• Login
    View Item 
    •   Home
    • Theses & Dissertations
    • 2018 - Mines Theses & Dissertations
    • View Item
    •   Home
    • Theses & Dissertations
    • 2018 - Mines Theses & Dissertations
    • View Item
    JavaScript is disabled for your browser. Some features of this site may not work without it.

    Browse

    All of Mines RepositoryCommunitiesPublication DateAuthorsTitlesSubjectsThis CollectionPublication DateAuthorsTitlesSubjects

    My Account

    Login

    Mines Links

    Arthur Lakes LibraryColorado School of Mines

    Statistics

    Display Statistics

    Symbolic computation of Lax pairs of nonlinear partial difference equations

    • CSV
    • RefMan
    • EndNote
    • BibTex
    • RefWorks
    Thumbnail
    Name:
    Bridgman_mines_0052E_11543.pdf
    Size:
    1.122Mb
    Format:
    PDF
    Download
    Author
    Bridgman, Terry J.
    Advisor
    Hereman, Willy A.
    Date issued
    2018
    Keywords
    integrability
    discrete difference equations
    Lax pairs
    
    Metadata
    Show full item record
    URI
    https://hdl.handle.net/11124/172403
    Abstract
    This thesis is primarily concerned with the symbolic computation of Lax pairs for nonlinear systems of partial difference equations (P∆Es) which are defined on a quadrilateral and consistent around a cube (CAC). A literature survey provides historical context for the results presented in this thesis. Particular attention is paid to the origins of integrable P∆Es which are central to this dissertation. Pioneering work of Ablowitz & Ladik as well as Hirota gave rise to nonlinear P∆Es as discretizations of completely integrable partial differential equations. Subsequent investigations by Nijhoff, Quispel & Capel and Adler, Bobenko & Suris provided a strong impetus to the modern and ongoing study of fully discrete integrable systems covered in this thesis. An algorithmic method due to Nijhoff and Bobenko & Suris to compute Lax pairs for scalar P∆Es is reviewed in detail. The extension and implementation of that algorithm for systems of P∆Es are part of the novel research in this thesis. The algorithm has been implemented in the syntax of Mathematica, a major and commonly used computer algebra system. A symbolic software package, LaxPairPartialDifferenceEquations.m accompanies the thesis. The code automatically (i) determines whether or not P∆Es have the CAC property, (ii) computes Lax pairs for nonlinear P∆Es that are CAC; and (iii) verifies if Lax pairs satisfy the Lax equation. Lax pairs are presented for the scalar integrable P∆Es classified by Adler, Bobenko, and Suris as well as for numerous systems of integrable P∆Es, including the lattice Boussinesq, Schwarzian Boussinesq, Toda-Modified Boussinesq systems, and the two-component potential Korteweg-de Vries system. Previously unknown Lax pairs are presented for systems of P∆Es derived by Hietarinta. Lax pairs are not unique. To the contrary, for any P∆E there exists an infinite number of Lax pairs due to gauge equivalence. The investigation of gauge and gauge-like transformations is a novel component of this thesis. A detailed discussion is given of how edge equations should be handled to obtain gauge and gauge-like equivalent Lax matrices of minimal size. The Lax pairs for Hietarinta’s systems presented in this thesis are compared with those computed by Zhang, Zhao, and Nijhoff via a direct linearization method.
    Rights
    Copyright of the original work is retained by the author.
    Collections
    2018 - Mines Theses & Dissertations

    entitlement

     
    DSpace software (copyright © 2002 - 2022)  DuraSpace
    Quick Guide | Contact Us
    Open Repository is a service operated by 
    Atmire NV
     

    Export search results

    The export option will allow you to export the current search results of the entered query to a file. Different formats are available for download. To export the items, click on the button corresponding with the preferred download format.

    By default, clicking on the export buttons will result in a download of the allowed maximum amount of items.

    To select a subset of the search results, click "Selective Export" button and make a selection of the items you want to export. The amount of items that can be exported at once is similarly restricted as the full export.

    After making a selection, click one of the export format buttons. The amount of items that will be exported is indicated in the bubble next to export format.