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(G'/G)-expansion method and novel fractal structures for high-dimensional nonlinear physical equation

Li Bang-Qing Ma Yu-Lan Xu Mei-Ping

(G'/G)-expansion method and novel fractal structures for high-dimensional nonlinear physical equation

Li Bang-Qing, Ma Yu-Lan, Xu Mei-Ping
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  • The (G′/G)-expansion method is extended to construct non-traveling wave solutions and explore the fractal structure for high dimensional nonlinear physical equation. As an example, a series of non-traveling solutions is obtained for the (2+1)-dimensional dispersive long wave system with variable coefficient. Furthermore, by setting properly the arbitrary functions in the solutions, a class of novel fractal structures, namely, the cross-like fractal structures are firstly observed.
    • Funds:
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    [1]Wang M L 1995 Phys. Lett. A 199 169

    [2]

    [2]Yan Z Y, Zhang H Q 2000 Acta Phys. Sin. 49 2113 (in Chinese)[闫振亚、张鸿庆 2000 物理学报 49 2113]

    [3]

    [3]Fan E G 2000 Acta Phys. Sin. 49 1409 (in Chinese)[范恩贵2000 物理学报 49 1409]

    [4]

    [4]Liu S K, Liu S D, Fu Z T 2001 Acta Phys. Sin. 50 2068 (in Chinese)[刘式适、刘适达、付遵涛 2001 物理学报 50 2068]

    [5]

    [5]Liu S D, Fu Z T, Liu S K, Zhao Q 2002 Acta Phys. Sin. 51 718 (in Chinese)[刘式达、付遵涛、刘适式、赵强 2002 物理学报 51 718 ]

    [6]

    [6]Wu G J, Han J H, Shi L M, Zhang M 2006 Acta Phys. Sin. 55 3858 (in Chinese)[吴国将、韩家骅、史良马、张苗 2006 物理学报 55 3858 ]

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    [7]Wang M L, Zhou Y B, Li Z B 2003 Phys. Lett. A 318 84

    [8]

    [8]Zhou Y B, Wang ML, Miao T D 2004 Phys. Lett. A 323 77

    [9]

    [9]Zhang J L,Wang Y M,Wang M L 2003 Acta Phys. Sin. 52 1574 (in Chinese)[张金良、王跃明、王明亮 2003物理学报 52 1574]

    [10]

    [10]Li X Z, Zhang J L, Wang Y M, Wang M L 2004 Acta Phys. Sin. 53 4045 (in Chinese)[李向正、张金良、王跃明、王明亮 2004 物理学报 53 4045]

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    [11]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]

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    [12]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]

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    [13]Li X Z, Li X Y, Zhao L Y, Zhang J L 2008 Acta Phys. Sin. 57 2203 (in Chinese)[李向正、李修勇、赵丽英、张金良 2008 物理学报 57 2203]

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    [14]Ma Y L, Li B Q 2009 Appl. Math. Comput. 211 102

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    [15]Lou S Y, Qu C Z, Zhang S L 2006 Chin. Phys. 15 2765

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    [16]Tang X Y, Liang Z F 2006 Phys. Lett. A 351 398

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    [17]Zhang S L, Lou S Y, Qu C Z 2006 Chin. Phys. 15 2765

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    [18]Ma H C, Ge D J, Yu Y D 2008 Chin. Phys. B 17 1448

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    [19]Ying J P, Lou S Y 2003 Chin. Phys. Lett. 20 1448

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    [20]Fang J P, Zheng C L, Zhu J M 2005 Commun. Theor. Phys. 44 203

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    [21]Fang J P, Zheng C L 2005 Chin. Phys. 14 670

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    [22]Fang J P, Zheng C L, Zhu J M 2005 Acta Phys. Sin. 54 2990 (in Chinese)[方建平、郑春龙、朱加民 2005 物理学报 54 2990]

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    [23]Ma S H, Wu X H, Fang J P, Zheng C L 2006 Z. Naturforsch. 61a 249

