Traveling waves to a Burgers-Korteweg-de Vries-type equation with higher-order nonlinearities

Journal of Mathematical Analysis and Applications

Volumn 328, Issue 2, 2007, Pages 1435-1450

  Feng, Zhaosheng ^a,b   Knobel, Roger ^a  

^a UNIVERSITY OF TEXAS PAN AMERICAN   (United States)

^b TIANJIN UNIVERSITY OF TECHNOLOGY AND EDUCATION   (China)

Author keywords

Autonomous system; Bendixson theorem; Burgers KdV equation; First integral; Painlev analysis; Solitary wave

Indexed keywords

EID: 33845905203     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2006.05.085     Document Type: Article

References (64)

1. Benney D.J. Long waves on liquid films. J. Math. Phys. 45 (1966) 150-155
2. Johnson R.S. Shallow water waves on a viscous fluid-the undular bore. Phys. Fluids 15 (1972) 1693-1699
3. van Wijngaarden L. On the motion of gas bubbles in a perfect fluid. Ann. Rev. Fluid Mech. 4 (1972) 369-373
4. Johnson R.S. A nonlinear equation incorporating damping and dispersion. J. Fluid Mech. 42 (1970) 49-60
5. Grad H., and Hu P.W. Unified shock profile in a plasma. Phys. Fluids 10 (1967) 2596-2602
6. Hu P.N. Collisional theory of shock and nonlinear waves in a plasma. Phys. Fluids 15 (1972) 854-864
7. Karahara T. Weak nonlinear magneto-acoustic waves in a cold plasma in the presence of effective electron-ion collisions. J. Phys. Soc. Japan 27 (1970) 1321-1329
8. Gao G. A theory of interaction between dissipation and dispersion of turbulence. Sci. Sinica (Ser. A) 28 (1985) 616-627
9. Liu S.D., and Liu S.K. KdV-Burgers equation modelling of turbulence. Sci. Sinica (Ser. A) 35 (1992) 576-586
10. Wadati M. Wave propagation in nonlinear lattice. J. Phys. Soc. Japan 38 (1975) 673-680
11. Feudel F., and Steudel H. Nonexistence of prolongation structure for the Korteweg-de Vries-Burgers equation. Phys. Lett. A 107 (1985) 5-8
12. Dodd R., and Fordy A. The prolongation structures of quasipolynomial flows. Proc. Roy. Soc. London A 385 (1983) 389-429
13. Burgers J.M. Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Trans. Roy. Neth. Acad. Sci. Amsterdam 17 (1939) 1-53
14. Korteweg D.J., and 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 (1895) 422-443
15. Dey B. Domain wall solution of KdV like equation higher-order nonlinearity. J. Phys. A 19 (1986) L9-L12
16. Coffey M.W. On series expansions giving closed form of Korteweg-de Vries-like equations. SIAM J. Appl. Math. 50 (1990) 1580-1592
17. Dey B. KdV like Equations with Domain Wall Solutions and Their Hamiltonians, Solitons. Springer Series on Nonlinear Dynamics (1988), Springer, New York pp. 188-194
18. Liu Z.R., and Li J.B. Bifurcations of solitary waves and domain wall waves for KdV-like equation with higher-order nonlinearity. Internat. J. Bifur. Chaos 12 (2002) 397-407
19. Miura R.M. The Korteweg-de Vries equation: A survey of results. SIAM. Rev. 8 (1976) 412-459
20. Dodd R.K., Eilbeck J.C., Gibbon J.D., and Morris H.C. Solitons and Nonlinear Wave Equations (1982), Academic Press, London
21. Bona J.L., and Schonbek M.E. Traveling wave solutions to Korteweg-de Vries-Burgers equation. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985) 207-226
22. Bona J.L., Souganidis P.E., and Strauss W.A. Stability and instability of solitary waves. Proc. R. Soc. Lond. Ser. A 411 (1987) 395-412
23. Laedke E.W., and Spatschek K.H. Stability theorem for KdV type equations. J. Plasma Phys. 32 (1984) 263-272
24. Weinstein M.I. On the solitary traveling wave of the generalized Korteweg-de Vries equation. In: Nicolaenko B., Holm D., and Hyman J. (Eds). Nonlinear Systems of Partial Differential Equations in Applied Mathematics. Lectures in Applied Mathematics vol. 23 (1986), Amer. Math. Soc.
25. Weinstein M.I. Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation. Comm. Partial Differential Equations 12 (1987) 1133-1173
26. Pego R.L., Smereka P., and Weinstein M.I. Oscillatory instability of traveling waves for a KdV-Burgers equation. Phys. D 67 (1993) 45-65
27. Pego R.L., and Weinstein M.I. Eigenvalues, and solitary wave instabilities. Philos. Trans. R. Soc. Lond. Ser. A 34 (1992) 47-94
28. Zhang W.G., Chang Q.S., and Jiang B.G. Explicit exact solitary-wave solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear term of any order. Chaos Solitons Fractals 13 (2002) 311-319
29. Feng Z. Traveling solitary wave solutions to evolution equations with nonlinear terms of any order. Chaos Solitons Fractals 17 (2003) 861-868
