Exact solutions to the KdV–Burgers’ equation. (English) Zbl 0833.35124

Summary: This paper presents two different methods for the construction of exact solutions to the KdVB equation. The first is a direct one based on a combination of solutions to the KdV equation and Burgers’ equation. In this approach, a number of unknown constants are involved, and it is shown that the equations leading to their determination are properly determined and are capable of solution. The second method involves a series, and is essentially an extension of Hirota’s method. This approach is capable of solving the KdVB equation exactly, and also of generalization to higher order equations with a KdVB-type nonlinearity.


35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
35C10 Series solutions to PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Full Text: DOI


[1] Wijngaarden, L. V., One-dimensional flow of liquids containing small gas bubbles, Ann. Rev. Fluid Mech., 4, 369-396 (1972) · Zbl 0243.76070
[2] Johnson, R. S., Nonlinear waves in fluid-filled elastic tubes and related problems, (PhD thesis (1969), University of London: University of London London)
[3] Johnson, R. S., A nonlinear equation incorporating damping and dispersion, J. Fluid Mech., 42, 49-60 (1970) · Zbl 0213.54904
[4] Grad, H.; Hu, P. W., Unified shock profile in a plasma, Phys. Fluids, 10, 2596-2602 (1967)
[5] Jeffrey, A., Some aspects of the mathematical modelling of long nonlinear waves, Arch. of Mechanics, 31, 559-574 (1979) · Zbl 0437.73010
[6] Canosa, J.; Gazdag, J., The Korteweg-de Vries-Burgers equation, J. Comput. Phys., 23, 393-403 (1977) · Zbl 0356.65107
[7] Bona, J. L.; Schonbek, M. E., Travelling wave solutions to Korteweg-de Vries-Burgers equation, Proc. Roy. Soc. Edinburgh, 101A, 207-226 (1985) · Zbl 0594.76015
[8] Gibbon, J. D.; Radmore, P.; Tabor, M.; Wood, D., The Painlevé and Hirota’s Method, Stud. Appl. Math., 72, 39-63 (1985) · Zbl 0581.35074
[9] Jeffrey, A.; Xu, S., Exact solutions to the Korteweg-de Vries-Burgers equation, Wave Motion, 11, 559-564 (1989) · Zbl 0698.35139
[10] Weiss, J.; Tabor, M.; Carnevale, G., The Painlevé property for partial differential equations, J. Math. Phys., 24, 522-526 (1983) · Zbl 0514.35083
[11] Cole, J. D., On a quasi-linear parabolic equation occuring in aerodynamics, Quart. Appl. Math., 9, 225-236 (1951) · Zbl 0043.09902
[12] Hopf, E., The partial differential equation \(u_t+ uu_x =μ u_{ xx } \), Comm. Pure Appl. Math., 3, 201-230 (1950) · Zbl 0039.10403
[13] Jeffrey, A.; Kakutani, T., Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation, SIAM Rev., 14, 582-643 (1972) · Zbl 0221.35038
[14] Jian-Jun, S., The proper analytical solution of the Korteweg-de Vries-Burgers equation, J. Phys. A: Math. Gen., 20, L49-L56 (1987) · Zbl 0663.35091
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