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Analytic approximation for the soliton solution of the discrete modified KdV equation

Yang Pei Chen Yong Li Zhi-Bin

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Analytic approximation for the soliton solution of the discrete modified KdV equation

Yang Pei, Chen Yong, Li Zhi-Bin
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  • We extend the homotopy analysis method to solving the discrete modified KdV equation. The bright soliton solution expressed by a series of exponential functions is obtained, which agrees well with the exact solution. It indicates the validity and great potential of the homotopy analysis method in solving complicated nonlinear problems.
    [1]

    [1]Fermi E, Pasta J, Ulam S 1965 Collected Papers of Enrico Fermi(Ⅱ) (Chicago: University of Chicago Press) p1

    [2]

    [2]Suris Y B 1997 J. Phys. A 30 1745

    [3]

    [3]Suris Y B 1999 Rev. Math. Phys. 11 727

    [4]

    [4]Suris Y B 2003 The Problem of Integrable Discretization: Hamitonian Approach, Progress in Mathemetics ( Vol. 219) (Basel: Birkhuser-Verlag) p1

    [5]

    [5]Blaszak M, Marciniak K 1994 J. Math. Phys. 35 4661

    [6]

    [6]Belov A A, Chaltikian K D 1993 Phys. Lett. B 309 268

    [7]

    [7]Hu X B, Zhu Z N 1998 J. Phys. A 31 4755

    [8]

    [8]Hu X B, Zhu Z N 1998 J. Math. Phys. 39 4766

    [9]

    [9]Ma W X, Hu X B, Zhu S M, Wu Y T 1999 J. Math. Phys. 40 6071

    [10]

    ]Zhang D J 2005 Chaos Solitons Fract. 23 1333

    [11]

    ]Sun Y P, Chen D Y, Xu X X 2006 Phys. Lett. A 359 47

    [12]

    ]Zhu J, Geng X 2007 Phys. Lett. A 368 464

    [13]

    ]Hu X B, Wu Y T 1998 Phys. Lett. A 246 523

    [14]

    ]Chou K S, Qu C Z 2001 Phys. Lett. A 280 303

    [15]

    ]Luo L, Fan E G 2007 Chin. Phys. Lett. 24 1444

    [16]

    ]Zhang W, Huang Y Z, Xiao Y 1998 Phys. Rev. E 57 7358

    [17]

    ]Xiao Y, Hai W H 1994 J. Phys. A 27 6873

    [18]

    ]Chow K W 1994 Phys. Scr. 50 233

    [19]

    ]Shek E C M, Chow K W 2008 Chaos Solitons Fract. 36 296

    [20]

    ]Yang P, Chen Y, Li Z B 2009 Appl. Math. Comput. 210 362

    [21]

    ]Yang P, Chen Y, Li Z B 2008 Chin. Phys. B 17 3953

    [22]

    ]Taogetusang, Sirendaoerji 2009 Acta Phys. Sin. 58 5894 (in Chinese) [套格图桑、斯仁道尔吉 2009 物理学报 58 5894]

    [23]

    ]Zhang S Q 2007 Acta Phys. Sin. 56 1870 (in Chinese) [张善卿 2007 物理学报 56 1870]

    [24]

    ]Liao S J 2003 Beyond Perturbation: Introduction to the Homotopy Analysis Method (Boca Raton: Chapman & Hall/CRC Press) p1

    [25]

    ]Liao S J 2002 J. Fluid Mech. 453 411

    [26]

    ]Liao S J 2003 Int. J. Non-Lin. Mech. 38 1173

    [27]

    ]Liao S J 2003 J. Fluid Mech. 488 189

    [28]

    ]Liao S J 2003 Appl. Math. Comput. 144 495

    [29]

    ]Liao S J 2004 Int. J. Non-Lin. Mech. 39 271

    [30]

    ]Liao S J 2004 Appl. Math. Comput. 147 499

    [31]

    ]Wu Y Y, Wang C, Liao S J 2005 Chaos Solitons Fract. 23 1733

    [32]

