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Explicit and exact traveling wave solutions to the nonlinear LC circuit equation

Shang Ya-Dong Huang Yong

Explicit and exact traveling wave solutions to the nonlinear LC circuit equation

Shang Ya-Dong, Huang Yong
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  • Traveling wave in a nonlinear LC circuit with dissipation have been investigated theoretically. With the aid of the extended hyperbolic function method,developed by the authors in recent works to solve nonlinear partial differential equations exactly, the fourth order nonlinear wave equation with dissipation, which models shock wave propagation in a nonlinear LC circuit, have been analytically studied. Abundant explicit and exact traveling wave solutions to the fourth order nonlinear wave equation with dissipation are obtained. These solutions include exact shock wave solutions, singular traveling wave solutions, and periodic wave solutions in a rational form of trigonometric functions.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 40890150, 40890153, 11271090) and the Scientific Program of Guangdong Province, China (Grant No. 2008B080701042).
    [1]

    Watanabe S, Miyakawa M, Tada M 1978 J. Phys. Soc. Jpn. 45 2030

    [2]

    Watanabe S, Miyakawa M, Muroya K 1980 J. Phys. Soc. Jpn. 49 825

    [3]

    Saitoh N , Watanabe S 1981 J. Phys. Soc. Jpn. 50 1774

    [4]

    Muroya K, Watanabe S 1981 J. Phys. Soc. Jpn. 50 2762

    [5]

    Muroya K, Watanabe S 1981 J. Phys. Soc. Jpn. 50 3159

    [6]

    Watanabe S, Muroya K 1981 J. Phys. Soc. Jpn. 50 3166

    [7]

    Watanabe S, Tada M 1981 J. Phys. Soc. Jpn. 50 3436

    [8]

    Watanabe S, Tada M 1981 J. Phys. Soc. Jpn. 50 3443

    [9]

    Muroya K, Saitoh N, Watanabe S 1982 J. Phys. Soc. Jpn. 51 1024

    [10]

    Watanabe S 1982 J. Phys. Soc. Jpn. 51 1030

    [11]

    Kako F, Miyakawa M, Watanabe S 1986 J. Phys. Soc. Jpn. 55 2919

    [12]

    Kako F, Miyakawa M, Watanabe S 1986 J. Phys. Soc. Jpn. 55 2928

    [13]

    Matsukawa M, Watanabe S, Tanaca H 1989 J. Phys. Soc. Jpn. 58 3081

    [14]

    Ishiwata S, Watanabe S, Tanaca H 1990 J. Phys. Soc. Jpn. 59 1163

    [15]

    Okada Y, Watanabe S, Tanaca H 1990 J. Phys. Soc. Jpn. 59 2647

    [16]

    Kawamura K, Watanabe S 1991 J. Phys. Soc. Jpn. 60 82

    [17]

    Hietarinta J, Kuusela T, Malomed B A 1995 J. Phys. A: Math. Gen. 28 3015

    [18]

    Oh H G, Watanabe S 1997 J. Phys. Soc. Jpn. 66 979

    [19]

    Watanabe S, Ishiwata S, Kawamura K, Oh H G 1997 J. Phys. Soc. Jpn. 66 984

    [20]

    Watanabe S, Kawaguchi M, Kawamura K, Ishiwata S, Ohta Y, Oh H G 1997 J. Phys. Soc. Jpn. 66 1231

    [21]

    Asano H, Kakei S, Ishiwata S, Watanabe S 1999 J. Phys. Soc. Jpn. 68 3208

    [22]

    Malfliet W, Rombouts B 2001 Math. Comput. Simulation 55 541

    [23]

    Zhu W Q 1980 Acta Solid Mechanics Sinica 1 247 (in Chinese) [朱位秋 1980 固体力学学报 1 247]

    [24]

    Taogetusang 2011 Acta Phys. Sin. 60 010202(in Chinese) [套格图桑 2011 物理学报 60 010202]

    [25]

    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]

    [26]

    Shang Y D, Qin J H, Huang Y, Yuan W J 2008 Appl. Math. Comput 202 532

    [27]

    Shang Y D, Huang Y, Yuan W J 2008 Appl. Math. Comput 200 110

    [28]

    Shang Y D, Huang Y, Yuan W J 2008 Chaos Solitons Fractals 36 762

    [29]

    Shang Y D, Huang Y, Yuan W J 2008 Comput Math. Appl. 56 1441

    [30]

    Huang Y, Shang Y D 2012 J. Appl. Math 2012 769843

    [31]

