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Novel double Casoratian solutions to a negative order four-potential isospectral Ablowitz-Ladik equation

  • School of Mathematics and Physical Science, Xuzhou Institute of Technology, Xuzhou 221111, China

Funds:  National Natural Science Foundation of China (11301454, 11271168), the Natural Science Foundation for Colleges and Universities in Jiangsu Province (13KJD110009), the Jiangsu Qing Lan Project for Excellent Young Teachers in University (2014) and XZIT (XKY 2013202).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.12.006
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  • Author Bio:

    XUE Yimin, male, born in 1977, master/lecturer. Research field: soliton theory and integrable system.

  • Corresponding author: CHEN Shouting
  • Received Date: 21 October 2014
  • Accepted Date: 18 January 2015
  • Rev Recd Date: 18 January 2015
  • Publish Date: 30 December 2015
    Abstract

  • Abstract

    A matrix method for constructing double Casoratian entries was applied to a negative order four-potential isospectral Ablowitz-Ladik equation. Novel double Casoratian solutions to it were obtained by allowing the general matrix to be some special cases, including Matveev solutions and mixed solutions.

    Abstract

    A matrix method for constructing double Casoratian entries was applied to a negative order four-potential isospectral Ablowitz-Ladik equation. Novel double Casoratian solutions to it were obtained by allowing the general matrix to be some special cases, including Matveev solutions and mixed solutions.
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    • References(21)
  • [1]
    Freeman N C,Nimmo J J C. Soliton solutions of the KdV and KP equations: The Wronskian technique[J]. Physics Letters A, 1983, 95(1): 1-3.
    [2]
    Nimmo J J C, Freeman N C. A method of obtainning the soliton solution of the Boussinesq equation in terms of a Wronskain[J]. Physics Letters A, 1983, 95(1): 4-6.
    [3]
    Nimmo J J C. Soliton solutions of three differential-difference equations in Wronskian form[J].Physics Letters A, 1983, 99(6-7): 281-286.
    [4]
    Satsuma J. A Wronskian representation of n-soliton solutions of nonlinear evolution equations[J].Journal of the Physical Society of Japan, 1979, 46(1): 359-360.
    [5]
    Hirota R, Ito M, Kato F. Two-dimensional Toda lattice equations[J]. Progress of Theoretical Physics Supplement, 1988, 94(94): 42-58.
    [6]
    Hirota R, Ohta Y, Satsuma J. Solutions of the KP equation and the two dimensional Toda equations[J]. Journal of the Physical Society of Japan, 1988, 57(6): 1 901-1 904.
    [7]
    Zhang D J.Notes on solutions in Wronskian form to soliton equations: KdV-type[DB/OL]. arXiv: nlin/0603008.
    [8]
    Darboux G. Lecons Surla Théorie Générale des Surfaces, Volume Ⅱ[M]. 3rd edition. New York: Chelsea Publishing Company, 1972.
    [9]
    Nimmo J J C, Freeman N C. A bilinear Bcklund transformation for the nonlinear Schrdinger equation[J]. Physics Letters A, 1983, 99(6-7): 279-281.
    [10]
    Liu Q M. DoubleWronskian solutions of the AKNS and the classical Boussinesq hierarchies[J]. Journal of the Physical Society of Japan, 1990, 59(10): 3 520-3 527.
    [11]
    Chen D Y, Zhang D J, Bi J B. New double Wronskian solutions of the AKNS equation[J]. Science in China Series A: Mathematics, 2008, 51(1): 55-69.
    [12]
    Zhai W, Chen D Y. Rational solutions of the general nonlinear Schrdinger equation with derivative[J]. Physics Letters A, 2008, 372(23): 4 217-4 221.
    [13]
    Yao Y Q, Zhang D J, Chen D Y. The double Wronskian solutions to the Kadomtset-Petviashvili equation[J]. Modern Physics Letters B, 2008, 22(9): 621-641.
    [14]
    Chen S T, Zhang J B, Chen D Y. Generalized double Casoratian solutions to the four-potential isospectral Ablowitz-Ladik equation[J]. Communications in Nonlinear Science and Numerical Simulation, 2013, 18(11): 2 949-2 959.
    [15]
    Chen S T, Li Q. Double Casoratian solutions of a negative order isospectral four-potential Ablowitz-Ladik equation[J]. Journal of Jiangsu Normal University: Natural Science Edition, 2013, 31(4):11-17.
    [16]
    Zhang D J, Chen S T. Symmetries for the Ablowitz-Ladik hierarchy: Ⅰ. Four-potential case[J]. Studies in Applied Mathematics, 2010, 125(4): 393-418.
    [17]
