mBBM方程和Vakhneoko方程的显式精确解

J4 ›› 2010, Vol. 27 ›› Issue (6): 683-687.

mBBM方程和Vakhneoko方程的显式精确解

郭鹏张磊王小云孙小伟

兰州交通大学数理与软件工程学院，甘肃兰州 730070

收稿日期:2010-02-05 修回日期:2010-04-05 出版日期:2010-11-28 发布日期:2010-11-19
通讯作者: 郭鹏(1978-), 男, 陕西富平人,主要从事非线性物理的研究工作。 E-mail:guopenglzjtu@126.com
基金资助:
甘肃省自然科学基金（0916RJZA047）、兰州交通大学“青蓝”人才工程（QL-06-22A）资助项目

Explicit and exact solutions to the mBBM and Vakhneoko equations

GUO Peng, Zhang-Lei, WANG Xiao-Yun, SUN Xiao-Wei

School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China

Received:2010-02-05 Revised:2010-04-05 Published:2010-11-28 Online:2010-11-19

摘要/Abstract

摘要：

应用试探函数方法求解了mBBM方程和Vakhneoko方程．通过引入试探函数，把难于求解的非线性偏微分方程化为易于求解的代数方程，然后用待定系数法确定相应的常数，从而简洁地求得了方程的精确解。

关键词: 非线性方程, 试探函数方法, mBBM方程, Vakhneoko方程, 精确解

Abstract:

The mBBM and Vakhneoko equations are solved by the trial function method. By introducing appropriate trial functions, the nonlinear partial differential equation that is hard to be solved by the usual ways can be reduced to a set of algebraic equation, which can be easily solved, and its related coefficients can be easily determined by the method of undetermined coefficient. Finally, the analytical solution to the equations are successfully derived.

Key words: nonlinear equation, trial function method, mBBM equation, Vakhneoko equation, exact solution

中图分类号:

O175.2

郭鹏张磊王小云孙小伟. mBBM方程和Vakhneoko方程的显式精确解[J]. J4, 2010, 27(6): 683-687.

GUO Peng, Zhang-Lei, WANG Xiao-Yun, SUN Xiao-Wei. Explicit and exact solutions to the mBBM and Vakhneoko equations[J]. J4, 2010, 27(6): 683-687.

参考文献

[1] Ablowitz M J , Clarkson P A , Soliton , Nonlinear Evolution Equations and Inverse Scattering [M]. New York: Cambridge University Press , 1991.
[2] Hasegawa A , Optical Solitons in Fibers [M]. Berlin: Springer , 1989.
[3] Zheng Bin , New soliton solutions to 2+1 dimensional breaking soliton equation [J]. Chinese Journal of Quantum Electronics (量子电子学报) , 2006 , 23(4): 451-455 (in Chinese).
[4] Tian Guichen , Liu Xiqiang , Exact solutions of the general variable coefficient KdV equation with external force term [J]. Chinese Journal of Quantum Electronics (量子电子学报) , 2005 , 22(3): 339-343 (in Chinese).
[5] Taogetusang , Sirendaoerji , Jacobi-like elliptic function exact solutions of (3+1)-dimensional Zakharov-Kuznetsov equation [J]. Chinese Journal of Quantum Electronics (量子电子学报) , 2010 , 27(1): 6-14 (in Chinese).
[6] Taogetusang , Sirendaoerji , New solitary wave solutions of the combined KdV equation with the variable coefficients [J]. Chinese Journal of Quantum Electronics (量子电子学报) , 2009 , 26(2): 148-154 (in Chinese).
[7] Li Bangqing , Ma Yulan , Xu Meiping , (G′/G)-expansion method and novel fractal structures for high-dimensional nonlinear physical equation [J]. Acta Physica Sinica (物理学报) , 2010 , 59(3): 1409-1415 (in Chinese).
[8] Xie Yuanxi , Tang Jiashi , New solitary wave solutions to the KdV-Burgers equation [J]. International Journal of Theoretical Physics , 2005 , 44(3): 304-312.
[9] Xie Yuanxi , Tang Jiashi , A simple fast method in finding the analytical solutions of a class of nonlinear partial differential equations [J]. Acta Physica Sinica (物理学报) , 2004 , 53(9): 2828-2830 (in Chinese).
[10] Liu Shikuo , Fu Zuntao , Liu Shida , Zhao Qiang , A simple fast method in finding particular solutions of some nonlinear PDE [J]. Applied Mathematics and Mechanics , 2001 , 22(3): 326-331.
[11] Benjamin T B , Bona J L , Mahony J J , Model equations for long waves in nonlinear dispersive systems [J]. Philosophical Transactions for the Royal Society of London: Series A , 1972 , 272: 47-78.
[12] Vakhnenko V O , Parkes E J , The two loop soliton solution of the Vakhnenko equation [J]. Nonlinearity , 1998 , 11(6): 1457-1464.
[13] Tso Taicheng , Existence of Solutions of the Modified Benjamin-Bone-Mahony Equation [J]. Chinese Journal of Mathematics , 1996 , 24(4): 327-336.
[14] Vakhnenko V O , Parkes E J , The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method [J]. Chaos, Soliton and Fractals , 2002 , 13(9): 1819-1836. [15] Yusufoglu E , Bekir A , The tanh and the sine-cosine methods for exact solutions of the MBBM and the Vakhnenko equations [J]. Chaos, Soliton and Fractals , 2008 , 38(4): 1126-1133.