    Zhang G X, Li Z B, Duan Y S 2000 Science in China (Series A) 301103(in Chinese) [张桂戌, 李志斌, 段一士 2000 中国科学A 30 1103]

    [32]

    Huang D J, Zhang H Q 2004 Acta Phys. Sin. 53 2434 (in Chinese) [黄定江, 张鸿庆 2004 物理学报 53 2434]

    [33]

    Wang Q, Chen Y, Li B, Zahng H Q 2005 Appl. Math. Comput 160 77

    [34]

    Yan Z Y 2003 Chaos Solitons Fractals 16 759

    [35]

    Chen Y, Li B 2004 Commun. Theor. Phys. 41 1

    [36]

    Chen Y Z, Ding X W 2005 Nonlinear Analysis 61 1005

    [37]

    Lu K P, Shi Y R, Duan W S 2001 Acta Phys. Sin. 50 2074 (in Chinese) [吕克璞, 石玉仁, 段文山等 2001 物理学报 50 2074]

    [38]

    Guo G P, Zhang J F 2002 Acta Phys. Sin. 51 1159 (in Chinese) [郭冠平, 张解放 2002 物理学报 51 1159]

    [39]

    Shi Y R, Zhang J, Yang H J, Duan W S 2010 Acta Phys. Sin. 59 7564 (in Chinese) [石玉仁, 张娟, 杨文娟, 段文山 2010 物理学报 59 7564]

    [40]

    Shi Y R, Zhang J, Yang H J, Duan W S 2011 Acta Phys. Sin. 60 020401 (in Chinese) [石玉仁, 张娟, 杨文娟, 段文山 2011 物理学报 60 020402]

    [41]

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

    [42]

    Ye C E, Zhang W G 2010 Acta Phys. Sin. 59 5229 (in Chinese) [叶采儿, 张卫国 2010 物理学报 59 5229]

    [43]

    Li X Z, Zhang W G, Yuan S L 2010 Acta Phys. Sin. 59 0744 (in Chinese) [李向正, 张卫国, 原三领 2010 物理学报 59 0744]

    [44]

    Feng Z S, Li Y 2006 Physica A 366 115

    [45]

    Feng Z S 2008 Chaos Solitons Fractals 38 481

  • [1]

    Watanabe S, Miyakawa M, Tada M 1978 J. Phys. Soc. Jpn. 45 2030

    [2]

    Watanabe S, Miyakawa M, Muroya K 1980 J. Phys. Soc. Jpn. 49 825

    [3]

    Saitoh N , Watanabe S 1981 J. Phys. Soc. Jpn. 50 1774

    [4]

    Muroya K, Watanabe S 1981 J. Phys. Soc. Jpn. 50 2762

    [5]

    Muroya K, Watanabe S 1981 J. Phys. Soc. Jpn. 50 3159

    [6]

    Watanabe S, Muroya K 1981 J. Phys. Soc. Jpn. 50 3166

    [7]

    Watanabe S, Tada M 1981 J. Phys. Soc. Jpn. 50 3436

    [8]

    Watanabe S, Tada M 1981 J. Phys. Soc. Jpn. 50 3443

    [9]

    Muroya K, Saitoh N, Watanabe S 1982 J. Phys. Soc. Jpn. 51 1024

    [10]

    Watanabe S 1982 J. Phys. Soc. Jpn. 51 1030

    [11]

    Kako F, Miyakawa M, Watanabe S 1986 J. Phys. Soc. Jpn. 55 2919

    [12]

    Kako F, Miyakawa M, Watanabe S 1986 J. Phys. Soc. Jpn. 55 2928

    [13]

    Matsukawa M, Watanabe S, Tanaca H 1989 J. Phys. Soc. Jpn. 58 3081

    [14]

    Ishiwata S, Watanabe S, Tanaca H 1990 J. Phys. Soc. Jpn. 59 1163

    [15]

    Okada Y, Watanabe S, Tanaca H 1990 J. Phys. Soc. Jpn. 59 2647

    [16]

    Kawamura K, Watanabe S 1991 J. Phys. Soc. Jpn. 60 82

    [17]

    Hietarinta J, Kuusela T, Malomed B A 1995 J. Phys. A: Math. Gen. 28 3015

    [18]

    Oh H G, Watanabe S 1997 J. Phys. Soc. Jpn. 66 979

    [19]

    Watanabe S, Ishiwata S, Kawamura K, Oh H G 1997 J. Phys. Soc. Jpn. 66 984

    [20]

