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The analytical evolution of NLS solitons due to the numerical discretization error

Journal Article


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Abstract


  • Soliton perturbation theory is used to obtain analytical solutions describing

    solitary wave tails or shelves, due to numerical discretization error, for

    soliton solutions of the nonlinear Schr¨odinger equation. Two important implicit

    numerical schemes for the nonlinear Schr¨odinger equation, with second-order

    temporal and spatial discretization errors, are considered. These are the Crank–

    Nicolson scheme and a scheme, due to Taha [1], based on the inverse scattering

    transform. The first-order correction for the solitary wave tail, or shelf, is in

    integral form and an explicit expression is found for large time. The shelf decays

    slowly, at a rate of t−12

    , which is characteristic of the nonlinear Schr¨odinger

    equation. Singularity theory, usually used for combustion problems, is applied

    to the explicit large-time expression for the solitary wave tail. Analytical results

    are then obtained, such as the parameter regions in which qualitatively different

    types of solitary wave tails occur, the location of zeros and the location and

    amplitude of peaks. It is found that three different types of tail occur for the

    Crank–Nicolson and Taha schemes and that the Taha scheme exhibits some

    unusual symmetry properties, as the tails for left and right moving solitary

    waves are different. Optimal choices of the discretization parameters for the

    numerical schemes are also found, whichminimize the amplitude of the solitary

    wave tail. The analytical solutions are compared with numerical simulations,

    and an excellent comparison is found.

Publication Date


  • 2011

Citation


  • Hoseini, S. & Marchant, T. R. (2011). The analytical evolution of NLS solitons due to the numerical discretization error. Journal of Physics A: Mathematical and Theoretical, 44 (50), 1-17.

Scopus Eid


  • 2-s2.0-82455209375

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=11129&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3793

Number Of Pages


  • 16

Start Page


  • 1

End Page


  • 17

Volume


  • 44

Issue


  • 50

Abstract


  • Soliton perturbation theory is used to obtain analytical solutions describing

    solitary wave tails or shelves, due to numerical discretization error, for

    soliton solutions of the nonlinear Schr¨odinger equation. Two important implicit

    numerical schemes for the nonlinear Schr¨odinger equation, with second-order

    temporal and spatial discretization errors, are considered. These are the Crank–

    Nicolson scheme and a scheme, due to Taha [1], based on the inverse scattering

    transform. The first-order correction for the solitary wave tail, or shelf, is in

    integral form and an explicit expression is found for large time. The shelf decays

    slowly, at a rate of t−12

    , which is characteristic of the nonlinear Schr¨odinger

    equation. Singularity theory, usually used for combustion problems, is applied

    to the explicit large-time expression for the solitary wave tail. Analytical results

    are then obtained, such as the parameter regions in which qualitatively different

    types of solitary wave tails occur, the location of zeros and the location and

    amplitude of peaks. It is found that three different types of tail occur for the

    Crank–Nicolson and Taha schemes and that the Taha scheme exhibits some

    unusual symmetry properties, as the tails for left and right moving solitary

    waves are different. Optimal choices of the discretization parameters for the

    numerical schemes are also found, whichminimize the amplitude of the solitary

    wave tail. The analytical solutions are compared with numerical simulations,

    and an excellent comparison is found.

Publication Date


  • 2011

Citation


  • Hoseini, S. & Marchant, T. R. (2011). The analytical evolution of NLS solitons due to the numerical discretization error. Journal of Physics A: Mathematical and Theoretical, 44 (50), 1-17.

Scopus Eid


  • 2-s2.0-82455209375

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=11129&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3793

Number Of Pages


  • 16

Start Page


  • 1

End Page


  • 17

Volume


  • 44

Issue


  • 50