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Riemann-Hilbert approach for a mixed coupled nonlinear Schrödinger equations and its soliton solutions

  • 1.

    School of Mathematics and Finance, Chuzhou University, Chuzhou 239000, China

  • 2.

    Department of Physics, Zhejiang Normal University, Jinhua 321004, China

  • 3.

    Department of Basical Courses, Shandong University of Science and Technology, Taian 271019, China

Cite this:
https://doi.org/10.52396/JUST-2021-0059
  • Received Date: 01 March 2021
  • Rev Recd Date: 20 March 2021
  • Publish Date: 31 March 2021
    Abstract

  • Abstract

    The integrable mixed coupled nonlinear Schrödinger (MCNLS) equations is studied, which describes the propagation of an optical pulse in a birefringent optical fiber. By the Riemann-Hilbert (RH) approach, the N-soliton solutions of the MCNLS equations can be expressed explicitly when the jump matrix of a constructed RH problem is a 3×3 unit matrix. As a special example, the expression of one soliton and two solitons are displayed explicitly. More generally, as a promotion, an integrable generalized multi-component NLS system with its linear spectral problem is discussed.

    Abstract

    The integrable mixed coupled nonlinear Schrödinger (MCNLS) equations is studied, which describes the propagation of an optical pulse in a birefringent optical fiber. By the Riemann-Hilbert (RH) approach, the N-soliton solutions of the MCNLS equations can be expressed explicitly when the jump matrix of a constructed RH problem is a 3×3 unit matrix. As a special example, the expression of one soliton and two solitons are displayed explicitly. More generally, as a promotion, an integrable generalized multi-component NLS system with its linear spectral problem is discussed.
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    • References(25)
  • [1]
    Gardner C S, Green J M, Kruskka M D, et al. Method for solving the Korteweg-de Vries equation. Physical Review Letters, 1967, 19: 1095-1097.
    [2]
    Ablowitz M J, Segur H. Solitons and the Inverse Scattering Transform. Philadelphia, PA: SIAM, 1981.
    [3]
    Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Physical Review Letters, 1971, 27: 1192-1194.
    [4]
    Matveev V B, Salle M A. Darboux Transformations and Solitons. Berlin: Springer-Verlag, 1991.
    [5]
    Lou S Y. A note on the new similarity reductions of the Boussinesq equation. Physics Letters A, 1990, 151: 133-135.
    [6]
    Yang J K. Nonlinear Wavesin Integrable and Nonintegrable Systems. Philadelphia, PA: SIAM, 2010.
    [7]
    Guo B L, Ling L M. Riemann-Hilbert approach and N-soliton formula for coupled derivative Schrödinger equation. Journal of Mathematical Physics, 2012, 53: 073506.
    [8]
    Wang D S, Yin S J, Liu Y F. Integrability and bright soliton solutions to the coupled nonlinear Schrödinger equation with higher-order effects. Applied Mathematics and Computation, 2014, 229: 296-309.
    [9]
    Zhang Y S, Cheng Y, He J S. Riemann-Hilbert method and N-soliton for two-component Gerdjikov-Ivanov equation. Journal of Nonlinear Mathematical Physics, 2017, 24: 210-223.
    [10]
    Wang Z, Qiao Z J. Riemann-Hilbert approach for the FQXL model: A generalized Camassa-Holm equation with cubic and quadratic nonlinearity. Journal of Mathematical Physics, 2016, 57: 073505.
    [11]
    Ma W X. The inverse scattering transform and soliton solutions of a combined modified Korteweg-de Vries equation. Journal of Mathematical Analysis and Applications, 2019, 471: 796-811.
    [12]
    Hu J, Xu J, Yu G F. Riemann-Hilbert approach and N-soliton formula for a higher-order Chen-Lee-Liu equation. Journal of Nonlinear Mathematical Physics, 2018, 25: 633-649.
    [13]
    Hu B B, Zhang L, Xia T C, et al. On the Riemann-Hilbert problem of the Kundu equation. Applied Mathematics and Computation, 2020, 381: 125262.
    [14]
    Hu B B, Xia T C, Zhang N, et al. Initial-boundary value problems for the coupled higher-order nonlinear Schrödinger equations on the half-line. International Journal of Nonlinear Sciences and Numerical Simulation, 2018, 19(1): 83-92.
    [15]
    Hu B B, Xia T C. A Riemann-Hilbert approach to the initial-boundary value problem for Kundu-Eckhaus equation on the half line. Complex Variables and Elliptic Equations, 2019, 64: 2019-2039.
    [16]
    Hu B B, Zhang L, Zhang N. On the Riemann-Hilbert problem for the mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation. Journal of Computational and Applied Mathematics, 2021, 390: 113393.
    [17]
    Tian S F. Initial-boundary value problems of the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method. Journal of Physics A: Mathematical and Theoretical, 2017, 50: 395204.
    [18]
    Tian S F. Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method. Journal of Differential Equations, 2017, 26: 506-558.
    [19]
    Deift P, Zhou X. A steepest descent method for oscillatory Riemann-Hilbert problems. Annals of Mathematics, 1993, 137: 295-368.
    [20]
    Tian S F, Zhang T T. Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition. Proceedings of the American Mathematical Society, 2018, 146(4): 1713-1729.
