第二类变系数KdV方程的新类型无穷序列精确解

套格图桑; 白玉梅

doi:10.7498/aps.61.060201

摘要

为了构造变系数非线性发展方程的无穷序列新精确解, 发掘第一种椭圆辅助方程的构造性和机械化性特点, 获得了该方程的新类型解和相应的 Bcklund 变换. 在符号计算系统 Mathematica 的帮助下, 以第二类变系数 KdV 方程为应用实例, 构造了三种类型的无穷序列新精确解. 这里包括无穷序列光滑类孤子解、无穷序列尖峰孤立子解和无穷序列紧孤立子解. 这种方法也可以获得其他变系数非线性发展方程的无穷序列新精确解.

关键词:

Abstract

To construct a number of new infinite sequence exact solutions of nonlinear evolution equations and to study the two characteristics of constructivity and mechanicalness of the first kind of elliptic equation, new types of solutions and the corresponding Bcklund transformation of the equation are presented. Then the second kind of KdV equation with variable coefficients is chosen as a practical example and three kinds of new infinite sequence exact solutions are obtained with the help of symbolic computation system Mathematica, where are included the smooth soliton-like solutions, the infinite sequence peak soliton solutions, and the infinite sequence compact soliton solutions. The method can be used to search for new infinite sequence exact solutions of other nonlinear evolution equations with variable coefficients.

Keywords:

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

通信作者: 套格图桑, tgts@imnu.edu.cn

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:NJZZ07031)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

Corresponding author: Taogetusang, tgts@imnu.edu.cn

Funds: Project supported by the National 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. NJZZ07031), and the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No. 2010MS0111).

