Exact solutions of a non-linear fifth-order equation for describing waves on water

Journal of Applied Mathematics and Mechanics

Volumn 65, Issue 5, 2001, Pages 855-865

  Kudryashov, N A ^a   Sukharev, M B ^a  

^a NONE   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0348201928     PISSN: 00218928     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-8928(01)00091-0     Document Type: Article

References (42)

1. Arnol'd V.I. Mathematical Models of Classical Mechanics. 1989;Nauka, Moscow.
2. Landa P.S. Non-linear Vibrations and Waves. 1977;Nauka. Fizmatlit, Moscow.
3. Perelomov A.M. Integrable Systems of Classical Mechanics and Lie Algebra. 1990;Nauka, Moscow.
4. Golubev V.V. Lectures on the Analytical Theory of Differential Equations. 1941;Gostekhizdat, Moscow.
5. Golubev V.V. Lectures on the Integration of Equations of Motion of a Heavy Rigid Body about a Stationary Point. 1953;Gostekhizdat, Moscow.
6. Ince E.L. Ordinary Differential Equations. 1927;Longmans, Green and Co. London-New York.
7. Conte R. The Painlevé approach to nonlinear ordinary differential equations. Conte R. The Painlevé Property. 1999;77-180 Springer, New York. CRM Series in Mathematical Physics.
8. Gromak V.I., Lukashevich N.A. Analytical Properties of the Solutions of Painlevé Equations. 1990;Universitetskoye, Minsk.
9. Ablowitz M., Ramani A., Segur H. Nonlinear evolution equations and ordinary differential equations of Painlevé type. Lett. Nuovo Cimento. 23:1978;333-338.
10. Ablowitz M., Ramani A., Segur H. A connection between non-linear evolution equations and ordinary differential equations of P-type. J. Math. Phys. 21:1980;715-721.
11. Ablowitz M., Ramani A., Segur H. A connection between non-linear evolution equations and ordinary differential equations of P-type. J. Math. Phys. 5:1980;1006-1015.
12. Weiss J., Tabor M., Carnevale G. The Painlevé property for partial differential equations. J. Math. Phys. 24:1983;522-526.
13. Weiss J. The Painlevé property for partial differential equations. II. Bäcklund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys. 24:1983;1405-1413.
14. Weiss J. Bäcklund transformation and the Painlevé property. J. Math. Phys. 27:1986;1293-1305.
15. Musette M., Conte R. Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of non-linear partial differential equations. J. Math. Phys. 32:1991;1450-1457.
16. Musette M., Conte R. The two-singular-manifold method: I. Modified Korteveg - de Vries and the sine-Gordon equations. J. Phys. A: Math. Gen. 27:(11):1994;3895-3913.
17. Conte R., Musette M. Painlevé analysis and Bäcklund transformation in the Kuramoto - Sivashinsky equation. J. Phys. A: Math. Gen. 22:(2):1989;169-177.
18. Conte R., Musette M. Link between solitary waves and projective Riccati equations. J. Phys. A: Math. Gen. 25:(21):1992;5609-5623.
19. Ablowitz M.J., Zeppetella A. Explicit solutions of Fisher's equation for a special wave speed. Bull. Math. Biology. 41:(6):1979;835-840.
20. Nozaki K. Hirota's method and the singular manifold expansion. J. Phys. Soc. Japan. 56:(9):1987;3052-3054.
21. Cariello F., Tabor M. Painlevé expansions for nonintegrable evolution equations. Physica D. 39:(1):1989;77-94.
22. Choudhury S.R. Painlevé analysis and special solutions of two families of reaction - diffusion equations. Phys. Lett. A. 159:(6-7):1991;311-317.
23. Pickering A. A new truncation in Painlevé analysis. J. Phys. A: Math. Gen. 26:(17):1993;4395-4405.
24. Kudryashov N.A. Bäcklund transformations for a fourth-order partial differential equation with Burgers - Korteveg - de Vries non-linearity. Dokl. Akad. Nauk SSSR. 300:(2):1988;342-345.
25. Kudryashov N.A. Exact soliton solutions of the generalized evolution equation of wave dynamics. Prikl. Mat. Mekh. 52:(3):1988;465-470.
26. Kudryashov N.A. Exact solutions of non-linear wave equations encountered in mechanics. Prikl. Mat. Mekh. 54:1990;450-453.
27. Kudryashov N.A. Exact solutions of an Nth-order equation with Burgers - Korteveg - de Vries non-linearity. Mat. Modelirovaniye. 1:1989;57-65.
28. Kudryashov N.A. Exact solutions of the generalized Ginzburg - Landau equation. Mat. Modelirovaniye. 1:(9):1989;151-158.
29. Kudryashov N.A. The method of Painlevé expansions for non-integrable non-linear equations. Mat. Modelirovaniye. 2:(12):1990;102-116.
30. Kudryashov N.A. Exact solutions of equations of the Fisher family. Teoret. Mat. Fizika. 94:(2):1993;296-306.
31. Kudryashov N.A. Exact solutions of the generalized Kuramoto - Sivashinsky equation. Phys. Lett. A. 147:(5-6):1990;287-291.
32. Kudryashov N.A. On types of nonlinear nonintegrable equations with exact solutions. Phys. Lett. A. 155:(4-5):1991;269-275.
33. Kudryashov N.A., Zargaryan E.D. Solitary waves in active-dissipative dispersive media. J. Phys. A: Math. Gen. 29:(24):1996;8067-8077.
34. Zemlyanukhin A.I. Exact soliton-like solution of a non-linear fifth-order evolution equation. Izv. Vysshikh Ucheb. Zaved. Prikl. Nelineinaya Dinamika. 7:(2-3):1999;29-32.
35. Gudkov V.V. Solutions of the travelling-wave type for two-component reaction - diffusion systems. Zh. Vychisl. Mat. Mat. Fiz. 35:(4):1995;615-623.
36. Lu H., Wang M. Exact soliton solutions of some nonliner physical models. Phys. Lett. A. 255:(4-6):1999;249-252.
37. Dai X., Dai J. Some solitary wave solutions for families of generalized higher order KdV equations. Phys. Lett. A. 142:(6-7):1989;367-370.
38. Lan H., Wang K. Exact soliton solution in ice-like structures. Phys. Lett. A. 139:(1-2):1989;61-64.
39. Huang G., Luo S., Dai X. Exact and explicit solitary wave solutions to a model equation for water waves. Phys. Lett. A. 139:(8):1989;373-374.
40. Lan H., Wang K. Exact solutions for some non linear equations. Phys. Lett. A. 137:(7-8):1989;369-372.
41. Porubov A.V. Exact travelling wave solutions of nonlinear evolution equation of surface waves in a convecting fluid. J. Phys. A: Math. Gen. 26:(17):1993;L797-L800.
42. Olver P.J. Hamilton and non-Hamilton models for water waves. Lecture Notes in Physics. 1984;273-290 Springer, New York. No. 195.