The inverse scattering transform and squared eigenfunctions for the nondegenerate 3 x 3 operator and its soliton structure

    Authors

    D. J. Kaup;R. A. Van Gorder

    Comments

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    Abbreviated Journal Title

    Inverse Probl.

    Keywords

    LINEAR 3-WAVE INTERACTIONS; SPACE-TIME EVOLUTION; RESONANT INTERACTION; NONLINEAR MEDIA; WAVE PACKETS; EQUATIONS; SYSTEM; OPTICS; Mathematics, Applied; Physics, Mathematical

    Abstract

    We develop a soliton perturbation theory for the non-degenerate 3x3 eigenvalue operator, with obvious applications to the three-wave resonant interaction. The key elements of an inverse scattering perturbation theory for integrable systems are the squared eigenfunctions and their adjoints. These functions serve as a mapping between variations in the potentials and variations in the scattering data. We also address the problem of the normalization of the Jost functions, how this affects the structure and solvability of the inverse scattering equations and the definition of the scattering data. We then explicitly provide the construction of the covering set of squared eigenfunctions and their adjoints, in terms of the Jost functions of the original eigenvalue problem. We also obtain, by a new and direct method (Yang and Kaup 2009 J. Math. Phys. 50 023504), the inner products and closure relations for these squared eigenfunctions and their adjoints. With this universal covering group, one would have tools to study the perturbations for any integrable system whose Lax pair contained the non-degenerate 3x3 eigenvalue operator, such as that found in the Lax pair of the integrable three-wave resonant interaction.

    Journal Title

    Inverse Problems

    Volume

    26

    Issue/Number

    5

    Publication Date

    1-1-2010

    Document Type

    Article

    Language

    English

    First Page

    34

    WOS Identifier

    WOS:000277085900005

    ISSN

    0266-5611

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