参考文献

[1]	Liu S K, Fu Z T, Liu S D, Zhao Q 2002 Acta Phys. Sin. 51 1923 (in Chinese)[刘式适, 付遵涛, 刘式达, 赵强 2002 51 1923]
[2]
[3]	Li D S, Zhang H Q 2003 Acta Phys. Sin. 52 1569 (in Chinese)[李德生, 张鸿庆 2003 52 1569]
[4]
[5]	Zhang J F, Chen F Y 2001 Acta Phys. Sin. 50 1648 (in Chinese)[张解放, 陈芳跃 2001 50 1648]
[6]
[7]	Zhu J M, Zheng C L, Ma Z Y 2004 Chin. Phys. 13 2008
[8]	Lou S Y, Ruan H Y 1992 Acta Phys. Sin. 41 182 (in Chinese)[楼森岳, 阮航宇 1992 41 182]
[9]
[10]	Chan W L, Li K S 1989J. Math. Phys. 30 2521
[11]
[12]
[13]	Tian C 1987 J. Phys. A: Math. Gen. 20 359
[14]
[15]	Zhang J L, Ren D F,Wang M L,Wang Y M, Fang Z D 2003 Chin. Phys. 12 825
[16]	Zhang L, Zhang L F, Li C Y 2008 Chin. Phys. B 17 403
[17]
[18]	Zhao X Q, Zhi H Y, Zhang H Q 2006 Chin. Phys. 15 2202
[19]
[20]
[21]	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 57 3312]
[22]	Taogetusang, Sirendaoerji 2010 Acta Phys. Sin. 59 4413 (in Chinese)[套格图桑, 斯仁道尔吉 2010 59 4413]
[23]
[24]	Camassa R, Holm D D 1993 Phys. Rev. Lett. 71 1661
[25]
[26]
[27]	Rosenau P, Hyman J M 1993 Phys. Rev. Lett. 70 564
[28]
[29]	Dullin H R, Gottwald G A, Holm D D 2002 Phys. Rev. Lett. 87 4501
[30]
[31]	Guo B L, Liu Z R 2003 Science in China A 33 325 (in Chinese)[郭柏灵, 刘正荣 2003 中国科学 A 33 325]
[32]	Yin J L, Tian L X 2009 Acta Phys. Sin. 58 3632 (in Chinese)[殷久利, 田立新 2009 58 3632]
[33]
[34]	Yu L Q, Tian L X 2006 Math. Practice. Theory 36 261 (in Chinese)[余丽琴, 田立新 2006 数学的实践与认识 36 261]
[35]
[36]
[37]	Yu L Q, Tian L X 2005 Pure. Appl. Math 21 310 (in Chinese)[余丽琴, 田立新 2005 纯粹数学与应用数学 21 310]
[38]
[39]	Yan Z Y 2002 Chaos, Solitons and Fractals 14 1151
[40]
[41]	Yin J L, Tian L X 2007 Acta Math. Phys. 27A 027 (in Chinese)[殷久利, 田立新 2007 数学 27A 027]
[42]
[43]	Fan E G 2000 Phys. Lett. A 277 212
[44]	Chen Y, Li B, Zhang H Q 2003 Chin. Phys. 12 940
[45]
[46]	Chen Y, Yan Z Y, Li B, Zhang H Q 2003 Chin. Phys. 12 1
[47]
[48]
[49]	Chen Y, Li B, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 40 137
[50]
[51]	Li D S, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 40 143
[52]
[53]	Li D S, Zhang H Q 2004 Chin. Phys. 13 984
[54]
[55]	Li D S, Zhang H Q 2004 Chin. Phys. 13 1377
[56]
[57]	Chen H T, Zhang H Q 2004 Commun. Theor. Phys. (Beijing) 42 497
[58]	Xie F D, Chen J, Lu Z S 2005 Commun. Theor. Phys. (Beijing) 43 585
[59]
[60]
[61]	Xie F D, Yuan Z T 2005 Commun. Theor. Phys. (Beijing) 43 39
[62]	Zhen X D, Chen Y, Li B, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 39 647
[63]
[64]	LU Z S, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 39 405
[65]
[66]
[67]	Xie F D, Gao X S 2004 Commun. Theor. Phys. (Beijing) 41 353
[68]	Chen Y, Li B 2004 Commun. Theor. Phys. (Beijing) 41 1
[69]
[70]
[71]	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 56 4319]
[72]
[73]	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 57 11]
[74]
[75]	Pan Z H, Ma S H, Fang J P 2010 Chin. Phys. B 19 100301(1)
[76]
[77]	Qiang J Y, Ma S H, Fang J P 2010 Chin. Phys. B 19 090305(1)
[78]	Taogetusang, Sirendaoerji, Li S M 2010 Chin. Phys. B 19 080303(1)
[79]
[80]	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]
[81]
[82]
[83]	Taogetusang, Sirendaoerji 2010 Acta Phys. Sin. 59 5194 (in Chinese)[套格图桑, 斯仁道尔吉 2010 59 5194]
[84]
[85]
[86]
[87]
[88]
[89]
[90]
[91]
[92]
[93]
[94]
[95]
[96]
[97]
[98]
[99]
[100]
[101]
[102]
[103]
[104]
[105]
[106]
[107]
[108]
[109]
[110]
[111]
[112]
[113]
[114]
[115]
[116]
[117]
[118]
[119]
[120]
[121]
[122]
[123]
[124]
[125]
[126]
[127]
[128]
[129]
[130]
[131]
[132]
[133]
[134]
[135]
[136]
[137]
[138]
[139]
[140]
[141]
[142]
[143]
[144]
[145]
[146]
[147]
[148]
[149]
[150]
[151]
[152]
[153]
[154]
[155]
[156]
[157]
[158]
[159]
[160]
[161]
[162]
[163]
[164]
[165]

