Stationary solutions and self-trapping in discrete quadratic nonlinear systems

Ole Bang, Peter Leth Christiansen, Carl A. Balslev Clausen

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    Abstract

    We consider the simplest equations describing coupled quadratic nonlinear (chi((2))) systems, which each consists of a fundamental mode resonantly interacting with its second harmonic. Such discrete equations apply, e.g., to optics, where they can describe arrays of chi((2)) waveguides, and to Solid state physics, where they can describe nonlinear interface waves under the conditions of Fermi resonance of the adjacent crystals. Focusing on the monomer and dimer we discuss their Hamiltonian structure and find all stationary solutions and their stability properties. in one limit the nonintegrable dimer reduce to the discrete nonlinear Schrodinger (DNLS) equation with two degrees of freedom, which is integrable. We show how the stationary solutions to the two systems correspond to each other and how the self-trapped DNLS solutions gradually develop chaotic dynamics in the chi((2)) system, when going away from the near integrable limit.
    Original languageEnglish
    JournalPhysical Review E. Statistical, Nonlinear, and Soft Matter Physics
    Volume56
    Issue number6
    Pages (from-to)7257-7266
    ISSN1063-651X
    DOIs
    Publication statusPublished - 1998

    Bibliographical note

    Copyright (1997) by the American Physical Society.

    Keywords

    • SPATIAL SOLITARY WAVES
    • GUIDES
    • INTERFACE
    • MEDIA
    • CRYSTALS
    • SOLITONS
    • FERMI-RESONANCE
    • 2ND-ORDER NONLINEARITIES
    • ARRAYS

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