变系数(3+1)维Zakharov-Kuznetsov方程的Jacobi椭圆函数精确解

J4 ›› 2010, Vol. 27 ›› Issue (1): 6-14.

变系数(3+1)维Zakharov-Kuznetsov方程的Jacobi椭圆函数精确解

套格图桑斯仁道尔吉

内蒙古师范大学数学科学学院,内蒙古呼和浩特, 010022

出版日期:2010-01-28 发布日期:2009-12-30
通讯作者: 套格图桑(1964 - ),理学硕士,副教授, 研究方向为孤立子与可积系统理论及其应用。 E-mail:tgts@imnu.edu.cn
基金资助:
国家自然科学基金(10461006)、内蒙古自治区高等学校科学研究基金(NJZZ07031)、内蒙古自治区自然科学基金(200408020103)和内蒙古师范大学自然科学青年研究计划(QN005023)资助

Jacobi-like elliptic function exact solutions of (3+1)-dimensional Zakharov-Kuznetsov equation

TAO Ge-Tu-Sang, SHI Ren-Dao-Er-Ji

College of Mathematical Science, Inner Mongolia Normal University, Huhhot 010022, China

Published:2010-01-28 Online:2009-12-30

摘要/Abstract

摘要：

给出了第一种椭圆方程的一些新解和解的非线性叠加公式,然后与一种函数变换相结合, 借助符号计算系统Mathematica,构造了变系数(3+1)维Zakharov-Kuznetsov方程的类Jacobi椭圆函数精确解以及无穷多个类孤子解和三角函数解。

关键词: 非线性方程, 辅助方程法, 非线性叠加公式, 函数变换, Jacobi椭圆函数, 精确解

Abstract:

Some new solutions of the first kind of elliptic equation and the formula of nonlinear superposition of the solutions are given. Combing the solutions with function transformation to construct Jacobi-like elliptic function exact solutions, a number of soliton-like solutions and triangular solutions of (3+1)-dimensional Zakharov-Kuznetsov equation with variable coefficients are given with the help of symbolic computation system Mathematica.

Key words: nonlinear equation, auxiliary equation method, formula of nonlinear superposition, function transformation, Jacobi elliptic function, exact solution

套格图桑斯仁道尔吉. 变系数(3+1)维Zakharov-Kuznetsov方程的Jacobi椭圆函数精确解[J]. J4, 2010, 27(1): 6-14.

TAO Ge-Tu-Sang, SHI Ren-Dao-Er-Ji. Jacobi-like elliptic function exact solutions of (3+1)-dimensional Zakharov-Kuznetsov equation[J]. J4, 2010, 27(1): 6-14.

参考文献

[1]Wang Mingliang.Solitary wave solutions for variant Boussinesq equations [J].Phys. Lett.,1995, A199: 169-172. [2]Sirendaoreji, Sun Jiong. Auxiliary equation method for solving nonlinear partial differential equations [J]. Phys. Lett.,2003, A309: 387-396.
[3]Fu Zuntao, Liu Shida, Liu Shikuo. Solving nonlinear wave equations by elliptic equation [J]. Commun. Theor. Phys.,2003, 39(5): 531-536.
[4]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(10): 2202-2209.
[5]Zhang Jinliang, Ren Dongfeng, Wang Mingliang, et al. The periodic wave solutions for the generalized Nizhnik-Novikov-Veselov equation [J]. Chin. Phys., 2003, 12(8): 825-830.
[6]Pan Junting, Gong Lunxun. Jacobi elliptic function solutions to the coupled KdV-mKdV equation[J]. Acta Physica Sinica(物理学报), 2007, 56(10): 5585-5590 (in Chinese).
[7]Zhang Liang, Zhang Lifeng, Li Chongyin. Some new exact solutions of Jacobian elliptic function about the generalized Boussinesq equation and Boussinesq-Burgers equation [J]. Chin. Phys. B,2008, 17(2): 403-410.
[8]Sirendaoerji, Sun jiong. Auxiliary equation method and its applications to nonlinear evolution equations [J]. J. Modern. Phys. C,2003, 14(8): 1075-1085.
[9]Chen Yong, Li Biao. New exact travelling wave solutions for generalized Zakharov-Kuzentsov equations using general projentive Riccati equation method [J]. Commun. Theor. Phys., 2004, 41(1): 1-6.
[10]Zhu Jiamin, Zheng Chunlong, Ma Zhengyi. A general mapping approach and new travelling wave solutions to the general variable coefficient KdV equation [J]. Chin. Phys., 2004, 13(12): 2008-2012.
[11]Li Huamei. Exact periodic wave and soliton solutions in two-component Bose-Einstein condensates [J]. Chin. Phys.,2007, 16(11): 3187-3191.
[12]Yan Zhenya, 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).
[13]Fu Zuntao, Liu Shida, Liu Shikuo, et al.New exact solutions to KdV equations with variable coefficients or forcing [J]. Appl. Math. Mech.(应用数学和力学), 2004, 25(1): 67-73 (in Chinese).
[14]Liu Shikuo, Fu Zuntao, Liu Shida, et al.Jacobi elliptic function expansion solution to the variable coefficient nonlinear equations [J]. Acta Physica Sinica(物理学报), 2002, 51(9): 1923-1926 (in Chinese).
[15]Liu Chengshi. Using trial equation method to solve the exact solutions for two kinds KdV equations with variable coefficients [J].Acta Physica Sinica(物理学报), 2005, 54(10): 4506-4510 (in Chinese).
[16]Zhang Jiefang, Chen Fangyue. Truncated expansion method and new exact soliton-like solution of the general variable coefficient KdV equation [J]. Acta Physica Sinica(物理学报),2001, 50(9): 1648-1650 (in Chinese). [17]Li Desheng, Zhang Hongqing. Improved tanh-function method and the new exact solutions for the general variable coefficient KdV equation and MKdV equation [J]. Acta Physica Sinica(物理学报),2003, 52(7): 1569-1573 (in Chinese).
[18]Lu Dianchen, Hong Baojian, Tian Lixin. Explicit and exact solutions to the variable coefficient combined KdV equation with forced term [J]. Acta Physica Sinica(物理学报), 2006, 55(11): 5617-5622 (in Chinese).
[19]Mao Jiejian,et al.New solitary wave-like solution and exact solution of variable coefficient KP equation [J]. Acta Physica Sinica(物理学报), 2005, 54(11): 4999-5002 (in Chinese).
[20]Tian Guichen, et al. 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).
[21]Shivamoggi B K, Rollins D K. Generalized painleve formulation of Lie group symmetries of the ZK equation [J]. Phys. Lett. A,1991, 160: 263-266.
[22]Hamza A M. Akinetic derivation of a generalized ZK equation for ion acoustic turbulence in magnetized plasma [J]. Phys. Lett. A, 1994, 190: 309-316.
[23]Yang Hongjuan, Duan Wenshan, et al. The solitary wave solutions with varying velocity for the (3+1)-dimensional variable-coefficients ZK equation [J]. J. Nor. Norm. Univ. Nat. Sci.(西北师范大学学报), 2007, 43(2):38-40(in Chinese).
[24]Yan Zhilian, Liu Xiqiang. Symmetry and similarity solutions of variable coefficients generalized Zakharov-Kuzentsov equation [J]. Appl. Math. Comput., 2006, 180: 288-294.
[25]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).