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## Riemann theta function and other several kinds of new solutions of nonlinear evolution equations

Taogetusang, Bai Yu Mei
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• #### 摘要

为了构造非线性发展方程的复合型无穷序列精确解, 获得了第二种椭圆方程的Riemann theta 函数等几种新解.在此基础上,利用第二种椭圆方程与Riccati方程的Bcklund变换和解的非线性叠加公式, 借助符号计算系统 Mathematica, 以mKdV方程为应用实例, 构造了该方程的复合型无穷序列新精确解.这里包括Riemann theta 函数、Jacobi椭圆函数、双曲函数、 三角函数和有理函数,通过几种形式构成的复合型无穷序列新精确解.

#### Abstract

Riemann theta function and other several kinds of new solutions to the second kind of elliptic equation are obtained to construct the infinite sequence complexiton solutions of nonlinear evolution equations. Based on this, applying Bcklund transformation and nonlinear superposition formula of the solutions to the second kind of elliptic equation and Riccati equation, mKdV equation is chosen as an example to seek infinite sequence new complexiton solutions with the help of symbolic computation system Mathematica, which are composed of Riemann theta function, Jacobi elliptic function, hyperbolic function, triangular function and rational function in several forms.

#### 作者及机构信息

###### 1. 内蒙古民族大学数学学院, 通辽 028043; 2. 内蒙古师范大学数学科学学院, 呼和浩特 010022
• 基金项目: 国家自然科学基金(批准号: 10862003)、内蒙古自治区高等学校科学研究基金 (批准号: NJZY12031)和 内蒙古自治区自然科学基金(批准号: 2010MS0111)资助的课题.

#### Authors and contacts

###### 1. The College of Mathematical, Inner Mongolia University for Nationalities, Tongliao 028043, China; 2. The College of Mathematical Science, Inner Mongolia Normal University, Huhhot 010022, China
• Funds: Project supported by the Natural Natural Science Foundation of China (Grant No. 10862003), the Science Research Foundation of Institution of Higher Education of Inner Mongolia Autonomous Region, China (Grant No. NJZY12031) and the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No. 2010MS0111).

