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Numerical simulation between long and short waves by multisymplectic method

  • 1.

    School of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China

  • 2.

    Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

  • 3.

    School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Funds:  Supported by the NNSFC (11301234, 11271171, 11101399), the Provincial Natural Science Foundation of Jiangxi (20142BCB23009, 20151BAB201012), State Key Laboratory of Scientific and Engineering Computing, CAS, and Jiangsu Key Lab for NSLSCS (201302).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.09.003
More Information
  • Author Bio:

    WANG Lan, female, born in 1979, master/lecturer. Research field: numerical methods for PDEs.

  • Corresponding author: KONG Linghua
  • Received Date: 06 March 2014
  • Accepted Date: 20 August 2014
  • Rev Recd Date: 20 August 2014
  • Publish Date: 30 September 2015
    Abstract

  • Abstract

    The multisymplectic structure-preserving scheme for the Schrdinger-KdV equation was investigated. First the canonical formulation of the equation was discussed. Then, it was discretized by the multisymplectic integrator, such as a midpoint integrator. Numerical results were presented to illustrate the validity of the new scheme.

    Abstract

    The multisymplectic structure-preserving scheme for the Schrdinger-KdV equation was investigated. First the canonical formulation of the equation was discussed. Then, it was discretized by the multisymplectic integrator, such as a midpoint integrator. Numerical results were presented to illustrate the validity of the new scheme.
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    • References(22)
  • [1]
    Abdou M A, Soliman A A. New application of variational iteration method [J]. Phys D, 2005, 211: 1-8.
    [2]
    Pava J A. Stability of solitary wave solution for equations of short and long dispersive waves [J]. Electr J Diff Equa, 2006,2006(72): 1-18.
    [3]
    Ascher U M, McLachlan R I. On symplectic and multisymplectic schemes for the KdV equation[J]. J Sci Comput, 2005, 25: 83-104.
    [4]
    Bai D M, Zhang L M. Numerical studies on a novel split-step quadratic B-spline finite element method for the coupled Schrdinger-KdV equations [J]. Commun Nonlinear Sci Numer Simulat, 2011, 16: 1 263-1 273.
    [5]
    Benney D J, A general theory for interactions between short and long waves [J]. Stud Appl Math, 1977, 56: 81-94.
    [6]
    Cai J X, Wang Y S, Liang H. Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schrdinger system[J]. J Comput Phys, 2013, 239: 30-50.
    [7]
    Cai W J, Wang Y S, Song Y Z. Numerical dispersion analysis of a multi-symplectic scheme for the three dimensional Maxwells equations[J]. J Comput Phys, 2013, 234: 330-352.
    [8]
    Bridges T J, Reich S. Numerical methods for Hamiltonian PDEs [J]. J Phys A: Math Gen, 2006, 39: 5 287-5 320.
    [9]
    Chang Q S, Wong Y S, Lin C K. Numerical computations for long-wave short-wave interaction equations in semi-classical limit [J]. J Comput Phys, 2008, 227: 8 489-8 507.
    [10]
    Erbay S, Nonlinear interaction between long and short waves in a generalized elastic solid [J]. Chaos Soliton Fract, 2000, 11: 1 789-1 798.
    [11]
    Gong Y Z, Cai J X, Wang Y S. Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs[J]. J Comput Phys, 2014, 279: 80-102.
    [12]
    Hong J L, Liu H Y, Sun G. The multisymplecticity of partitioned Runge-Kutta methods for Hamiltonian systems[J]. Math Comput, 2005, 75: 167-181.
    [13]
    Hong J L, Li C. Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations[J]. J Comput Phys, 2006, 211: 448-472.
    [14]
    Hong J L, Kong L H. Novel multi-symplectic integrators for nonlinear fourth-order Schrdinger equation with trapped term[J]. Commun Comput Phys, 2010, 7: 613-630.
    [15]
    Hong J L, Sun Y J. Generating functions of multi-symplectic RK methods via DW Hamilton-Jacobi equations[J]. Numer Math, 2008, 110: 491-519.
    [16]
    Kong L H, Zeng W P, Liu R X, et al. Multi-symplectic scheme of SRLW equation and conservation laws [J]. Journal of University of Science and Technology of China, 2005, 35: 770-776.
    [17]