    [21]
    Manakov S V. On the theory of two-dimensional stationary self-focusing of electromagenic waves. Soviet Physics JETP, 1974, 38(2): 248-253.
    [22]
    Kanna T, Lakshmanan M, Dinda P T, et al. Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations. Physical Review E, 2006, 73: 026604.
    [23]
    Vijayajayanthi M, Kanna T, Lakshmanan M. Bright-dark solitons and their collisions in mixed N-coupled nonlinear Schrödinger equations. Physical Review A, 2008, 77: 013820.
    [24]
    Ling L M, Zhao L C, Guo B L. Darboux transformation and classification of solution for mixed coupled nonlinear Schrödinger equations. Communications in Nonlinear Science and Numerical Simulation, 2016, 32: 285-304.
    [25]
    Tian S F. The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method. Proceedings of the Royal Society A, 2016, 472: 20160588.
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    [1]
    Gardner C S, Green J M, Kruskka M D, et al. Method for solving the Korteweg-de Vries equation. Physical Review Letters, 1967, 19: 1095-1097.
    [2]
    Ablowitz M J, Segur H. Solitons and the Inverse Scattering Transform. Philadelphia, PA: SIAM, 1981.
    [3]
    Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Physical Review Letters, 1971, 27: 1192-1194.
    [4]
    Matveev V B, Salle M A. Darboux Transformations and Solitons. Berlin: Springer-Verlag, 1991.
    [5]
    Lou S Y. A note on the new similarity reductions of the Boussinesq equation. Physics Letters A, 1990, 151: 133-135.
    [6]
    Yang J K. Nonlinear Wavesin Integrable and Nonintegrable Systems. Philadelphia, PA: SIAM, 2010.
    [7]
    Guo B L, Ling L M. Riemann-Hilbert approach and N-soliton formula for coupled derivative Schrödinger equation. Journal of Mathematical Physics, 2012, 53: 073506.
    [8]
    Wang D S, Yin S J, Liu Y F. Integrability and bright soliton solutions to the coupled nonlinear Schrödinger equation with higher-order effects. Applied Mathematics and Computation, 2014, 229: 296-309.
    [9]
    Zhang Y S, Cheng Y, He J S. Riemann-Hilbert method and N-soliton for two-component Gerdjikov-Ivanov equation. Journal of Nonlinear Mathematical Physics, 2017, 24: 210-223.
    [10]
    Wang Z, Qiao Z J. Riemann-Hilbert approach for the FQXL model: A generalized Camassa-Holm equation with cubic and quadratic nonlinearity. Journal of Mathematical Physics, 2016, 57: 073505.
    [11]
    Ma W X. The inverse scattering transform and soliton solutions of a combined modified Korteweg-de Vries equation. Journal of Mathematical Analysis and Applications, 2019, 471: 796-811.
    [12]
    Hu J, Xu J, Yu G F. Riemann-Hilbert approach and N-soliton formula for a higher-order Chen-Lee-Liu equation. Journal of Nonlinear Mathematical Physics, 2018, 25: 633-649.
    [13]
    Hu B B, Zhang L, Xia T C, et al. On the Riemann-Hilbert problem of the Kundu equation. Applied Mathematics and Computation, 2020, 381: 125262.
    [14]
    Hu B B, Xia T C, Zhang N, et al. Initial-boundary value problems for the coupled higher-order nonlinear Schrödinger equations on the half-line. International Journal of Nonlinear Sciences and Numerical Simulation, 2018, 19(1): 83-92.
    [15]
    Hu B B, Xia T C. A Riemann-Hilbert approach to the initial-boundary value problem for Kundu-Eckhaus equation on the half line. Complex Variables and Elliptic Equations, 2019, 64: 2019-2039.
    [16]
    Hu B B, Zhang L, Zhang N. On the Riemann-Hilbert problem for the mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation. Journal of Computational and Applied Mathematics, 2021, 390: 113393.
    [17]
    Tian S F. Initial-boundary value problems of the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method. Journal of Physics A: Mathematical and Theoretical, 2017, 50: 395204.
    [18]
    Tian S F. Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method. Journal of Differential Equations, 2017, 26: 506-558.
    [19]
    Deift P, Zhou X. A steepest descent method for oscillatory Riemann-Hilbert problems. Annals of Mathematics, 1993, 137: 295-368.
    [20]
    Tian S F, Zhang T T. Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition. Proceedings of the American Mathematical Society, 2018, 146(4): 1713-1729.
    [21]
    Manakov S V. On the theory of two-dimensional stationary self-focusing of electromagenic waves. Soviet Physics JETP, 1974, 38(2): 248-253.
    [22]
    Kanna T, Lakshmanan M, Dinda P T, et al. Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations. Physical Review E, 2006, 73: 026604.
    [23]
    Vijayajayanthi M, Kanna T, Lakshmanan M. Bright-dark solitons and their collisions in mixed N-coupled nonlinear Schrödinger equations. Physical Review A, 2008, 77: 013820.
    [24]
    Ling L M, Zhao L C, Guo B L. Darboux transformation and classification of solution for mixed coupled nonlinear Schrödinger equations. Communications in Nonlinear Science and Numerical Simulation, 2016, 32: 285-304.
    [25]
    Tian S F. The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method. Proceedings of the Royal Society A, 2016, 472: 20160588.
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