Approximate damped oscillatory solutions for compound KdVBurgers equation and their error estimates
 Weiguo Zhang,
 Yan Zhao,
 Xiaoyan Teng
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Abstract
In this paper, we focus on studying approximate solutions of damped oscillatory solutions of the compound KdVBurgers equation and their error estimates. We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdVBurgers 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 KdVBurgers 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.
 Ablowitz, M.J., Segur, H. Solitons and the inverse scattering transform. SIAM, Philiadelphia, 1981 CrossRef
 Aronson, D.G., Weiberger, H.F. Multidimentional nonlinear diffusion arising in population genetics. Adv. in Math., 30: 33–76 (1978) CrossRef
 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) CrossRef
 Benney, D.J. Long waves on liquid films. J. Math. Phys., 45: 150–155 (1966)
 Bona, J.L., Dougalia, V.A. An initialand boundaryvalue problem for a model equation for propagation of long waves. J. Math. Anal. Appl., 75: 513–522 (1980) CrossRef
 Bona, J.L., Schonbek, M.E. Travelling wave solutions to the Kortewegde VriesBurgers equation. Proc. R. Soc. Edin., 101A: 207–226 (1985) CrossRef
 Canosa, J., Gazdag, J. The Kortewegde VriesBurgers equation. J. Comput. Phys., 23: 393–403 (1977) CrossRef
 Coffey, M.W. On series expansions giving closedform solutions of Kortewegde Vrieslike equations. SIAM J. Appl. Math., 50: 1580–1592 (1990) CrossRef
 Dai, S.Q. Approximate analytical solutions for some strong nonlinear problems. Science in China Series A, 2: 43–52 (1990)
 Dai, S.Q. Solitary waves at the interface of a twolayer fluid. Appl. Math. Mech., 3: 721–731 (1982)
 Dey, B. Domain wall solutions of KdV like equations with higher order nonlinearity. J. Phys. A, 19: L9–L12 (1986) CrossRef
 Dood, R.K. Solitons and nonlinear wave equations. Academic Press Inc Ltd, London, 1982
 Feng, Z.S. On explicit exact solutions to the compound BurgersKdV equations, Phys. Lett. A, 293: 57–66 (2002) CrossRef
 Fife, P.C. Mathematical aspects of reacting and diffusing systems. Lect. notes in biomath., 28, SpringerVerlag, New York, 1979 CrossRef
 Gao, G. A theory of interaction between dissipation and dispersion of turbulence. Science in China Series A, 28: 616–627 (1985)
 Grad, H., Hu, P.N. Unified shock profile in a plasma. Phys. Fluids, 10: 2596–2602 (1967) CrossRef
 Guan, K.Y., Gao, G. Qualitative Analysis for the travelling wave solutions of BurgersKdV mixed type equation. Science in China Series A, 30: 64–73 (1987)
 Hu, P.N. Collisional theory of shock and nonlinear waves in a plasma. Phys. Fluids, 15: 854–864 (1972) CrossRef
 Johnson, R.S. A modern introduction to the mathematical theory of water waves. Cambridge University Press, Cambridge, 1997 CrossRef
 Johnson, R.S. A nonlinear incorporating damping and dispersion. J. Fluid Mech., 42: 49–60 (1970) CrossRef
 Johnson, R.S. Shallow water waves on a viscous fluidthe undular bore. Phys. Fluids, 15: 1693–1699 (1972) CrossRef
 Karahara, T. Weak nonlinear magnetoacoustic waves in a cold plasma in the presence of effective electronion collisions. J. Phys. Soc. Japan, 27: 1321–1329 (1970) CrossRef
 Konno, K., Ichikawa, Y.H. A modified Kortewegde Vries equation for ion acoustic waves. J. Phys. Soc. Japan, 37: 1631–1636 (1974) CrossRef
 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) CrossRef
 Liu, S.D., Liu, S.K. KdVBurgers equation modelling of turbulence. Science in China Series A, 35: 576–586 (1992)
 Liu, S.D., Liu, S.K. Solitons and Turbulent Flows. Shanghai Science and Education Press, Shanghai, 1994 (in Chinese)
 Narayanamurti, V., Varma, C.M. Nonlinear propagation of heat pulses in solids. Phys. Rev. Lett., 25: 1105–1108 (1970) CrossRef
 Nemytskii, V., Stepanov, V. Qualitative theory of differential equations. Dover, New York, 1989
 Pan, X.D. Solitary wave and similarity solutions of the combined KdV equation. Appl. Math. Mech., 9: 281–285 (1988)
 Raadu, M., Chanteur, G. Formation of double layers: shock like solutions of an mKdVequation. Phys. Scripta, 33: 240–245 (1986) CrossRef
 Tappert, F.D., Varma, C.M. Asymptotic theory of selftrapping of heat pulses in solids. Phys. Rev. Lett., 25: 1108–1111 (1970) CrossRef
 Wadati, M. Wave propagation in nonlinear lattice, I, II. J. Phys. Soc. Japan, 38: 673–686 (1975) CrossRef
 Wang, M.L. Exact solution for a compound KdVBurgers equation. Phys. Lett. A, 213: 279–287 (1996) CrossRef
 Whitham, G.B. Linear and nonlinear wave. SpringerVerlag, New York, 1974
 Wijngaarden, L.V. On the motion of gas bubbles in a perfect fluid. Ann. Rev. Fluid Mech., 4: 369–373 (1972) CrossRef
 Xiong, S.L. An analytic solution of BurgersKdV equation. Chinese Science Bulletin, 1: 26–29 (1989)
 Ye, Q.X., Li, Z.Y. Introduction of reaction diffusion equations. Science Press, Beijing, 1990 (in Chinese)
 Zhang, W.G. Exact solutions of the Burgerscombined KdV mixed type equation. Acta Math. Sci., 16: 241–248 (1996)
 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) CrossRef
 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
 Title
 Approximate damped oscillatory solutions for compound KdVBurgers equation and their error estimates
 Journal

Acta Mathematicae Applicatae Sinica, English Series
Volume 28, Issue 2 , pp 305324
 Cover Date
 20120401
 DOI
 10.1007/s1025501201475
 Print ISSN
 01689673
 Online ISSN
 16183932
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 compound KdVBurgers equation
 qualitative analysis
 solitary wave solution
 damped oscillatory solution
 error estimate
 34C05
 34C37
 35Q51
 37C29
 Authors

 Weiguo Zhang ^{(1)}
 Yan Zhao ^{(1)} ^{(2)}
 Xiaoyan Teng ^{(3)}
 Author Affiliations

 1. School of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China
 2. College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing, 210044, China
 3. College of Foundation, Shanghai University of Engineering Science, Shanghai, 201600, China