]
    Ye, Q.X., Li, Z.Y. Introduction of reaction diffusion equations. Science Press, Beijing, 1990 (in Chinese)Google Scholar
  38. [38]
    Zhang, W.G. Exact solutions of the Burgers-combined KdV mixed type equation. Acta Math. Sci., 16: 241–248 (1996)MATHGoogle Scholar
  39. [39]
    Zhang, W.G., Li, S.W., Zhang, L., Ning, T.K. New solitary wave solutions and periodic wave solutions for the compound KdV equation. Chaos, Solitons and Fractals, 39: 143–149 (2009)MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    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, 1992MATHGoogle Scholar

Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of ScienceUniversity of Shanghai for Science and TechnologyShanghaiChina
  2. 2.College of Mathematics and PhysicsNanjing University of Information Science and TechnologyNanjingChina
  3. 3.College of FoundationShanghai University of Engineering ScienceShanghaiChina

Personalised recommendations