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    [24]Ma S H, Fang J P 2006 Acta Phys. Sin. 55 5611 (in Chinese)[马松华、方建平 2006 物理学报 55 5611]

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    [25]Ma S H, Fang J P, Zheng C L 2008 Chin. Phys. B 17 2767

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    [26]Ma S H, Fang J P,Hong B H, Zheng C L 2008 Commun. Theor. Phys. 49 1245

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    [27]Huang L, Sun J A, Dou F Q, Duan W S, Liu X X 2007 Acta Phys. Sin. 56 611 (in Chinese)[黄磊、孙建安、豆福全、段文山 2007 物理学报 56 611]

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    [28]Ma S H, Fang J P, Zheng C L 2007 Z. Naturforsch. 62a 8

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    [29]Ma S H, Fang J P, Zheng C L 2008 Z. Naturforsch. 63a 121

    [30]

    [30]Li J B, Zheng C L, Ma S H 2008 Z. Naturforsch. 63a 641

    [31]

    [31]Ma S H, Wu X H, Fang J P, Zheng C L 2008 Acta Phys. Sin. 57 0011 (in Chinese)[马松华、吴小红、方建平、郑春龙 2008 物理学报 57 0011]

    [32]

    [32]Lou S Y, Tang X Y 2006 Methods of Nonlinear Mathematical Physics(Beijing: Science Press) p120(in Chinese)[楼森岳、唐晓艳 2006 非线性数学物理方法(北京: 科学出版社) 第120页]

    [33]

    [33]Ma Z Y, Zhu J M, Zheng C L 2004 Chin. Phys. 13 1382

    [34]

    [34]Zhu J M, Ma Z Y, Zheng C L 2004 Acta Phys. Sin. 53 3248 (in Chinese)[朱加民、马正义、郑春龙 2004 物理学报 53 3248]

    [35]

    [35] Ma Z Y, Zheng C L 2006 Chin. Phys. 15 0045

    [36]

    [36]Ma S H, Fang J P, Ren Q B 2007 Acta Phys. Sin. 56 6784 (in Chinese)[马松华、方建平 2007 物理学报 56 6784]

    [37]

    [37]Ma Z Y, Hu Y H 2007 Chaos, Solitons Fractals 34 1667

    [38]

    [38]Wang M L, Li X Z,Zhang J L 2008 Phys. Lett. A 372 417

    [39]

    [39]Zhang S, Tong J L, Wang W 2008 Phys. Lett. A 372 2254

    [40]

    [40]Bekir A. 2008 Phys. Lett. A 372 3400

    [41]

    [41]Zhang J, Wei X L, Lu Y J 2008 Phys. Lett. A 372 3653

    [42]

    [42]Li B Q, Ma Y L 2009 Acta Phys. Sin. 58 4373 (in Chinese)[李帮庆、马玉兰 2009 物理学报58 4373]

    [43]

    [43]Ma Y L, Li B Q, Sun J Z 2009 Acta Phys. Sin. 58 7402 (in Chinese, in press)[ 马玉兰、李帮庆、孙践知 2009 物理学报 58 7402]

    [44]

    [44]Boiti P, Leon J J, Manna M, Pempinelli F 1987 Inverse Problems 3 25

    [45]

    [45]Zeng X, Zhang H Q 2005 Acta Phys. Sin. 54 504 (in Chinese)[曾昕、张鸿庆 2005 物理学报 54 504]

    [46]

    [46]Naranmandula 2002 Acta Phys. Sin. 51 1671 (in Chinese)[那仁满都拉 2009 物理学报 51 1671]

    [47]

    [47]Liu C S 2005 Chin. Phys. 14 1710

    [48]

    [48]Zhang W L, Wu G J, Zhang M, Wang J M, Han J H 2008 Chin. Phys. B 17 1156

  • [1]

    [1]Wang M L 1995 Phys. Lett. A 199 169

    [2]

