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Discrete Systems and Integrability

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Part of Cambridge Texts in Applied Mathematics

  • Date Published: September 2016
  • availability: Available
  • format: Paperback
  • isbn: 9781107669482

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  • This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. While treating the material at an elementary level, the book also highlights many recent developments. Topics include: Darboux and Bäcklund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Padé approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painlevé equations; and the novel theory of Lagrangian multiforms. The book builds the material in an organic way, emphasizing interconnections between the various approaches, while the exposition is mostly done through explicit computations on key examples. Written by respected experts in the field, the numerous exercises and the thorough list of references will benefit upper-level undergraduate, and beginning graduate students as well as researchers from other disciplines.

    • A generous and up-to-date bibliography allows the reader to use the text as a source for branching out into more specialized directions, according to their own interests
    • Readers will more easily absorb the ideas and techniques by working through explicit computations built up on the basis of key examples while at the same time being presented with the general theory
    • Based on tried and tested lecture notes - the material has been used for teaching undergraduate courses and is tailored for student use
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    Product details

    • Date Published: September 2016
    • format: Paperback
    • isbn: 9781107669482
    • length: 458 pages
    • dimensions: 246 x 173 x 25 mm
    • weight: 0.83kg
    • contains: 68 b/w illus. 2 tables 98 exercises
    • availability: Available
  • Table of Contents

    Preface
    1. Introduction to difference equations
    2. Discrete equations from transformations of continuous equations
    3. Integrability of P∆Es
    4. Interlude: lattice equations and numerical algorithms
    5. Continuum limits of lattice P∆Es
    6. One-dimensional lattices and maps
    7. Identifying integrable difference equations
    8. Hirota's bilinear method
    9. Multi-soliton solutions and the Cauchy matrix scheme
    10. Similarity reductions of integrable P∆Es
    11. Discrete Painlevé equations
    12. Lagrangian multiform theory
    Appendix A. Elementary difference calculus and difference equations
    Appendix B. Theta functions and elliptic functions
    Appendix C. The continuous Painlevé equations and the Garnier system
    Appendix D. Some determinantal identities
    References
    Index.

  • Authors

    J. Hietarinta, University of Turku, Finland
    J. Hietarinta is Professor Emeritus of Theoretical Physics at the University of Turku, Finland. His work has focused on the search for integrable systems of various forms, including Hamiltonian mechanics, Hirota bilinear form, Yang-Baxter and tetrahedron equations, as well as lattice equations. He was instrumental in the setting up of the nlin.SI category in arxiv.org and created the web pages for the SIDE (Symmetries and Integrability of Difference Equations) conference series (http://side-conference.net).

    N. Joshi, University of Sydney
    N. Joshi is Professor of Applied Mathematics at the University of Sydney. She is best known for her work on the Painlevé equations and works at the leading edge of international efforts to analyse discrete and continuous integrable systems in the geometric setting of their initial-value spaces, constructed by resolving singularities in complex projective space. She was elected as a Fellow of the Australian Academy of Science in 2008, holds a Georgina Sweet Australian Laureate Fellowship and was awarded the special Hardy Fellowship of the London Mathematical Society in 2015.

    F. W. Nijhoff, University of Leeds
    F. W. Nijhoff is Professor of Mathematical Physics in the School of Mathematics at the University of Leeds. His research focuses on nonlinear difference and differential equations, symmetries and integrability of discrete systems, variational calculus, quantum integrable systems and linear and nonlinear special functions. He was the principal organizer of the 2009 six-month programme on Discrete Integrable Systems at the Isaac Newton Institute, and was awarded a Royal Society Leverhulme Trust Senior Research Fellow in 2011.

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