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Symmetries and Integrability of Difference Equations

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Part of London Mathematical Society Lecture Note Series

A. K. Common, E. Hesameddini, M. Musette, X.-B. Hu, P. Clarkson, S. Kharchev, A. Mironov, A. Zhedanov, A. Fordy, J. P. Francoise, O. Ragnisco, F. Nijhoff, V. Z. Enolskii, Y. Suris, A. Bobenko, J. Cieslinski, A. Doliwa, W. K. Schief, M. Ablowitz, P. Bleher, A. Its, P. C. Bressloff, A. Fokas, C. Cresswell, N. Joshi, J. Hietarinta, K. Kajiwara, A. Magnus, B. Fuchssteiner, W. X. Ma, D. Levi, P. Winternitz, A. Fordy, A. Pickering, W. Oevel, C. Gilson, B. Grammaticos, A. Ramani, S. Moriwaki, A. Nagaki, J. Satsuma, T. Tokihiro, M. Torii, D. Takahashi, J. Matsukidaira, N. Atakishiyev, F. A. Grünbaum, L. Haine, V. B. Kuztensov, E. K. Skylanin, R. Kashaev, F. Bonechi, R. Giachetti, E. Sorace, M. Tarlini, C. Viallet, A. Volkov
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  • Date Published: May 2012
  • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • format: Adobe eBook Reader
  • isbn: 9781139244107

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  • There has in recent years been a remarkable growth of interest in the area of discrete integrable systems. Much progress has been made by applying symmetry groups to the study of differential equations, and connections have been made to other topics such as numerical methods, cellular automata and mathematical physics. This volume comprises state of the art articles from almost all the leading workers in this important and rapidly developing area, making it a necessary resource for all researchers interested in discrete integrable systems or related subjects.

    • Exciting area of research
    • Articles by top names
    • Links with mathematical physics
    Read more

    Reviews & endorsements

    '… this book will form an inspiration for further research and so help to establish the links between the various communities working on discrete systems.' European Mathematical Society

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    Product details

    • Date Published: May 2012
    • format: Adobe eBook Reader
    • isbn: 9781139244107
    • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • Table of Contents

    Part I. Partial Differential Equations:
    1. Discrete linearisable Gambier equations A. K. Common, E. Hesameddini and M. Musette
    2. Generalized Bäcklund transformation and new explicit solutions of the two-dimensional Toda equation X.-B. Hu and P. Clarkson
    3. Different aspects of relativistic Toda Chain S. Kharchev, A. Mironov, A. Zhedanov
    Part II. Integrable Mappings:
    4. Integrable symplectic maps A. Fordy
    5. An iterative process on quartics and integrable symplectic maps J. P. Francoise, and O. Ragnisco
    6. Integrable mappings of KdV type and hyperelliptic addition theorems F. Nijhoff and V. Z. Enolskii
    7. R-matrix hierarchies, integrable lattice systems and their integrable discretizations Y. Suris
    Part III. Discrete Geometry:
    8. Discrete conformal maps and surfaces A. Bobenko
    9. The Bäcklund transformation for discrete isothermic surfaces J. Cieslinski
    10. Integrable discrete geometry with ruler and compass A. Doliwa
    11. Self-dual Einstein spaces and a discrete Tzitzeica equation. A permutability theorem link W. K. Schief
    Part IV. Asymptotic Analysis:
    12. New solutions of the non-stationary Schrödinger and Kadomtsev-Petviashvili equation M. Ablowitz
    13. On asymptotic analysis of the orthogonal polynomials via the Riemann-Hilbert method P. Bleher and A. Its
    14. A new spectral transform for solving the continuous and spatially discrete heat equation on simple trees P. C. Bressloff and A. Fokas
    Part V. Discrete Painlevé Equations:
    15. The discrete Painlevé hierarchy C. Cresswell and N. Joshi
    16. Rational solutions to d-PIV J. Hietarinta and K. Kajiwara
    17. The discrete Painlevé II equation and the classical special functions K. Kajiwara
    18. Freud's equations for orthogonal polynomials as discrete Painlevé equations A. Magnus
    Part VI. Symmetries of Difference Equations:
    19. An approach to master symmetries of lattice equations B. Fuchssteiner and W. X. Ma
    20. Symmetries and generalized symmetries for discrete dynamical systems D. Levi
    21. Nonlinear difference equations with superposition formulas P. Winternitz
    Part VII. Numerical Methods and Miscellaneous:
    22. Generalising the Painlevé truncation: expansions in Riccati pseudopotentials A. Fordy and A. Pickering
    23. Symplectic Runge-Kutta schemes W. Oevel
    Part VIII. Cellular Automata:
    24. Soliton cellular automata C. Gilson
    25. Painlevé equations and cellular automata B. Grammaticos and A. Ramani
    26. 2+1 Dimensional soliton cellular automaton S. Moriwaki, A. Nagaki, J. Satsuma, T. Tokihiro, M. Torii, D. Takahashi and J. Matsukidaira
    Part IX. q-Special Functions and q-Difference Equations:
    27. Fourier–Gauss transforms of q-exponential and q-Bessel functions N. Atakishiyev
    28. The Wilson bispectral involution: some elementary examples F. A. Grünbaum and L. Haine
    29. Factorisation of Macdonald polynomials V. B. Kuztensov and E. K. Skylanin
    30. Local Yang-Baxter relations associated with Hirota's discrete equation R. Kashaev
    31. Ultra-discrete soliton systems J. Satsuma
    32. Schrödinger equation on quantum homogeneous spaces F. Bonechi, R. Giachetti, E. Sorace, M. Tarlini
    33. Some algebraic solutions of discrete equations from anticommuting variables C. Viallet
    34. Q-combinatorics and quantum integrability A. Volkov.

  • Editors

    Peter A. Clarkson, University of Kent, Canterbury

    Frank W. Nijhoff, University of Leeds

    Contributors

    A. K. Common, E. Hesameddini, M. Musette, X.-B. Hu, P. Clarkson, S. Kharchev, A. Mironov, A. Zhedanov, A. Fordy, J. P. Francoise, O. Ragnisco, F. Nijhoff, V. Z. Enolskii, Y. Suris, A. Bobenko, J. Cieslinski, A. Doliwa, W. K. Schief, M. Ablowitz, P. Bleher, A. Its, P. C. Bressloff, A. Fokas, C. Cresswell, N. Joshi, J. Hietarinta, K. Kajiwara, A. Magnus, B. Fuchssteiner, W. X. Ma, D. Levi, P. Winternitz, A. Fordy, A. Pickering, W. Oevel, C. Gilson, B. Grammaticos, A. Ramani, S. Moriwaki, A. Nagaki, J. Satsuma, T. Tokihiro, M. Torii, D. Takahashi, J. Matsukidaira, N. Atakishiyev, F. A. Grünbaum, L. Haine, V. B. Kuztensov, E. K. Skylanin, R. Kashaev, F. Bonechi, R. Giachetti, E. Sorace, M. Tarlini, C. Viallet, A. Volkov

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