New type infinite sequence exact solutions of the second KdV equation with variable coefficients

Taogetusang; Bai Yu-Mei

doi:10.7498/aps.61.060201

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:

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).

References

[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 物理学报 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 物理学报 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 物理学报 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 物理学报 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 物理学报 57 3312]
[22]	Taogetusang, Sirendaoerji 2010 Acta Phys. Sin. 59 4413 (in Chinese)[套格图桑, 斯仁道尔吉 2010 物理学报 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 物理学报 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 数学物理学报 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 物理学报 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 物理学报 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 物理学报 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]

Cited By

[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 物理学报 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 物理学报 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 物理学报 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 物理学报 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 物理学报 57 3312]
[22]	Taogetusang, Sirendaoerji 2010 Acta Phys. Sin. 59 4413 (in Chinese)[套格图桑, 斯仁道尔吉 2010 物理学报 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 物理学报 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 数学物理学报 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 物理学报 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 物理学报 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 物理学报 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]	Taogetusang. Construction of new infinite sequence complexion soliton-like solutions of nonlinear evolution equations. Acta Physica Sinica, 2013, 62(7): 070202. doi: 10.7498/aps.62.070202
[2]	Wan Hui. Exact solutions to the nonlinear diffusion-convection equation with variable coefficients and source term. Acta Physica Sinica, 2013, 62(9): 090203. doi: 10.7498/aps.62.090203
[3]	Pang Jing, Jin Ling-Hua, Zhao Qiang. Nonlinear evolution equation with variable coefficient G'/G expansion solution. Acta Physica Sinica, 2012, 61(14): 140201. doi: 10.7498/aps.61.140201
[4]	Bai Yu-Mei, Taogetusang, Han Yuan-Chun. Infinite sequence new exact solutions of K(m,n) equation and B(m,n) equation. Acta Physica Sinica, 2012, 61(20): 200205. doi: 10.7498/aps.61.200205
[5]	Taogetusang, Bai Yu-Mei. A method of constructing infinite sequence soliton-like solutions of nonlinear evolution equations. Acta Physica Sinica, 2012, 61(13): 130202. doi: 10.7498/aps.61.130202
[6]	Taogetusang, Bai Yu-Mei. Infinite sequence soliton-like exact solutions of Nizhnik-Novikov-Vesselov equation. Acta Physica Sinica, 2012, 61(11): 110203. doi: 10.7498/aps.61.110203
[7]	Qian Cun, Wang Liang-Liang, Zhang Jie-Fang. Solitons of nonlinear Schrödinger equation withvariable-coefficients and interaction. Acta Physica Sinica, 2011, 60(6): 064214. doi: 10.7498/aps.60.064214
[8]	Taogetusang. A method for constructing infinite sequence complexiton solutions to nonlinear evolution equations. Acta Physica Sinica, 2011, 60(1): 010202. doi: 10.7498/aps.60.010202
[9]	Taogetusang. Several auxiliary equations and infinite sequence exactsolutions to nonlinear evolution equations. Acta Physica Sinica, 2011, 60(5): 050201. doi: 10.7498/aps.60.050201
[10]	Taogetusang. New infinite sequences exact solutions to sine-Gordon-type equations. Acta Physica Sinica, 2011, 60(7): 070203. doi: 10.7498/aps.60.070203
[11]	Taogetusang, Narenmandula. New infinite sequence exact solutions of nonlinear evolution equations with variable coefficients by the second kind of elliptic equation. Acta Physica Sinica, 2011, 60(9): 090201. doi: 10.7498/aps.60.090201
[12]	Taogetusang, Sirendaoerji. New exact infinite sequence solutions to generalized Boussinesq equation. Acta Physica Sinica, 2010, 59(7): 4413-4419. doi: 10.7498/aps.59.4413
[13]	Taogetusang. New infinite sequence exact solutions to the general lattice. Acta Physica Sinica, 2010, 59(10): 6712-6718. doi: 10.7498/aps.59.6712
[14]	Taogetusang, Sirendaoerji. A method for constructing exact solutions of nonlinear evolution equation with variable coefficients. Acta Physica Sinica, 2009, 58(4): 2121-2126. doi: 10.7498/aps.58.2121
[15]	Zong Feng-De, Dai Chao-Qing, Yang Qin, Zhang Jie-Fang. Soliton solutions for variable coefficient nonlinear Schr?dinger equation for optical fiber and their application. Acta Physica Sinica, 2006, 55(8): 3805-3812. doi: 10.7498/aps.55.3805
[16]	Taogetusang, Sirendaoerji. The method for constructing the exact solutions to the nonlinear evolution equation. Acta Physica Sinica, 2006, 55(12): 6214-6221. doi: 10.7498/aps.55.6214
[17]	Han Zhao-Xiu. New exact solutions for nonlinear Klein-Gordon equations. Acta Physica Sinica, 2005, 54(4): 1481-1484. doi: 10.7498/aps.54.1481
[18]	Lü Da-Zhao. Abundant Jacobi elliptic function solutions of nonlinear evolution equations. Acta Physica Sinica, 2005, 54(10): 4501-4505. doi: 10.7498/aps.54.4501
[19]	Liu Cheng-Shi. Using trial equation method to solve the exact solutions for two kinds of KdV equations with variable coefficients. Acta Physica Sinica, 2005, 54(10): 4506-4510. doi: 10.7498/aps.54.4506
[20]	Ruan Hang-Yu, Chen Yi-Xin. . Acta Physica Sinica, 2000, 49(2): 177-180. doi: 10.7498/aps.49.177

Catalog

Vol. 61, Issue 6 - 2012-03-20

Metrics

Abstract views: 6372
PDF Downloads: 891
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板