﻿ 修正的简单方程法与sine-Godon方程和广义的变系数KdV-mKdV方程的精确解 The Modified Simple Equation Method and the Exact Solutions for the sine-Gordon Equation and the Generalized Variable-Coefficient KdV-mKdV Equation

Advances in Applied Mathematics
Vol.05 No.03(2016), Article ID:18443,7 pages
10.12677/AAM.2016.53055

The Modified Simple Equation Method and the Exact Solutions for the sine-Gordon Equation and the Generalized Variable-Coefficient KdV-mKdV Equation

Lingfeng Xiao, Sirendaoerji

College of Mathematics Science, Inner Mongolia Normal University, Hohhot Inner Mongolia

Received: Aug. 16th, 2016; accepted: Aug. 27th, 2016; published: Aug. 30th, 2016

Copyright © 2016 by authors and Hans Publishers Inc.

ABSTRACT

The modified simple equation method is used to construct the exact solutions for the sine-Gordon equation and the generalized variable-coefficient KdV-mKdV equation. Some exact solutions of the hyperbolic function form for the sine-Gordon equation and the generalized variable-coefficient KdV-mKdV equation are derived by the method. When taking special values of the parameters, the exact traveling wave solutions of the equations are derived from the exact solutions.

Keywords:The Modified Simple Equation Method, sine-Gordon Equation, Variable-Coefficient KdV-mKdV Equation, Exact Solutions

1. 引言

sine-Gordon方程在非线性光学、等离子物理、固体物理等自然科学领域中有着广泛的应用；在流体力学和等离子体中，广义变系数非等谱KdV-mKdV方程用来刻画弱非线性长波在KdV介质中传播，因此这两个方程都是有重要物理背景的模型方程。因此寻找sine-Gordon方程和广义的变系数KdV-mKdV方程的精确解在理论和实际应用中有着至关重要的意义。

2. sine-Gordon方程

(2.1)

，则 (2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

1) 当时，(2.5)无(2.6)形式的解

2) 当时，(2.8)~(2.12)化为常微分方程组

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

，则 (2.19)

，则 (2.20)

，则 (2.21)

，则 (2.22)

(2.23)

(2.24)

(2.25)

(2.26)

(2.27)

(2.28)

，得到sine-Gordon方程的精确孤波解

(2.29)

(2.30)

3) 当时，(2.8)~(2.12)化为常微分方程组

(2.31)

(2.32)

(2.33)

(2.34)

(2.35)

(2.36)

(2.37)

(2.38)

(2.39)

(2.40)

，得到sine-Gordon方程的精确孤波解

(2.41)

(2.42)

3. 广义变系数KdV-mKdV方程

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

The Modified Simple Equation Method and the Exact Solutions for the sine-Gordon Equation and the Generalized Variable-Coefficient KdV-mKdV Equation[J]. 应用数学进展, 2016, 05(03): 443-449. http://dx.doi.org/10.12677/AAM.2016.53055

1. 1. Alowitz, M.J., Ramani, A. and Segur, H. (1978) Nonlinear Evolution Equations and Ordinary Differential Equations of Painlevé Type. Lettere al Nuovo Cimento, 23, 333-338. http://dx.doi.org/10.1007/BF02824479

2. 2. Weiss, J., Tabor, M. and Carnevale, G. (1983) The Painlevé Property for Partial Differential Equations. Journal of Mathematical Physics, 24, 522. http://dx.doi.org/10.1063/1.525721

3. 3. Wang, Y.-H. and Chen, Y. (2012) Bäcklund Transformations and Solutions of a Gene-ralized Kadomtsev-Petviashvili Equation. Communications in Theoretical Physics, 57, 217-322. http://dx.doi.org/10.1088/0253-6102/57/2/10

4. 4. Sirendaoerji, T. (2006) New Exact Solitary Wave for Nonlinear Wave Equation with Fifth-Order Strong Nonlinear Term Constructed by Hyperbolic Function Type of Auxiliary Equation. Acta Physica Sinica, 55, 13-18.

5. 5. Wang, M.L., Li, X.Z. and Zhang, J.L. (2008) The G'/G-Expansion Method and Travelling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics. Physics Letters A, 372, 417-423. http://dx.doi.org/10.1016/j.physleta.2007.07.051

6. 6. Khan, K. and Akbar, M.A. (2013) Exact and Solitary Wave Solutions for the Tzitzeica-Dodd-Bullough and the Modified KdV-Zakharov-Kuznetsov Equations Using the Modified Simple Equation Method. Ain Shams Engineering Journal, 4, 903-909. http://dx.doi.org/10.1016/j.asej.2013.01.010

7. 7. 张哲, 李德生. 修正的BBM方程新的精确解[J]. 原子与分子物理学报, 2013, 30(5).

8. 8. Rubinstein, J. (1970) sine-Gordon Equation. Journal of Mathematical Physics, 11, 258-266. http://dx.doi.org/10.1063/1.1665057

9. 9. Bratsos, A.G. (2007) The Solution of the Two-Dimensional sine-Gordon Equation Using the Method of Lines. Journal of Computational and Applied Mathematics, 206, 251-277.

10. 10. Meng, G.-Q., Gao, Y.-T., Yu, X., Shen, Y.-J. and Qin, Y. (2012) Painlevé Analysis, Lax Pair, Bäcklund Transformation and Multi-Soliton Solutions for a Generalized Variable-Coefficient KdV-mKdV Equation in Fluids and Plasmas. Physica Scripta, 85, 055010. http://dx.doi.org/10.1088/0031-8949/85/05/055010