施引文献

[1]	Liu S K, Fu Z T, Liu S D, Zhao Q 2002 Acta Phys. Sin. 51 1923 (in Chinese)[刘式适, 付遵涛, 刘式达, 赵强 2002 51 1923]
[2]
[3]	Li D S, Zhang H Q 2003 Acta Phys. Sin. 52 1569 (in Chinese)[李德生, 张鸿庆 2003 52 1569]
[4]
[5]	Zhang J F, Chen F Y 2001 Acta Phys. Sin. 50 1648 (in Chinese)[张解放, 陈芳跃 2001 50 1648]
[6]
[7]	Zhu J M, Zheng C L, Ma Z Y 2004 Chin. Phys. 13 2008
[8]	Lou S Y, Ruan H Y 1992 Acta Phys. Sin. 41 182 (in Chinese)[楼森岳, 阮航宇 1992 41 182]
[9]
[10]	Chan W L, Li K S 1989J. Math. Phys. 30 2521
[11]
[12]
[13]	Tian C 1987 J. Phys. A: Math. Gen. 20 359
[14]
[15]	Zhang J L, Ren D F,Wang M L,Wang Y M, Fang Z D 2003 Chin. Phys. 12 825
[16]	Zhang L, Zhang L F, Li C Y 2008 Chin. Phys. B 17 403
[17]
[18]	Zhao X Q, Zhi H Y, Zhang H Q 2006 Chin. Phys. 15 2202
[19]
[20]
[21]	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 57 3312]
[22]	Taogetusang, Sirendaoerji 2010 Acta Phys. Sin. 59 4413 (in Chinese)[套格图桑, 斯仁道尔吉 2010 59 4413]
[23]
[24]	Camassa R, Holm D D 1993 Phys. Rev. Lett. 71 1661
[25]
[26]
[27]	Rosenau P, Hyman J M 1993 Phys. Rev. Lett. 70 564
[28]
[29]	Dullin H R, Gottwald G A, Holm D D 2002 Phys. Rev. Lett. 87 4501
[30]
[31]	Guo B L, Liu Z R 2003 Science in China A 33 325 (in Chinese)[郭柏灵, 刘正荣 2003 中国科学 A 33 325]
[32]	Yin J L, Tian L X 2009 Acta Phys. Sin. 58 3632 (in Chinese)[殷久利, 田立新 2009 58 3632]
[33]
[34]	Yu L Q, Tian L X 2006 Math. Practice. Theory 36 261 (in Chinese)[余丽琴, 田立新 2006 数学的实践与认识 36 261]
[35]
[36]
[37]	Yu L Q, Tian L X 2005 Pure. Appl. Math 21 310 (in Chinese)[余丽琴, 田立新 2005 纯粹数学与应用数学 21 310]
[38]
[39]	Yan Z Y 2002 Chaos, Solitons and Fractals 14 1151
[40]
[41]	Yin J L, Tian L X 2007 Acta Math. Phys. 27A 027 (in Chinese)[殷久利, 田立新 2007 数学 27A 027]
[42]
[43]	Fan E G 2000 Phys. Lett. A 277 212
[44]	Chen Y, Li B, Zhang H Q 2003 Chin. Phys. 12 940
[45]
[46]	Chen Y, Yan Z Y, Li B, Zhang H Q 2003 Chin. Phys. 12 1
[47]
[48]
[49]	Chen Y, Li B, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 40 137
[50]
[51]	Li D S, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 40 143
[52]
[53]	Li D S, Zhang H Q 2004 Chin. Phys. 13 984
[54]
[55]	Li D S, Zhang H Q 2004 Chin. Phys. 13 1377
[56]
[57]	Chen H T, Zhang H Q 2004 Commun. Theor. Phys. (Beijing) 42 497
[58]	Xie F D, Chen J, Lu Z S 2005 Commun. Theor. Phys. (Beijing) 43 585
[59]
[60]
[61]	Xie F D, Yuan Z T 2005 Commun. Theor. Phys. (Beijing) 43 39
[62]	Zhen X D, Chen Y, Li B, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 39 647
[63]
[64]	LU Z S, Zhang H Q 2003 Commun. Theor. Phys. (Beijing) 39 405
[65]
[66]
[67]	Xie F D, Gao X S 2004 Commun. Theor. Phys. (Beijing) 41 353
[68]	Chen Y, Li B 2004 Commun. Theor. Phys. (Beijing) 41 1
[69]
[70]
[71]	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 56 4319]
[72]
[73]	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 57 11]
[74]
[75]	Pan Z H, Ma S H, Fang J P 2010 Chin. Phys. B 19 100301(1)
[76]
[77]	Qiang J Y, Ma S H, Fang J P 2010 Chin. Phys. B 19 090305(1)
[78]	Taogetusang, Sirendaoerji, Li S M 2010 Chin. Phys. B 19 080303(1)
[79]
[80]	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]
[81]
[82]
[83]	Taogetusang, Sirendaoerji 2010 Acta Phys. Sin. 59 5194 (in Chinese)[套格图桑, 斯仁道尔吉 2010 59 5194]
[84]
[85]
[86]
[87]
[88]
[89]
[90]
[91]
[92]
[93]
[94]
[95]
[96]
[97]
[98]
[99]
[100]
[101]
[102]
[103]
[104]
[105]
[106]
[107]
[108]
[109]
[110]
[111]
[112]
[113]
[114]
[115]
[116]
[117]
[118]
[119]
[120]
[121]
[122]
[123]
[124]
[125]
[126]
[127]
[128]
[129]
[130]
[131]
[132]
[133]
[134]
[135]
[136]
[137]
[138]
[139]
[140]
[141]
[142]
[143]
[144]
[145]
[146]
[147]
[148]
[149]
[150]
[151]
[152]
[153]
[154]
[155]
[156]
[157]
[158]
[159]
[160]
[161]
[162]
[163]
[164]
[165]