    Mclachlan R I, Wilkins M C. The multisymplectic diamond schems[J]. SIAM J Sci Comput, 2015, 37: A369-A390.
    [18]
    Reich S. Multisymplecitc Runge-Kutta collocation methods for Hamilton wave equation [J]. J Comput Phys, 2000, 157: 473-499.
    [19]
    Yoshinaga T. Kakutani T. Solitary and E-shock waves in a resonant system between long and short waves [J]. J Phys Soc Jpn, 1994, 63: 445-459.
    [20]
    Zhang S Q. Li Z B. New explicit exact solutions to nonlinearly coupled Schrdinger-KdV equations [J]. Acta Phys Sinica, 2002, 51: 2 197-2 201.
    [21]
    Zhao P F, Qin M Z. Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation [J]. J Phys A: Math Gen, 2000, 33: 3 613-3 626.
    [22]
    Zhu P F, Kong L H, Wang L. The conservation laws of Schrdinger-KdV equations [J]. J Jiangxi Normal Univ, 2012, 36: 495-498.
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    [1]
    Abdou M A, Soliman A A. New application of variational iteration method [J]. Phys D, 2005, 211: 1-8.
    [2]
    Pava J A. Stability of solitary wave solution for equations of short and long dispersive waves [J]. Electr J Diff Equa, 2006,2006(72): 1-18.
    [3]
    Ascher U M, McLachlan R I. On symplectic and multisymplectic schemes for the KdV equation[J]. J Sci Comput, 2005, 25: 83-104.
    [4]
    Bai D M, Zhang L M. Numerical studies on a novel split-step quadratic B-spline finite element method for the coupled Schrdinger-KdV equations [J]. Commun Nonlinear Sci Numer Simulat, 2011, 16: 1 263-1 273.
    [5]
    Benney D J, A general theory for interactions between short and long waves [J]. Stud Appl Math, 1977, 56: 81-94.
    [6]
    Cai J X, Wang Y S, Liang H. Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schrdinger system[J]. J Comput Phys, 2013, 239: 30-50.
    [7]
    Cai W J, Wang Y S, Song Y Z. Numerical dispersion analysis of a multi-symplectic scheme for the three dimensional Maxwells equations[J]. J Comput Phys, 2013, 234: 330-352.
    [8]
    Bridges T J, Reich S. Numerical methods for Hamiltonian PDEs [J]. J Phys A: Math Gen, 2006, 39: 5 287-5 320.
    [9]
    Chang Q S, Wong Y S, Lin C K. Numerical computations for long-wave short-wave interaction equations in semi-classical limit [J]. J Comput Phys, 2008, 227: 8 489-8 507.
    [10]
    Erbay S, Nonlinear interaction between long and short waves in a generalized elastic solid [J]. Chaos Soliton Fract, 2000, 11: 1 789-1 798.
    [11]
    Gong Y Z, Cai J X, Wang Y S. Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs[J]. J Comput Phys, 2014, 279: 80-102.
    [12]
    Hong J L, Liu H Y, Sun G. The multisymplecticity of partitioned Runge-Kutta methods for Hamiltonian systems[J]. Math Comput, 2005, 75: 167-181.
    [13]
    Hong J L, Li C. Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations[J]. J Comput Phys, 2006, 211: 448-472.
    [14]
    Hong J L, Kong L H. Novel multi-symplectic integrators for nonlinear fourth-order Schrdinger equation with trapped term[J]. Commun Comput Phys, 2010, 7: 613-630.
    [15]
    Hong J L, Sun Y J. Generating functions of multi-symplectic RK methods via DW Hamilton-Jacobi equations[J]. Numer Math, 2008, 110: 491-519.
    [16]
    Kong L H, Zeng W P, Liu R X, et al. Multi-symplectic scheme of SRLW equation and conservation laws [J]. Journal of University of Science and Technology of China, 2005, 35: 770-776.
    [17]
    Mclachlan R I, Wilkins M C. The multisymplectic diamond schems[J]. SIAM J Sci Comput, 2015, 37: A369-A390.
    [18]
    Reich S. Multisymplecitc Runge-Kutta collocation methods for Hamilton wave equation [J]. J Comput Phys, 2000, 157: 473-499.
    [19]
    Yoshinaga T. Kakutani T. Solitary and E-shock waves in a resonant system between long and short waves [J]. J Phys Soc Jpn, 1994, 63: 445-459.
    [20]
    Zhang S Q. Li Z B. New explicit exact solutions to nonlinearly coupled Schrdinger-KdV equations [J]. Acta Phys Sinica, 2002, 51: 2 197-2 201.
    [21]
    Zhao P F, Qin M Z. Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation [J]. J Phys A: Math Gen, 2000, 33: 3 613-3 626.
    [22]
    Zhu P F, Kong L H, Wang L. The conservation laws of Schrdinger-KdV equations [J]. J Jiangxi Normal Univ, 2012, 36: 495-498.
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