    Ablowitz M J, Ladik J F. Nonlinear differential-difference equations[J]. Journal of Mathematical Physics, 1975, 16(3): 598-603.
    [18]
    Chen S T, Zhu X M, Li Q, Chen D Y. N-Soliton solutions for the four-potential isospectral Ablowitz-Ladik equation[J]. Chinese Physics Letters, 2011, 28(6): 060202.
    [19]
    Gegenhasi, Hu X B, Levi D. On a discrete Davey-Stewartson system[J]. Inverse Problems, 2006, 22(5): 1 677-1 688.
    [20]
    Wu H, Zhang D J.Mixed rational-soliton solutions of two differential-difference equations in Casorati determinant form[J]. Journal of Physics A: Mathematical and General, 2003, 36(17): 4 867-4 873.
    [21]
    Chen Y. Rational-like solutions for a negative order isospectral four-potential Ablowitz-Ladik equation[J]. Journal of Jiangsu Normal University (Natural Science Edition), 2014, 32(4): 36-39.
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    [1]
    Freeman N C,Nimmo J J C. Soliton solutions of the KdV and KP equations: The Wronskian technique[J]. Physics Letters A, 1983, 95(1): 1-3.
    [2]
    Nimmo J J C, Freeman N C. A method of obtainning the soliton solution of the Boussinesq equation in terms of a Wronskain[J]. Physics Letters A, 1983, 95(1): 4-6.
    [3]
    Nimmo J J C. Soliton solutions of three differential-difference equations in Wronskian form[J].Physics Letters A, 1983, 99(6-7): 281-286.
    [4]
    Satsuma J. A Wronskian representation of n-soliton solutions of nonlinear evolution equations[J].Journal of the Physical Society of Japan, 1979, 46(1): 359-360.
    [5]
    Hirota R, Ito M, Kato F. Two-dimensional Toda lattice equations[J]. Progress of Theoretical Physics Supplement, 1988, 94(94): 42-58.
    [6]
    Hirota R, Ohta Y, Satsuma J. Solutions of the KP equation and the two dimensional Toda equations[J]. Journal of the Physical Society of Japan, 1988, 57(6): 1 901-1 904.
    [7]
    Zhang D J.Notes on solutions in Wronskian form to soliton equations: KdV-type[DB/OL]. arXiv: nlin/0603008.
    [8]
    Darboux G. Lecons Surla Théorie Générale des Surfaces, Volume Ⅱ[M]. 3rd edition. New York: Chelsea Publishing Company, 1972.
    [9]
    Nimmo J J C, Freeman N C. A bilinear Bcklund transformation for the nonlinear Schrdinger equation[J]. Physics Letters A, 1983, 99(6-7): 279-281.
    [10]
    Liu Q M. DoubleWronskian solutions of the AKNS and the classical Boussinesq hierarchies[J]. Journal of the Physical Society of Japan, 1990, 59(10): 3 520-3 527.
    [11]
    Chen D Y, Zhang D J, Bi J B. New double Wronskian solutions of the AKNS equation[J]. Science in China Series A: Mathematics, 2008, 51(1): 55-69.
    [12]
    Zhai W, Chen D Y. Rational solutions of the general nonlinear Schrdinger equation with derivative[J]. Physics Letters A, 2008, 372(23): 4 217-4 221.
    [13]
    Yao Y Q, Zhang D J, Chen D Y. The double Wronskian solutions to the Kadomtset-Petviashvili equation[J]. Modern Physics Letters B, 2008, 22(9): 621-641.
    [14]
    Chen S T, Zhang J B, Chen D Y. Generalized double Casoratian solutions to the four-potential isospectral Ablowitz-Ladik equation[J]. Communications in Nonlinear Science and Numerical Simulation, 2013, 18(11): 2 949-2 959.
    [15]
    Chen S T, Li Q. Double Casoratian solutions of a negative order isospectral four-potential Ablowitz-Ladik equation[J]. Journal of Jiangsu Normal University: Natural Science Edition, 2013, 31(4):11-17.
    [16]
    Zhang D J, Chen S T. Symmetries for the Ablowitz-Ladik hierarchy: Ⅰ. Four-potential case[J]. Studies in Applied Mathematics, 2010, 125(4): 393-418.
    [17]
    Ablowitz M J, Ladik J F. Nonlinear differential-difference equations[J]. Journal of Mathematical Physics, 1975, 16(3): 598-603.
    [18]
    Chen S T, Zhu X M, Li Q, Chen D Y. N-Soliton solutions for the four-potential isospectral Ablowitz-Ladik equation[J]. Chinese Physics Letters, 2011, 28(6): 060202.
    [19]
    Gegenhasi, Hu X B, Levi D. On a discrete Davey-Stewartson system[J]. Inverse Problems, 2006, 22(5): 1 677-1 688.
    [20]
    Wu H, Zhang D J.Mixed rational-soliton solutions of two differential-difference equations in Casorati determinant form[J]. Journal of Physics A: Mathematical and General, 2003, 36(17): 4 867-4 873.
    [21]
    Chen Y. Rational-like solutions for a negative order isospectral four-potential Ablowitz-Ladik equation[J]. Journal of Jiangsu Normal University (Natural Science Edition), 2014, 32(4): 36-39.
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