    ]Wu W, Liao S J 2005 Chaos Solitons Fract. 26 177

    [33]

    ]Liao S J 2009 Commun. Nonlin. Sci. Numer. Simulat. 14 983

    [34]

    ]Liao S J 2009 Commun. Nonlin. Sci. Numer. Simulat. 14 2144

    [35]

    ]Liao S J, Tan Y 2007 Stud. Appl. Math. 119 297

    [36]

    ]Abbasbandy S, Parkes E J 2008 Chaos Solitons Fract. 36 581

    [37]

    ]Abbasbandy S 2009 Chaos Solitons Fract. 39 428

    [38]

    ]Liu Y P, Li Z B 2008 Z. Naturforsch. A 63 241

    [39]

    ]Liu Y P, Li Z B 2009 J. Phys. A 42 125205

    [40]

    ]Liu Y P, Li Z B 2009 Chaos Solitons Fract. 39 1

    [41]

    ]Yang H J, Shi Y R, Duan W S,Lü K P 2007 Acta Phys. Sin. 56 3064 (in Chinese) [杨红娟、石玉仁、段文山、吕克璞 2007 物理学报 56 3064]

    [42]

    ]Shi Y R, Wang Y H, Yang H J, Duan W S 2007 Acta Phys. Sin. 56 6791 (in Chinese) [石玉仁、汪映海、杨红娟、段文山 2007 物理学报 56 6791]

    [43]

    ]Shi Y R, Xu X J, Wu Z X, Wang Y H, Yang H J, Duan W S, Lü K P 2006 Acta Phys. Sin. 55 1555 (in Chinese) [石玉仁、许新建、吴枝喜、汪映海、杨红娟、段文山、吕克璞 2006 物理学报 55 1555]

    [44]

    ]Xu W, Sun Z K, Yang X L 2005 Acta Phys. Sin. 54 5069 (in Chinese) [徐伟、孙中奎、杨晓丽 2005 物理学报 54 5069]

    [45]

    ]Wang Z, Zou L, Zhang H Q 2008 Commun. Theor. Phys. 49 1373

    [46]

    ]Zou L, Zong Z, Wang Z, He L 2007 Phys. Lett. A 370 287

  • [1]

    [1]Fermi E, Pasta J, Ulam S 1965 Collected Papers of Enrico Fermi(Ⅱ) (Chicago: University of Chicago Press) p1

    [2]

    [2]Suris Y B 1997 J. Phys. A 30 1745

    [3]

    [3]Suris Y B 1999 Rev. Math. Phys. 11 727

    [4]

    [4]Suris Y B 2003 The Problem of Integrable Discretization: Hamitonian Approach, Progress in Mathemetics ( Vol. 219) (Basel: Birkhuser-Verlag) p1

    [5]

    [5]Blaszak M, Marciniak K 1994 J. Math. Phys. 35 4661

    [6]

    [6]Belov A A, Chaltikian K D 1993 Phys. Lett. B 309 268

    [7]

    [7]Hu X B, Zhu Z N 1998 J. Phys. A 31 4755

    [8]

    [8]Hu X B, Zhu Z N 1998 J. Math. Phys. 39 4766

    [9]

    [9]Ma W X, Hu X B, Zhu S M, Wu Y T 1999 J. Math. Phys. 40 6071

    [10]

    ]Zhang D J 2005 Chaos Solitons Fract. 23 1333

    [11]

    ]Sun Y P, Chen D Y, Xu X X 2006 Phys. Lett. A 359 47

    [12]

    ]Zhu J, Geng X 2007 Phys. Lett. A 368 464

    [13]

    ]Hu X B, Wu Y T 1998 Phys. Lett. A 246 523

    [14]

    ]Chou K S, Qu C Z 2001 Phys. Lett. A 280 303

    [15]

    ]Luo L, Fan E G 2007 Chin. Phys. Lett. 24 1444

    [16]

    ]Zhang W, Huang Y Z, Xiao Y 1998 Phys. Rev. E 57 7358

    [17]