    Watanabe S, Kawaguchi M, Kawamura K, Ishiwata S, Ohta Y, Oh H G 1997 J. Phys. Soc. Jpn. 66 1231

    [21]

    Asano H, Kakei S, Ishiwata S, Watanabe S 1999 J. Phys. Soc. Jpn. 68 3208

    [22]

    Malfliet W, Rombouts B 2001 Math. Comput. Simulation 55 541

    [23]

    Zhu W Q 1980 Acta Solid Mechanics Sinica 1 247 (in Chinese) [朱位秋 1980 固体力学学报 1 247]

    [24]

    Taogetusang 2011 Acta Phys. Sin. 60 010202(in Chinese) [套格图桑 2011 物理学报 60 010202]

    [25]

    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]

    [26]

    Shang Y D, Qin J H, Huang Y, Yuan W J 2008 Appl. Math. Comput 202 532

    [27]

    Shang Y D, Huang Y, Yuan W J 2008 Appl. Math. Comput 200 110

    [28]

    Shang Y D, Huang Y, Yuan W J 2008 Chaos Solitons Fractals 36 762

    [29]

    Shang Y D, Huang Y, Yuan W J 2008 Comput Math. Appl. 56 1441

    [30]

    Huang Y, Shang Y D 2012 J. Appl. Math 2012 769843

    [31]

    Zhang G X, Li Z B, Duan Y S 2000 Science in China (Series A) 301103(in Chinese) [张桂戌, 李志斌, 段一士 2000 中国科学A 30 1103]

    [32]

    Huang D J, Zhang H Q 2004 Acta Phys. Sin. 53 2434 (in Chinese) [黄定江, 张鸿庆 2004 物理学报 53 2434]

    [33]

    Wang Q, Chen Y, Li B, Zahng H Q 2005 Appl. Math. Comput 160 77

    [34]

    Yan Z Y 2003 Chaos Solitons Fractals 16 759

    [35]

    Chen Y, Li B 2004 Commun. Theor. Phys. 41 1

    [36]

    Chen Y Z, Ding X W 2005 Nonlinear Analysis 61 1005

    [37]

    Lu K P, Shi Y R, Duan W S 2001 Acta Phys. Sin. 50 2074 (in Chinese) [吕克璞, 石玉仁, 段文山等 2001 物理学报 50 2074]

    [38]

    Guo G P, Zhang J F 2002 Acta Phys. Sin. 51 1159 (in Chinese) [郭冠平, 张解放 2002 物理学报 51 1159]

    [39]

    Shi Y R, Zhang J, Yang H J, Duan W S 2010 Acta Phys. Sin. 59 7564 (in Chinese) [石玉仁, 张娟, 杨文娟, 段文山 2010 物理学报 59 7564]

    [40]

    Shi Y R, Zhang J, Yang H J, Duan W S 2011 Acta Phys. Sin. 60 020401 (in Chinese) [石玉仁, 张娟, 杨文娟, 段文山 2011 物理学报 60 020402]

    [41]

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

    [42]

    Ye C E, Zhang W G 2010 Acta Phys. Sin. 59 5229 (in Chinese) [叶采儿, 张卫国 2010 物理学报 59 5229]

    [43]

    Li X Z, Zhang W G, Yuan S L 2010 Acta Phys. Sin. 59 0744 (in Chinese) [李向正, 张卫国, 原三领 2010 物理学报 59 0744]

    [44]

    Feng Z S, Li Y 2006 Physica A 366 115

    [45]

    Feng Z S 2008 Chaos Solitons Fractals 38 481

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  • Received Date:  26 September 2012
  • Accepted Date:  28 November 2012
  • Published Online:  05 April 2013

Explicit and exact traveling wave solutions to the nonlinear LC circuit equation

  • 1. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China;
  • 2. Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China;
  • 3. School of Computer Science and Educational Software, Guangzhou University, Guangzhou 510006, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant Nos. 40890150, 40890153, 11271090) and the Scientific Program of Guangdong Province, China (Grant No. 2008B080701042).

Abstract: Traveling wave in a nonlinear LC circuit with dissipation have been investigated theoretically. With the aid of the extended hyperbolic function method,developed by the authors in recent works to solve nonlinear partial differential equations exactly, the fourth order nonlinear wave equation with dissipation, which models shock wave propagation in a nonlinear LC circuit, have been analytically studied. Abundant explicit and exact traveling wave solutions to the fourth order nonlinear wave equation with dissipation are obtained. These solutions include exact shock wave solutions, singular traveling wave solutions, and periodic wave solutions in a rational form of trigonometric functions.

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