    [2]Yan Z Y, Zhang H Q 2000 Acta Phys. Sin. 49 2113 (in Chinese)[闫振亚、张鸿庆 2000 物理学报 49 2113]

    [3]

    [3]Fan E G 2000 Acta Phys. Sin. 49 1409 (in Chinese)[范恩贵2000 物理学报 49 1409]

    [4]

    [4]Liu S K, Liu S D, Fu Z T 2001 Acta Phys. Sin. 50 2068 (in Chinese)[刘式适、刘适达、付遵涛 2001 物理学报 50 2068]

    [5]

    [5]Liu S D, Fu Z T, Liu S K, Zhao Q 2002 Acta Phys. Sin. 51 718 (in Chinese)[刘式达、付遵涛、刘适式、赵强 2002 物理学报 51 718 ]

    [6]

    [6]Wu G J, Han J H, Shi L M, Zhang M 2006 Acta Phys. Sin. 55 3858 (in Chinese)[吴国将、韩家骅、史良马、张苗 2006 物理学报 55 3858 ]

    [7]

    [7]Wang M L, Zhou Y B, Li Z B 2003 Phys. Lett. A 318 84

    [8]

    [8]Zhou Y B, Wang ML, Miao T D 2004 Phys. Lett. A 323 77

    [9]

    [9]Zhang J L,Wang Y M,Wang M L 2003 Acta Phys. Sin. 52 1574 (in Chinese)[张金良、王跃明、王明亮 2003物理学报 52 1574]

    [10]

    [10]Li X Z, Zhang J L, Wang Y M, Wang M L 2004 Acta Phys. Sin. 53 4045 (in Chinese)[李向正、张金良、王跃明、王明亮 2004 物理学报 53 4045]

    [11]

    [11]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]

    [12]

    [12]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]

    [13]

    [13]Li X Z, Li X Y, Zhao L Y, Zhang J L 2008 Acta Phys. Sin. 57 2203 (in Chinese)[李向正、李修勇、赵丽英、张金良 2008 物理学报 57 2203]

    [14]

    [14]Ma Y L, Li B Q 2009 Appl. Math. Comput. 211 102

    [15]

    [15]Lou S Y, Qu C Z, Zhang S L 2006 Chin. Phys. 15 2765

    [16]

    [16]Tang X Y, Liang Z F 2006 Phys. Lett. A 351 398

    [17]

    [17]Zhang S L, Lou S Y, Qu C Z 2006 Chin. Phys. 15 2765

    [18]

    [18]Ma H C, Ge D J, Yu Y D 2008 Chin. Phys. B 17 1448

    [19]

    [19]Ying J P, Lou S Y 2003 Chin. Phys. Lett. 20 1448

    [20]

    [20]Fang J P, Zheng C L, Zhu J M 2005 Commun. Theor. Phys. 44 203

    [21]

    [21]Fang J P, Zheng C L 2005 Chin. Phys. 14 670

    [22]

    [22]Fang J P, Zheng C L, Zhu J M 2005 Acta Phys. Sin. 54 2990 (in Chinese)[方建平、郑春龙、朱加民 2005 物理学报 54 2990]

    [23]

    [23]Ma S H, Wu X H, Fang J P, Zheng C L 2006 Z. Naturforsch. 61a 249

    [24]

    [24]Ma S H, Fang J P 2006 Acta Phys. Sin. 55 5611 (in Chinese)[马松华、方建平 2006 物理学报 55 5611]

    [25]

    [25]Ma S H, Fang J P, Zheng C L 2008 Chin. Phys. B 17 2767

    [26]

    [26]Ma S H, Fang J P,Hong B H, Zheng C L 2008 Commun. Theor. Phys. 49 1245

    [27]

    [27]Huang L, Sun J A, Dou F Q, Duan W S, Liu X X 2007 Acta Phys. Sin. 56 611 (in Chinese)[黄磊、孙建安、豆福全、段文山 2007 物理学报 56 611]

