Solitonlike solutions of the generalized discrete nonlinear Schrödinger equation

Kim Rasmussen, D. Henning, H. Gabriel, A. Bülow

    Research output: Contribution to journalJournal articleResearchpeer-review

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    Abstract

    We investigate the solution properties oi. a generalized discrete nonlinear Schrodinger equation describing a nonlinear lattice chain. The generalized equation interpolates between the integrable discrete Ablowitz-Ladik equation and the nonintegrable discrete Schrodinger equation. Special interest is paid to the creation of stationary localized solutions called breathers. To tackle this problem we apply a map approach and illuminate the linkage of homoclinic and heteroclinic map orbits with localized lattice solutions. The homoclinic and heteroclinic orbits correspond to exact nonlinear solitonlike eigenstates of the lattice. Normal forms and the Melnikov method are used for analytical determinations of homoclinic orbits. Nonintegrability of the map leads to soliton pinning on the lattice. The soliton pinning energy is calculated and it is shown that it can be tuned by varying the ratio of the nonintegrability parameter versus the integrability parameter. The heteroclinic map orbit is derived on the basis of a variational principle. Finally, we use homoclinic and heteroclinic orbits as initial conditions to excite designed stationary localized solutions of desired width in the dynamics of the discrete nonlinear Schrodinger equation. In this way eve are able to construct coherent solitonlike structures of profile determined by the map parameters.
    Original languageEnglish
    JournalPhysical Review E. Statistical, Nonlinear, and Soft Matter Physics
    Volume54
    Issue number5
    Pages (from-to)5788-5801
    ISSN1063-651X
    DOIs
    Publication statusPublished - 1996

    Bibliographical note

    Copyright (1996) American Physical Society.

    Keywords

    • SYSTEMS
    • INTRINSIC LOCALIZED MODES
    • FRENKEL-KONTOROVA MODEL
    • LATTICES
    • EXCITATIONS
    • PROPAGATION
    • MOLECULAR CHAINS
    • GAP SOLITONS
    • INTEGRABLE MAPPINGS
    • ANTIINTEGRABILITY

    Cite this

    Rasmussen, Kim ; Henning, D. ; Gabriel, H. ; Bülow, A. / Solitonlike solutions of the generalized discrete nonlinear Schrödinger equation. In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics. 1996 ; Vol. 54, No. 5. pp. 5788-5801.
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    abstract = "We investigate the solution properties oi. a generalized discrete nonlinear Schrodinger equation describing a nonlinear lattice chain. The generalized equation interpolates between the integrable discrete Ablowitz-Ladik equation and the nonintegrable discrete Schrodinger equation. Special interest is paid to the creation of stationary localized solutions called breathers. To tackle this problem we apply a map approach and illuminate the linkage of homoclinic and heteroclinic map orbits with localized lattice solutions. The homoclinic and heteroclinic orbits correspond to exact nonlinear solitonlike eigenstates of the lattice. Normal forms and the Melnikov method are used for analytical determinations of homoclinic orbits. Nonintegrability of the map leads to soliton pinning on the lattice. The soliton pinning energy is calculated and it is shown that it can be tuned by varying the ratio of the nonintegrability parameter versus the integrability parameter. The heteroclinic map orbit is derived on the basis of a variational principle. Finally, we use homoclinic and heteroclinic orbits as initial conditions to excite designed stationary localized solutions of desired width in the dynamics of the discrete nonlinear Schrodinger equation. In this way eve are able to construct coherent solitonlike structures of profile determined by the map parameters.",
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    author = "Kim Rasmussen and D. Henning and H. Gabriel and A. B{\"u}low",
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    language = "English",
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    pages = "5788--5801",
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    Solitonlike solutions of the generalized discrete nonlinear Schrödinger equation. / Rasmussen, Kim; Henning, D.; Gabriel, H.; Bülow, A.

    In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, Vol. 54, No. 5, 1996, p. 5788-5801.

    Research output: Contribution to journalJournal articleResearchpeer-review

    TY - JOUR

    T1 - Solitonlike solutions of the generalized discrete nonlinear Schrödinger equation

    AU - Rasmussen, Kim

    AU - Henning, D.

    AU - Gabriel, H.

    AU - Bülow, A.

    N1 - Copyright (1996) American Physical Society.

    PY - 1996

    Y1 - 1996

    N2 - We investigate the solution properties oi. a generalized discrete nonlinear Schrodinger equation describing a nonlinear lattice chain. The generalized equation interpolates between the integrable discrete Ablowitz-Ladik equation and the nonintegrable discrete Schrodinger equation. Special interest is paid to the creation of stationary localized solutions called breathers. To tackle this problem we apply a map approach and illuminate the linkage of homoclinic and heteroclinic map orbits with localized lattice solutions. The homoclinic and heteroclinic orbits correspond to exact nonlinear solitonlike eigenstates of the lattice. Normal forms and the Melnikov method are used for analytical determinations of homoclinic orbits. Nonintegrability of the map leads to soliton pinning on the lattice. The soliton pinning energy is calculated and it is shown that it can be tuned by varying the ratio of the nonintegrability parameter versus the integrability parameter. The heteroclinic map orbit is derived on the basis of a variational principle. Finally, we use homoclinic and heteroclinic orbits as initial conditions to excite designed stationary localized solutions of desired width in the dynamics of the discrete nonlinear Schrodinger equation. In this way eve are able to construct coherent solitonlike structures of profile determined by the map parameters.

    AB - We investigate the solution properties oi. a generalized discrete nonlinear Schrodinger equation describing a nonlinear lattice chain. The generalized equation interpolates between the integrable discrete Ablowitz-Ladik equation and the nonintegrable discrete Schrodinger equation. Special interest is paid to the creation of stationary localized solutions called breathers. To tackle this problem we apply a map approach and illuminate the linkage of homoclinic and heteroclinic map orbits with localized lattice solutions. The homoclinic and heteroclinic orbits correspond to exact nonlinear solitonlike eigenstates of the lattice. Normal forms and the Melnikov method are used for analytical determinations of homoclinic orbits. Nonintegrability of the map leads to soliton pinning on the lattice. The soliton pinning energy is calculated and it is shown that it can be tuned by varying the ratio of the nonintegrability parameter versus the integrability parameter. The heteroclinic map orbit is derived on the basis of a variational principle. Finally, we use homoclinic and heteroclinic orbits as initial conditions to excite designed stationary localized solutions of desired width in the dynamics of the discrete nonlinear Schrodinger equation. In this way eve are able to construct coherent solitonlike structures of profile determined by the map parameters.

    KW - SYSTEMS

    KW - INTRINSIC LOCALIZED MODES

    KW - FRENKEL-KONTOROVA MODEL

    KW - LATTICES

    KW - EXCITATIONS

    KW - PROPAGATION

    KW - MOLECULAR CHAINS

    KW - GAP SOLITONS

    KW - INTEGRABLE MAPPINGS

    KW - ANTIINTEGRABILITY

    U2 - 10.1103/PhysRevE.54.5788

    DO - 10.1103/PhysRevE.54.5788

    M3 - Journal article

    VL - 54

    SP - 5788

    EP - 5801

    JO - Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)

    JF - Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)

    SN - 2470-0045

    IS - 5

    ER -