#### 参考文献

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#### 施引文献

•  [1] Fan E G 2000 Phys. Lett. A 277 212 [2] Chen Y, Li B, Zhang H Q 2003 Chin. Phys. 12 940 [3] Chen Y, Yan Z Y, Li B, Zhang H Q 2003 Chin. Phys. 12 1 [4] Chen Y, Li B, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 40 137 [5] Li D S, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 40 143 [6] Li D S, Zhang H Q 2004 Chin. Phys. 13 984 [7] Li D S, Zhang H Q 2004 Chin. Phys. 13 1377 [8] Chen H T, Zhang H Q 2004 Commun. Theor. Phys. (Beijing) 42 497 [9] Xie F D, Chen J, L Z S 2005 Commun. Theor. Phys. (Beijing) 43 585 [10] Xie F D, Yuan Z T 2005 Commun. Theor. Phys. (Beijing) 43 39 [11] Zhen X D, Chen Y, Li B, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 39 647 [12] L Z S, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 39 405 [13] Xie F D, Gao X S 2004 Commun. Theor. Phys. (Beijing) 41 353 [14] Chen Y, Li B 2004 Commun. Theor. Phys. (Beijing) 41 1 [15] Ma S H, Fang J P, Zhu H P 2007 Acta Phys. Sin. 56 4319 (in Chinese) [马松华,方建平,朱海平 2007 物理学报 56 4319] [16] Ma S H, Wu X H, Fang J P, Zheng C L 2008 Acta Phys. Sin. 57 11 (in Chinese) [马松华,吴小红,方建平,郑春龙 2008 物理学报 57 11] [17] Pan J T, Gong L X 2008 Chin. Phys. B 17 399 [18] Jiao X Y, Wang J H, Zhang H Q 2005 Commun. Theor. Phys. (Beijing) 44 407 [19] Liu Y P, Li Z B 2003 Chin. Phys. Lett. 20 317 [20] Xu G Q, Li Z B 2003 Commun. Theor. Phys. (Beijing) 39 39 [21] Li Z L 2009 Chin. Phys. B 18 4074 [22] Wang Z, Li D S, Lu H F, Zhang H Q 2005 Chin. Phys. 14 2158 [23] Li D S, Zhang H Q 2004 Chin. Phys. 13 1377 [24] Lu B, Zhang H Q 2008 Chin. Phys. B 17 3974 [25] Wang Z, Zhang H Q 2006 Chin. Phys. 15 2210 [26] Zhang J L, Ren D F, Wang M L, Wang Y M, Fang Z D 2003 Chin. Phys. 12 825 [27] Zhang L, Zhang L F, Li C Y 2008 Chin. Phys. B 17 403 [28] Zhao X Q, Zhi H Y, Zhang H Q 2006 Chin. Phys. 15 2202 [29] Li J B 2007 Sci. Chin. Math. A 50 153 [30] Li H M 2003 Commun. Theor. Phys. (Beijing) 39 395 [31] Li H M 2005 Chin. Phys. 14 251 [32] Li H M 2002 Chin. Phys. 11 1111 [33] Taogetusang, Sirendaoerji 2006 Chin. Phys. 15 2809 [34] Liu C S 2005 Chin. Phys. 14 1710 [35] Zhu J M, Zheng C L, Ma Z Y 2004 Chin. Phys. 13 2008 [36] Fu Z T, Liu S D, Liu S K 2003 Commun. Theor. Phys. (Beijing) 39 531 [37] Fu Z T, Liu S K, Liu S D 2004 Commun. Theor. Phys. (Beijing) 42 343 [38] Taogetusang, Sirendaoerji 2008 Acta Phys. Sin. 57 1295 (in Chinese) [套格图桑,斯仁道尔吉 2008 物理学报 57 1295] [39] Sirendaoerji, Sun J 2003 Phys. Lett. A 309 387 [40] Taogetusang, Sirendaoerji 2006 Acta Phys. Sin. 55 3246 (in Chinese) [套格图桑,斯仁道尔吉 2006 物理学报 55 3246] [41] Taogetusang, Sirendaoerji 2006 Chin. Phys. 15 1143 [42] Taogetusang, Sirendaoerji 2007 Acta Phys. Sin. 56 627 (in Chinese) [套格图桑,斯仁道尔吉 2007 物理学报 56 627] [43] Taogetusang, Sirendaoerji 2006 Acta Phys. Sin. 55 13 (in Chinese) [套格图桑,斯仁道尔吉 2006 物理学报 55 13] [44] Fu Z T, Liu S K, Liu S D 2003 Commun. Theor. Phys. (Beijing) 39 27 [45] Yu J, Ke Y Q, Zhang W J 2004 Commun. Theor. Phys. (Beijing) 40 493 [46] Sirendaoerji, Sun J 2002 Phys. Lett. A 298 133 [47] Taogetusang, Sirendaoerji,Wang Q P 2009 Acta Sci. J. Nat. Univ. Neimongol. 38 387 (in Chinese) [套格图桑,斯仁道尔吉,王庆鹏 2009 内蒙古师范大学学报 38 387] [48] Wang J M 2012 Acta Phys. Sin. 61 080201 (in Chinese) [王军民 2012 物理学报 61 080201] [49] Lawden D F 1989 Elliptic Functions and Applications (Berlin: Springer-Verlag) p496 [50] Li D S, Zhang H Q 2006 Acta Phys. Sin. 55 1565 (in Chinese) [李德生, 张鸿庆 2006 物理学报 55 1565] [51] Wu H Y, Zhang L, Tan Y K, Zhou X T 2008 Acta Phys. Sin. 57 3312 (in Chinese) [吴海燕, 张亮, 谭言科, 周小滔 2008 物理学报 57 3312] [52] Liu S K, Fu Z T, Liu S D, Zhao Q 2002 Acta Phys. Sin. 51 10 (in Chinese) [刘式适,付遵涛,刘式达,赵强 2002 物理学报 51 10] [53] Liu S K, Fu Z T, Liu S D, Zhao Q 2001 Acta Phys. Sin. 50 2069 (in Chinese) [刘式适,付遵涛,刘式达,赵强 2001 物理学报 50 2069] [54] Liu S K, Fu Z T, Wang Z G, Liu S D 2003 Acta Phys. Sin. 52 1838 (in Chinese) [刘式适, 付遵涛,王彰贵,刘式达 2003 物理学报 52 1838] [55] Liu S K, Chen H, Fu Z T, Liu S D 2003 Acta Phys. Sin. 52 1843 (in Chinese) [刘式适,陈华,付遵涛,刘式达 2003 物理学报 52 1843] [56] Shi Y R, Guo P, L K P, Duan W S 2004 Acta Phys. Sin. 53 3265 (in Chinese) [石玉仁, 郭鹏,吕克璞,段文山 2004 物理学报 53 3265] [57] Taogetusang, Sirendaoerji, Li S M 2011 Commun. Theor. Phys. (Beijing) 55 949
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##### 出版历程
• 收稿日期:  2012-08-16
• 修回日期:  2013-01-25
• 刊出日期:  2013-05-05

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