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

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Abstract

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.

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References

  1. Ablowitz, M.J., Segur, H. Solitons and the inverse scattering transform. SIAM, Philiadelphia, 1981

    Book  MATH  Google Scholar 

  2. Aronson, D.G., Weiberger, H.F. Multidimentional nonlinear diffusion arising in population genetics. Adv. in Math., 30: 33–76 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benney, D.J. Long waves on liquid films. J. Math. Phys., 45: 150–155 (1966)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  7. Canosa, J., Gazdag, J. The Korteweg-de Vries-Burgers equation. J. Comput. Phys., 23: 393–403 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dai, S.Q. Approximate analytical solutions for some strong nonlinear problems. Science in China Series A, 2: 43–52 (1990)

    Google Scholar 

  10. Dai, S.Q. Solitary waves at the interface of a two-layer fluid. Appl. Math. Mech., 3: 721–731 (1982)

    Google Scholar 

  11. Dey, B. Domain wall solutions of KdV like equations with higher order nonlinearity. J. Phys. A, 19: L9–L12 (1986)

    Article  MATH  Google Scholar 

  12. Dood, R.K. Solitons and nonlinear wave equations. Academic Press Inc Ltd, London, 1982

    Google Scholar 

  13. Feng, Z.S. On explicit exact solutions to the compound Burgers-KdV equations, Phys. Lett. A, 293: 57–66 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fife, P.C. Mathematical aspects of reacting and diffusing systems. Lect. notes in biomath., 28, Springer-Verlag, New York, 1979

    Book  MATH  Google Scholar 

  15. Gao, G. A theory of interaction between dissipation and dispersion of turbulence. Science in China Series A, 28: 616–627 (1985)

    MATH  Google Scholar 

  16. Grad, H., Hu, P.N. Unified shock profile in a plasma. Phys. Fluids, 10: 2596–2602 (1967)

    Article  Google Scholar 

  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. Hu, P.N. Collisional theory of shock and nonlinear waves in a plasma. Phys. Fluids, 15: 854–864 (1972)

    Article  Google Scholar 

  19. Johnson, R.S. A modern introduction to the mathematical theory of water waves. Cambridge University Press, Cambridge, 1997

    Book  MATH  Google Scholar 

  20. Johnson, R.S. A nonlinear incorporating damping and dispersion. J. Fluid Mech., 42: 49–60 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  21. Johnson, R.S. Shallow water waves on a viscous fluid-the undular bore. Phys. Fluids, 15: 1693–1699 (1972)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  23. Konno, K., Ichikawa, Y.H. A modified Korteweg-de Vries equation for ion acoustic waves. J. Phys. Soc. Japan, 37: 1631–1636 (1974)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  25. Liu, S.D., Liu, S.K. KdV-Burgers equation modelling of turbulence. Science in China Series A, 35: 576–586 (1992)

    MATH  Google Scholar 

  26. Liu, S.D., Liu, S.K. Solitons and Turbulent Flows. Shanghai Science and Education Press, Shanghai, 1994 (in Chinese)

    Google Scholar 

  27. Narayanamurti, V., Varma, C.M. Nonlinear propagation of heat pulses in solids. Phys. Rev. Lett., 25: 1105–1108 (1970)

    Article  Google Scholar 

  28. Nemytskii, V., Stepanov, V. Qualitative theory of differential equations. Dover, New York, 1989

    Google Scholar 

  29. Pan, X.D. Solitary wave and similarity solutions of the combined KdV equation. Appl. Math. Mech., 9: 281–285 (1988)

    Google Scholar 

  30. Raadu, M., Chanteur, G. Formation of double layers: shock like solutions of an mKdV-equation. Phys. Scripta, 33: 240–245 (1986)

    Article  Google Scholar 

  31. Tappert, F.D., Varma, C.M. Asymptotic theory of self-trapping of heat pulses in solids. Phys. Rev. Lett., 25: 1108–1111 (1970)

    Article  MathSciNet  Google Scholar 

  32. Wadati, M. Wave propagation in nonlinear lattice, I, II. J. Phys. Soc. Japan, 38: 673–686 (1975)

    Article  MathSciNet  Google Scholar 

  33. Wang, M.L. Exact solution for a compound KdV-Burgers equation. Phys. Lett. A, 213: 279–287 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Whitham, G.B. Linear and nonlinear wave. Springer-Verlag, New York, 1974

    Google Scholar 

  35. Wijngaarden, L.V. On the motion of gas bubbles in a perfect fluid. Ann. Rev. Fluid Mech., 4: 369–373 (1972)

    Article  Google Scholar 

  36. Xiong, S.L. An analytic solution of Burgers-KdV equation. Chinese Science Bulletin, 1: 26–29 (1989)

    Google Scholar 

  37. Ye, Q.X., Li, Z.Y. Introduction of reaction diffusion equations. Science Press, Beijing, 1990 (in Chinese)

    Google Scholar 

  38. Zhang, W.G. Exact solutions of the Burgers-combined KdV mixed type equation. Acta Math. Sci., 16: 241–248 (1996)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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, 1992

    MATH  Google Scholar 

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Authors and Affiliations

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

    Wei-guo Zhang & Yan Zhao

  2. College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Yan Zhao

  3. College of Foundation, Shanghai University of Engineering Science, Shanghai, 201600, China

    Xiao-yan Teng

Authors
  1. Wei-guo Zhang
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  2. Yan Zhao
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  3. Xiao-yan Teng
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Correspondence to Wei-guo Zhang.

Additional information

Supported by the National Natural Science Foundation of China (No. 11071164), Shanghai Natural Science Foundation Project (No. 10ZR1420800) and Leading Academic Discipline Project of Shanghai Municipal Government (No. S30501).

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Zhang, Wg., Zhao, Y. & Teng, Xy. Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates. Acta Math. Appl. Sin. Engl. Ser. 28, 305–324 (2012). https://doi.org/10.1007/s10255-012-0147-5

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  • DOI: https://doi.org/10.1007/s10255-012-0147-5

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