Jacobi elliptic function exact solutions of sine-Gordon equation

Abstract

Abstract:

The auxiliary equation of the triangle function type and its solutions are given. The equation is directly applied to sine-Gordon equation, double sine-Gordon and mKdV sine-Gordon, with the help of symbolic computation system Mathematica, the Jacobi elliptic function exact solutions and degenerated solitary wave solutions and triangle function wave solutions of the equations are obtained.

Key words: auxiliary equation of triangle function type, sine-Gordon equation, Jacobi elliptic function, exact solution

CLC Number:

O175.29

TAOGeTuSang, SIRenDaoErJi. Jacobi elliptic function exact solutions of sine-Gordon equation[J]. J4, 2009, 26(3): 278-287.

References

[1]Sirendaoreji,Sun Jiong. A direct method for solving sine-Gordon type equations [J]. Phys. Lett., 2002, A298: 133-139.
[2]Xie Yuanxi, Tang Jiashi. A unified approach in seeking the solitary wave solutions to sine-Gordon type equations [J].,2005, 14: 1303-1306.
[3] Taogetusang,Sirendaoerji. New exact solitary wave solutions for the sine-Gordon type equations[J]. Journal of inner mongolia normal university(内蒙古师范大学学报), 2004, 33: 361-365(in Chinese).
[4] Zhang Jinliang, Ren D F, Wang M L et al. The periodic wave solution for the generalized Nizhnik-Novikov-Veselov equation[J]. Chin.Phys., 2003, 12: 825-830.
[5]Fu Zuntao, Liu Shikuo, Liu Shida. Elliptic Equation and New Solutions to Nonlinear Wave Equations[J]. Commun. Theor. Phys., 2004, 42:343-346.
[6]Li Huamei. New exact solutions of nonlinear Gross-Pitaevskii equation with weak bias magnetic and time-dependent laser fields[J]. Chin.Phys., 2005, 14: 251-256.
[7]Zhao Xueqin, Zhi Hongyan, Zhang Hongqing. Construction of doubly-Periodic solutions to nonlinear partial differential equations using improved Jacobi elliptic function expansion method and symbolic computation[J]. Chin.Phys. , 2006, 15: 2202-2209.
[8]Sirendaoreji, Sun Jiong .Auxiliary equation method for solving nonlinear partial differential equations[J]. Phys. Lett., 2003, A309:387-396.
[9]Yan Zenya, Zhang Hongqing. Exact soliton solutions of the variable coeficient KdV-MKdV equation with three arbitrary funtions[J]. Acta Physica Sinica(物理学报), 1999, 48(11):1957-1961(in Chinese).
[10]Chen Yong, Li Biao. New Exact Travelling Wave Solutions for Generalized Zakharov-Kuzentsov Equations Using General Projective Riccati Equation Method [J]. Commun.Theor. Phys., 2004, 41:1-6.
[11]Li Desheng, Zhang Hongqing. The new doubly-periodic solutions for nonlinear coupled scalar field equations(I)[J]. Acta Physica Sinica(物理学报), 2003, 52(10):2373-2377(in Chinese).
[12]Fu Zuntao, Liu Shikuo, Liu Shida. A New Approach to Solve Nonlinear Wave Equations [J]. Commun. Theor. Phys., 2003, 39:27-30.
[13]Fu Zuntao, Liu Shida, Liu Shikuo. Solving Nonlinear Wave Equations Elliptic Equation [J]. Commun.Theor. Phys., 2003, 39:531-536.
[14] Fu Zuntao, Liu Shikuo, Liu Shida. A new method to construct solutions to nonliear wave equations[J]. Acta Physica Sinica(物理学报), 2004, 53 :343-348(in Chinese).
[15] Xu Bingzhen, He Jinyu, Wang Wenzheng. Solitary Wave Solutions of a New Hamiltonian Amplititude Equation[J]. Chinese Journal of quantum electronics(量子电子学报) ,1996, 13(1) :46-50(in Chinese).
[16] Liu Shanliang, Wang Wenzheng, Xu Jingzhi. The Solitary Waves Solution of the Extended Nonlinear Schr\"{o}dinger Equation with the Bandwidth-Limited Optical Gain and Loss Terms[J]. Chinese Journal of quantum electronics(量子电子学报),1994, 11(1) :55-60(in Chinese).
[17] Tian Guichen, Liu Xiqiang. Exact solutions of the general variable coefficient KdV equation with external term [J]. Chinese Journal of quantum electronics(量子电子学报), 2005, 22(3):339-343(in Chinese).
[18] Zhou Jianping, Yan Zhilian. New explicit solutions for a class of generalized nonlinear sine-Gordon equation[J]. Chinese Journal of quantum electronics(量子电子学报), 2007, 24(5):535-538(in Chinese).