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

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

doi:10.7498/aps.59.1409

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.

Keywords:

Authors and contacts

1.
北京工商大学计算机与信息工程学院,北京 100048

References

[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 49 2113］
[3]	［3]Fan E G 2000 Acta Phys. Sin. 49 1409 （in Chinese）［范恩贵2000 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 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 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 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 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 53 4045］
[11]	［11]Taogetusang, Sirendaoerji 2006 Acta Phys. Sin. 55 13 （in Chinese）［套格图桑、斯仁道尔吉 2006 55 13］
[12]	［12]Taogetusang, Sirendaoerji 2006 Acta Phys. Sin. 55 3246 （in Chinese）［套格图桑、斯仁道尔吉 2006 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 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 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 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 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 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 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 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 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 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 54 504］
[46]	［46]Naranmandula 2002 Acta Phys. Sin. 51 1671 （in Chinese）［那仁满都拉 2009 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

Cited By

[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 49 2113］
[3]	［3]Fan E G 2000 Acta Phys. Sin. 49 1409 （in Chinese）［范恩贵2000 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 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 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 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 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 53 4045］
[11]	［11]Taogetusang, Sirendaoerji 2006 Acta Phys. Sin. 55 13 （in Chinese）［套格图桑、斯仁道尔吉 2006 55 13］
[12]	［12]Taogetusang, Sirendaoerji 2006 Acta Phys. Sin. 55 3246 （in Chinese）［套格图桑、斯仁道尔吉 2006 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 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 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 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 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 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 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 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 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 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 54 504］
[46]	［46]Naranmandula 2002 Acta Phys. Sin. 51 1671 （in Chinese）［那仁满都拉 2009 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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Vol. 59, Issue 3 - 2010-02-05

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

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

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