    ]Xiao Y, Hai W H 1994 J. Phys. A 27 6873

    [18]

    ]Chow K W 1994 Phys. Scr. 50 233

    [19]

    ]Shek E C M, Chow K W 2008 Chaos Solitons Fract. 36 296

    [20]

    ]Yang P, Chen Y, Li Z B 2009 Appl. Math. Comput. 210 362

    [21]

    ]Yang P, Chen Y, Li Z B 2008 Chin. Phys. B 17 3953

    [22]

    ]Taogetusang, Sirendaoerji 2009 Acta Phys. Sin. 58 5894 (in Chinese) [套格图桑、斯仁道尔吉 2009 物理学报 58 5894]

    [23]

    ]Zhang S Q 2007 Acta Phys. Sin. 56 1870 (in Chinese) [张善卿 2007 物理学报 56 1870]

    [24]

    ]Liao S J 2003 Beyond Perturbation: Introduction to the Homotopy Analysis Method (Boca Raton: Chapman & Hall/CRC Press) p1

    [25]

    ]Liao S J 2002 J. Fluid Mech. 453 411

    [26]

    ]Liao S J 2003 Int. J. Non-Lin. Mech. 38 1173

    [27]

    ]Liao S J 2003 J. Fluid Mech. 488 189

    [28]

    ]Liao S J 2003 Appl. Math. Comput. 144 495

    [29]

    ]Liao S J 2004 Int. J. Non-Lin. Mech. 39 271

    [30]

    ]Liao S J 2004 Appl. Math. Comput. 147 499

    [31]

    ]Wu Y Y, Wang C, Liao S J 2005 Chaos Solitons Fract. 23 1733

    [32]

    ]Wu W, Liao S J 2005 Chaos Solitons Fract. 26 177

    [33]

    ]Liao S J 2009 Commun. Nonlin. Sci. Numer. Simulat. 14 983

    [34]

    ]Liao S J 2009 Commun. Nonlin. Sci. Numer. Simulat. 14 2144

    [35]

    ]Liao S J, Tan Y 2007 Stud. Appl. Math. 119 297

    [36]

    ]Abbasbandy S, Parkes E J 2008 Chaos Solitons Fract. 36 581

    [37]

    ]Abbasbandy S 2009 Chaos Solitons Fract. 39 428

    [38]

    ]Liu Y P, Li Z B 2008 Z. Naturforsch. A 63 241

    [39]

    ]Liu Y P, Li Z B 2009 J. Phys. A 42 125205

    [40]

    ]Liu Y P, Li Z B 2009 Chaos Solitons Fract. 39 1

    [41]

    ]Yang H J, Shi Y R, Duan W S,Lü K P 2007 Acta Phys. Sin. 56 3064 (in Chinese) [杨红娟、石玉仁、段文山、吕克璞 2007 物理学报 56 3064]

    [42]

    ]Shi Y R, Wang Y H, Yang H J, Duan W S 2007 Acta Phys. Sin. 56 6791 (in Chinese) [石玉仁、汪映海、杨红娟、段文山 2007 物理学报 56 6791]

    [43]

    ]Shi Y R, Xu X J, Wu Z X, Wang Y H, Yang H J, Duan W S, Lü K P 2006 Acta Phys. Sin. 55 1555 (in Chinese) [石玉仁、许新建、吴枝喜、汪映海、杨红娟、段文山、吕克璞 2006 物理学报 55 1555]

    [44]

    ]Xu W, Sun Z K, Yang X L 2005 Acta Phys. Sin. 54 5069 (in Chinese) [徐伟、孙中奎、杨晓丽 2005 物理学报 54 5069]

    [45]

    ]Wang Z, Zou L, Zhang H Q 2008 Commun. Theor. Phys. 49 1373

    [46]

    ]Zou L, Zong Z, Wang Z, He L 2007 Phys. Lett. A 370 287

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Publishing process
  • Received Date:  17 September 2009
  • Accepted Date:  03 December 2009
  • Published Online:  05 March 2010

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