[1]	套格图桑. 构造非线性发展方程的无穷序列复合型类孤子新解. , 2013, 62(7): 070202. doi: 10.7498/aps.62.070202
[2]	万晖. 带源项的变系数非线性反应扩散方程的精确解. , 2013, 62(9): 090203. doi: 10.7498/aps.62.090203
[3]	庞晶, 靳玲花, 赵强. 变系数非线性发展方程的G'/G展开解. , 2012, 61(14): 140201. doi: 10.7498/aps.61.140201
[4]	白玉梅, 套格图桑, 韩元春. K(m,n)方程与B(m,n)方程的无穷序列新精确解. , 2012, 61(20): 200205. doi: 10.7498/aps.61.200205
[5]	套格图桑, 白玉梅. 构造非线性发展方程无穷序列类孤子精确解的一种方法. , 2012, 61(13): 130202. doi: 10.7498/aps.61.130202
[6]	套格图桑, 白玉梅. Nizhnik-Novikov-Vesselov方程的无穷序列类孤子精确解. , 2012, 61(11): 110203. doi: 10.7498/aps.61.110203
[7]	钱存, 王亮亮, 张解放. 变系数非线性Schrödinger方程的孤子解及其相互作用. , 2011, 60(6): 064214. doi: 10.7498/aps.60.064214
[8]	套格图桑. 构造非线性发展方程无穷序列复合型精确解的一种方法. , 2011, 60(1): 010202. doi: 10.7498/aps.60.010202
[9]	套格图桑. 几种辅助方程与非线性发展方程的无穷序列精确解. , 2011, 60(5): 050201. doi: 10.7498/aps.60.050201
[10]	套格图桑. sine-Gordon型方程的无穷序列新精确解. , 2011, 60(7): 070203. doi: 10.7498/aps.60.070203
[11]	套格图桑, 那仁满都拉. 第二种椭圆方程构造变系数非线性发展方程的无穷序列新精确解. , 2011, 60(9): 090201. doi: 10.7498/aps.60.090201
[12]	套格图桑, 斯仁道尔吉. 广义Boussinesq方程的无穷序列新精确解. , 2010, 59(7): 4413-4419. doi: 10.7498/aps.59.4413
[13]	套格图桑. 一般格子方程新的无穷序列精确解. , 2010, 59(10): 6712-6718. doi: 10.7498/aps.59.6712
[14]	套格图桑, 斯仁道尔吉. 构造变系数非线性发展方程精确解的一种方法. , 2009, 58(4): 2121-2126. doi: 10.7498/aps.58.2121
[15]	宗丰德, 戴朝卿, 杨琴, 张解放. 光纤中变系数非线性Schr?dinger方程的孤子解及其应用. , 2006, 55(8): 3805-3812. doi: 10.7498/aps.55.3805
[16]	套格图桑, 斯仁道尔吉. 构造非线性发展方程精确解的一种方法. , 2006, 55(12): 6214-6221. doi: 10.7498/aps.55.6214
[17]	韩兆秀. 非线性Klein-Gordon方程新的精确解. , 2005, 54(4): 1481-1484. doi: 10.7498/aps.54.1481
[18]	吕大昭. 非线性发展方程的丰富的Jacobi椭圆函数解. , 2005, 54(10): 4501-4505. doi: 10.7498/aps.54.4501
[19]	刘成仕. 用试探方程法求变系数非线性发展方程的精确解. , 2005, 54(10): 4506-4510. doi: 10.7498/aps.54.4506
[20]	阮航宇, 陈一新. 寻找变系数非线性方程精确解的新方法. , 2000, 49(2): 177-180. doi: 10.7498/aps.49.177

计量

文章访问数: 7760
PDF下载量: 898
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板