On explicit exact solutions to the compound Burgers-KdV equation. (English) Zbl 0984.35138
Summary: Applying the theory of commutative algebra, we propose a new approach which is currently called the first integral method to study the compound Burgers-KdV equation and Burgers-KdV equation. The results indicate that the solutions obtained in the previous literature contain errors.
MSC:
35Q53 | KdV equations (Korteweg-de Vries equations) |
37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
References:
[1] | Wang, M., Phys. Lett. A, 213, 279 (1996) · Zbl 0972.35526 |
[2] | Johnson, R. S., J. Fluid Mech., 42, 49 (1970) · Zbl 0213.54904 |
[3] | van Wijngaarden, L., Ann. Rev. Fluid Mech., 4, 369 (1972) · Zbl 0243.76070 |
[4] | Gao, G., Sci. Sinica Ser. A, 28, 616 (1985) · Zbl 0589.76067 |
[5] | Grua, H.; Hu, P. W., Phys. Fluids, 10, 2596 (1967) |
[6] | Jeffrey, A., Arch. Mech., 31, 559 (1979) · Zbl 0437.73010 |
[7] | Canosa, J.; Gazdag, J., J. Comp. Phys., 23, 393 (1977) · Zbl 0356.65107 |
[8] | Dauletiyarov, K. Z., Zh. Vychisl. Mat. Mat. Fiz., 24, 383 (1984), in Russian |
[9] | Turetaev, I. D., Comput. Math. Math. Phys., 31, 69 (1991) |
[10] | Bona, J. L.; Schonbeck, M. E., Proc. R. Soc. Edinburgh, 101A, 207 (1985) · Zbl 0594.76015 |
[11] | Guan, K. Y.; Gao, G., Sci. Sinica Ser. A, 30, 64 (1987) |
[12] | Shu, J. J., J. Phys. A: Math. Gen., 20, L49 (1987) · Zbl 0663.35091 |
[13] | Gibbon, J. D.; Radmore, P.; Tabor, M.; Wood, D., Stud. Appl. Math., 72, 39 (1985) · Zbl 0581.35074 |
[14] | Xiong, S. L., Chin. Sci. Bull., 34, 1158 (1989) · Zbl 0704.35126 |
[15] | Liu, S. D.; Liu, S. K.; Ye, Q. X., Math. Pract. Theory, 28, 289 (1998) · Zbl 1493.35091 |
[16] | Jeffrey, A.; Xu, S., Wave Motion, 11, 559 (1989) · Zbl 0698.35139 |
[17] | Jeffrey, A.; Mohamad, M. N.B., Wave Motion, 14, 369 (1991) · Zbl 0732.76012 |
[18] | Halford, W. D.; Vlieg-Hulstman, M., Wave Motion, 14, 267 (1991) · Zbl 0832.35129 |
[19] | Parkes, E. J.; Duffy, B. R., Phys. Lett. A, 229, 217 (1997) · Zbl 1043.35521 |
[20] | Pan, X. D., Appl. Math. Mech., 9, 281 (1988) |
[21] | Ding, T. R.; Li, C. Z., Ordinary Differential Equations (1996), Peking University Press: Peking University Press Peking |
[22] | Bourbaki, N., Commutative Algebra (1972), Addison-Wesley: Addison-Wesley Paris |
[23] | Zhang, W. G., Acta Math. Sci., 16, 241 (1996) · Zbl 0958.35125 |
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