|
|
耦合kdv方程的相互作用解
|
Abstract:
随着人们对非线性方程的深入研究，经常通过探求耦合kdv方程的相互作用解来描述水波等自然运动规律。本文提出了一种求耦合kdv方程相互作用解的辅助方程新方法。该新方法可以很容易得到三角函数、指数函数、双曲函数和其他函数的混合函数解。利用该方法，我们成功地得到了耦合kdv方程的相互作用解。这些解在帮助物理学家准确分析相关领域中的特殊现象方面具有十分重要的理论意义和应用价值。
In this paper, based on the auxiliary equation method, we obtain new interaction solutions of the coupled KdV equations, these solutions are degenerated to the solitary wave solutions, the triangle function solutions and other function solutions. It is significant to help physicists to analyze special phenomena in their relevant fields accurately.
| [1] | Gardner, C.S., Greene, J.M., Kruskal, M.D. and Miura, R.M. (1967) Method for Solving the Korteweg-deVries Equa-tion. Physical Review Letters, 19, 1095-1097. https://doi.org/10.1103/PhysRevLett.19.1095 |
| [2] | Hirota, R. (1971) Exact Solution of the Korteweg-deVries Equation for Multiple Collisions of Solutions. Physics Review Letters, 27, 1192-1194. 7 https://doi.org/10.1103/PhysRevLett.27.1192 |
| [3] | Weiss, J., Tabor, M. and Garnevale, G. (1983) The Painlevé Property for Partial Differential Equations. Journal of Mathematical Physics, 24, 522-526. https://doi.org/10.1063/1.525721 |
| [4] | Li, Y.S. and Zhang, J.E. (2001) Darboux Transformations of Classical Boussinesq System and Its Multi-Soliton Solutions. Physics Letters A, 284, 253-258. https://doi.org/10.1016/S0375-9601(01)00331-0 |
| [5] | Wang, M.L., Zhou, Y.B. and Li, Z.B. (1996) Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Equations in Mathematical Physics. Physics Letters A, 216, 67-75.
https://doi.org/10.1016/0375-9601(96)00283-6 |
| [6] | 徐炳振, 李悦科, 阎循领. 一类五阶非线性演化方程的新孤波解[J]. 物理学报, 1998, 47(12): 1946-1951. |
| [7] | 陈德芳, 楼森岳. KdV方程与高阶KdV方程行波解之间的形变理论[J]. 物理学报, 1991, 40(4): 513-521. |
| [8] | Ma, W.X. and Fuchssteiner, B. (1996) Explicit and Exact Solutions to a Kolmogorov-Petrovskii-Piskunov Equation. International Journal of Non-Linear Mechanics, 31, 329-338. https://doi.org/10.1016/0020-7462(95)00064-X |
| [9] | Blickle, T., Lakatos, B.G, Mihálykó, C. and Ulbert, Z. (1998) The Hyperbolic Tangent Distribution Family. Powder Technology, 97, 100-108. https://doi.org/10.1016/S0032-5910(97)03393-7 |
| [10] | Ma, W.X. and Zhu, Z.N. (2012) Solving the (3 + 1)-Dimensional Generalized KP and BKP Equations by the Multiple Exp-Function Algorithm. Applied Mathematics and Computation, 24, 11871-11879.
https://doi.org/10.1016/j.amc.2012.05.049 |
| [11] | Ma, W.-X., Huang, T. and Zhang, Y. (2010) A Multiple Exp-Function Method for Nonlinear Differential Equations and Its Application. Physica Scripta, 82, 5468-5478. https://doi.org/10.1088/0031-8949/82/06/065003 |
| [12] | Fu, Z.T., Liu, S.K., Liu, S.D. and Zhao, Q. (2001) New Ja-cobi Elliptic Function Expansion and New Periodic Solutions of Nonlinear Wave Equations. Physics Letters A, 290, 72-76. https://doi.org/10.1016/S0375-9601(01)00644-2 |
| [13] | Chen, H.T. and Zhang, H.Q. (2004) New Double Pe-riodic and Multiple Soliton Solutions of the Generalized (2 + 1)-Dimensional Boussinesq Equation. Chaos, Solitons & Fractals, 20, 765-769.
https://doi.org/10.1016/j.chaos.2003.08.006 |
| [14] | 许斌, 刘希强, 刘玉堂. 耦合KdV方程组的对称,精确解和守恒律[J]. 应用数学学报, 2010, 33(1): 118-123. |
| [15] | Chen, H.T. and Zhang, H.Q. (2003) Improved Jacobin Elliptic Function Method and Its Applications. Chaos, Solitons & Fractals, 15, 585-591. https://doi.org/10.1016/S0960-0779(02)00147-9 |