    [28]

    [28]Ma S H, Fang J P, Zheng C L 2007 Z. Naturforsch. 62a 8

    [29]

    [29]Ma S H, Fang J P, Zheng C L 2008 Z. Naturforsch. 63a 121

    [30]

    [30]Li J B, Zheng C L, Ma S H 2008 Z. Naturforsch. 63a 641

    [31]

    [31]Ma S H, Wu X H, Fang J P, Zheng C L 2008 Acta Phys. Sin. 57 0011 (in Chinese)[马松华、吴小红、方建平、郑春龙 2008 物理学报 57 0011]

    [32]

    [32]Lou S Y, Tang X Y 2006 Methods of Nonlinear Mathematical Physics(Beijing: Science Press) p120(in Chinese)[楼森岳、唐晓艳 2006 非线性数学物理方法(北京: 科学出版社) 第120页]

    [33]

    [33]Ma Z Y, Zhu J M, Zheng C L 2004 Chin. Phys. 13 1382

    [34]

    [34]Zhu J M, Ma Z Y, Zheng C L 2004 Acta Phys. Sin. 53 3248 (in Chinese)[朱加民、马正义、郑春龙 2004 物理学报 53 3248]

    [35]

    [35] Ma Z Y, Zheng C L 2006 Chin. Phys. 15 0045

    [36]

    [36]Ma S H, Fang J P, Ren Q B 2007 Acta Phys. Sin. 56 6784 (in Chinese)[马松华、方建平 2007 物理学报 56 6784]

    [37]

    [37]Ma Z Y, Hu Y H 2007 Chaos, Solitons Fractals 34 1667

    [38]

    [38]Wang M L, Li X Z,Zhang J L 2008 Phys. Lett. A 372 417

    [39]

    [39]Zhang S, Tong J L, Wang W 2008 Phys. Lett. A 372 2254

    [40]

    [40]Bekir A. 2008 Phys. Lett. A 372 3400

    [41]

    [41]Zhang J, Wei X L, Lu Y J 2008 Phys. Lett. A 372 3653

    [42]

    [42]Li B Q, Ma Y L 2009 Acta Phys. Sin. 58 4373 (in Chinese)[李帮庆、马玉兰 2009 物理学报58 4373]

    [43]

    [43]Ma Y L, Li B Q, Sun J Z 2009 Acta Phys. Sin. 58 7402 (in Chinese, in press)[ 马玉兰、李帮庆、孙践知 2009 物理学报 58 7402]

    [44]

    [44]Boiti P, Leon J J, Manna M, Pempinelli F 1987 Inverse Problems 3 25

    [45]

    [45]Zeng X, Zhang H Q 2005 Acta Phys. Sin. 54 504 (in Chinese)[曾昕、张鸿庆 2005 物理学报 54 504]

    [46]

    [46]Naranmandula 2002 Acta Phys. Sin. 51 1671 (in Chinese)[那仁满都拉 2009 物理学报 51 1671]

    [47]

    [47]Liu C S 2005 Chin. Phys. 14 1710

    [48]

    [48]Zhang W L, Wu G J, Zhang M, Wang J M, Han J H 2008 Chin. Phys. B 17 1156

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  • Received Date:  25 May 2009
  • Accepted Date:  26 June 2009
  • Published Online:  15 March 2010

(G'/G)-expansion method and novel fractal structures for high-dimensional nonlinear physical equation

  • 1. 北京工商大学计算机与信息工程学院,北京 100048

Abstract: The (G′/G)-expansion method is extended to construct non-traveling wave solutions and explore the fractal structure for high dimensional nonlinear physical equation. As an example, a series of non-traveling solutions is obtained for the (2+1)-dimensional dispersive long wave system with variable coefficient. Furthermore, by setting properly the arbitrary functions in the solutions, a class of novel fractal structures, namely, the cross-like fractal structures are firstly observed.

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