30. Li B., Chen Y., and Qing Z.H. Explicit exact solutions for new general two-dimensional KdV-type and two-dimensional KdV-Burgers-type equations with nonlinear term of any order. J. Phys. A 35 (2002) 8253-8265
31. Jeffrey A. Some aspects of the mathematical modelling of long nonlinear waves. Arch. Mech. 31 (1979) 559-574
32. Canosa J., and Gazdag J. The Korteweg-de Vries-Burgers equation. J. Comput. Phys. 23 (1977) 393-403
33. Dauletiyarov K.Z. Investigation of the difference method for the Bona-Smith and Burgers-Korteweg-de Vries equations. Zh. Vychisl. Mat. Mat. Fiz. 24 (1984) 383-402
34. Avilov V.V., Krichever I.M., and Novikov S.P. Evolution of the Whiteham zone in the Korteweg-de Vries theory. Soviet Phys. Dokl. 32 (1987) 345-349
35. Guan K.Y., and Gao G. The qualitative theory of the mixed Korteweg-de Vries-Burgers equation. Sci. Sinica (Ser. A) 30 (1987) 64-73
36. Guan K.Y., and Lei J.Z. Intergrability of second-order antonomous system. Ann. Differential Equations 10 (2002) 117-135
37. Gao J.X., Lei J.Z., and Guan K.Y. Integrable condition on traveling wave solutions of Burgers-KdV equation. J. Northern Jiaotong University 27 (2003) 38-42
38. Zhu G.W., Qin B.Q., and Gao G. Direct evidence of phosphorus outbreak release from sediment to overlying water in a large shallow lake caused by strong wind wave disturbance. Chinese Sci. Bull. 50 (2005) 577-582
39. Shu J.J. The proper analytical solution of the Korteweg-de Vries equation. J. Phys. A 20 (1987) L49-L56
40. Gibbon J.D., Radmore P., Tabor M., and Wood D. The Painlevé and Hirota's method. Stud. Appl. Math. 72 (1985) 39-63
41. Feng Z. Qualitative analysis and exact solutions to the Burgers-KdV equation. Dyn. Contin. Discrete Impuls. Syst. Ser. A 9 (2002) 563-580
42. Xiong S.L. An analytic solution of Burgers-KdV equation. Chinese Sci. Bull. 34 (1989) 1158-1162
43. McIntosh I. Single phase averaging and traveling wave solutions of the modified Burgers-Korteweg-de Vries equation. Phys. Lett. A 143 (1990) 57-61
44. Zhang W.G. Exact solutions of the Burgers-combined KdV mixed type equation. Acta. Math. Sci. 16 (1996) 241-248
45. Liu S.D., Liu S.K., and Ye Q.X. Explicit traveling wave solutions of nonlinear evolution equations. Math. Practice Theory 28 (1998) 289-301
46. Jeffrey A., and Xu S. Exact solutions to the Korteweg-de Vries-Burgers equation. Wave Motion 11 (1989) 559-564
47. Jeffrey A., and Mohamad M.N.B. Exact solutions to the KdV-Burgers equation. Wave Motion 14 (1991) 369-375
48. Parkes E.J., and Duffy B.R. An automated tanh-fucntion method for finding solitary wave solutions to nonlinear evolution equations. Comput. Phys. Comm. 98 (1996) 288-300
49. Parkes E.J., and Duffy B.R. Traveling solitary wave solutions to a compound KdV-Burgers equation. Phys. Lett. A 229 (1997) 217-220
50. Wang M. Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A 213 (1996) 279-287
51. Demiray H. A note on the exact traveling wave solution to the KdV-Burgers equation. Wave Motion 38 (2003) 367-369
52. Halford W.D., and Vlieg-Hulstman M. The Korteweg-de Vries-Burgers equation: A reconstruction of exact solutions. Wave Motion 14 (1991) 267-271
53. Ince E.I. Ordinary Differential Equations (1956), Dover, New York
54. Weiss J., Tabor M., and Carnevale G. The Painlevé property for partial differential equations. J. Math. Phys. 24 (1983) 522-526
55. Halford W.D., and Vlieg-Hulstman M. The Korteweg-de Vries-Burgers equation and Painlevé property. J. Phys. A 25 (1992) 2375-2379
56. Feng Z. The first-integral method to the Burgers-Korteweg-de Vries equation. J. Phys. A 35 (2002) 343-350
57. Feng Z. On explicit exact solutions to the compound Burgers-Korteweg-de Vries equation. Phys. Lett. A 293 (2002) 57-66
58. Feng Z. Exact solutions in terms of elliptic functions for the Burgers-Korteweg-de Vries equation. Wave Motion 38 (2003) 109-115
59. Feng Z. Traveling solitary wave solutions to several nonlinear differential equations. Amer. Math. Soc. Contemp. Math. 357 (2004) 269-286
60. Ablowitz M.J., and Zeppetella A. Explicit solution of Fisher's equation for a special wave speed. Bull. Math. Biol. 41 (1979) 835-840
61. Lawden D.F. Elliptic Functions and Applications (1989), Springer-Verlag, New York
62. Abramowitz M., and Stegun I. Handbook of Mathematical Functions (1965), Dover, New York
63. Feng Z. Algebraic curve solution for second-order polynomial autonomous systems. Electron J. Linear Algebra 8 (2001) 14-25
64. Bourbaki N. Commutative Algebra (1972), Addison-Wesley Publishing, Paris