Symmetries and Integrability of Difference Equations

doi:10.1017/CBO9780511569432

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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.

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

Source: European Mathematical Society

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Contents

Frontmatter
pp i-iv
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Contents
pp v-viii
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Preface
pp ix-xiv
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List of Participants
pp xv-xvi
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1 - Partial Difference Equations
pp 1-2
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Discrete linearisable Gambier equations
pp 3-14
- By A.K. Common, Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, UK., E. Hesameddini, Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, UK., M. Musette, Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B–1050 Brussels, Belgium
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Generalized Bäcklund transformation and new explicit solutions of the two-dimensional Toda equation
pp 15-22
- By Xing-Biao Hu, State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific Engineering Computing, Academia Sinca, P.O. Box 2719, Beijing 100080, P.R. China, Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent CT2 7NF, UK, Peter A. Clarkson, Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent CT2 7NF, UK
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Different aspects of relativistic Toda chain
pp 23-40
- By S. Kharchev, ITEP, Bol. Cheremushkinskaya, 25, Moscow, 117 259, Russia, A. Mironov, Theory Department, P.N. Lebedev Physics Institute, Leninsky prospect, 53, Moscow, 117924, Russia, A. Zhedanov, Physics Department, Donetsk State University, Donetsk, 340 055, Ukraine
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2 - Integrable Mappings
pp 41-42
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Integrable symplectic maps
pp 43-55
- By A.P. Fordy, Department of Applied Mathematical Studies and Centre for Nonlinear Studies, University of Leeds, Leeds LS2 9JT, UK
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An iterative process on quartics and integrable symplectic maps
pp 56-63
- By J.P. Francoise, Laboratoire de Geométrie Differentielle et Appliquée Université Pierre et Marie Curie Place Jussieu, Paris, France, O. Ragnisco, Dipartimento di Fisica, Università di Roma Tre Via Vasca Navale 84, Roma, Italy
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Integrable mappings of KdV type and hyperelliptic addition formulae
pp 64-78
- By F.W. Nijhoff, Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, UK, V.Z. Enolskii, Institute of Magnetism of NASU 36 Vernadskii str., 252142 Kiev, Ukraine
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R-matrix hierarchies, integrable lattice systems and their integrable discretizations
pp 79-94
- By Yuri B. Suris, Centre for Complex Systems and Visualization, University of Bremen, Universitätsallee 29, 28359 Bremen, Germany
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3 - Discrete Geometry
pp 95-96
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Discrete conformal maps and surfaces
pp 97-108
- By Alexander I. Bobenko, FB Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, Berlin, Germany and Department of Mathematics, University of Massachusetts, Amherst, MA 01003
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The Bäcklund transformation for discrete isothermic surfaces
pp 109-121
- By Jan Cieśliński, Filia Uniwersytetu Warszawskiego w Bialymstoku Instytut Fizyki, ul. Lipowa 41, 15-424 Biaiystkok, Poland
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Integrable discrete geometry with ruler and compass
pp 122-136
- By Adam Doliwa, Istituto Nazionale di Fisica Nucleare, Sezione di Roma P-le Aldo Moro 2, I–00185 Roma, Italy, Instytut Fizyki Teoretycznej, Uniwersytet Warszawski ul. Hoża 69, 00–681 Warszawa, Poland
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Self-dual Einstein spaces and a discrete Tzitzeica equation, a permutability theorem link
pp 137-148
- By W.K. Schief, School of Mathematics, The University of New South Wales, Sydney, NSW 2052, Australia
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4 - Asymptotic Analysis
pp 149-150
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New solutions of nonstationary Schrödinger and Kadomtsev-Petviashvili equations
pp 151-164
- By Mark J. Ablowitz, Department of Applied Mathematics University of Colorado Boulder, CO 80309-0526, USA, Javier Villarroel, Universidad de Salamanca Dept. de Matematicos puras y aplicadas 37008 Salamanca, SPAIN
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On asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert method
pp 165-177
- By Pavel Bleher, Department of Mathematical Sciences, Indiana University-Purdue University at Indianapolis, Alexander Its, Department of Mathematical Sciences, Indiana University-Purdue University at Indianapolis
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A new spectral transform for solving the continuous and spatially discrete heat equations on simple trees
pp 178-194
- By P.C. Bressloff, Department of Mathematical Sciences, Loughborough University, Loughborough, Leics. LE11 3TU, U.K., A.S. Fokas, Department of Mathematics, Imperial College, University of London, London, SW7 2BZ, U.K.
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5 - Discrete Painlevé Equations
pp 195-196
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The discrete Painlevé I hierarchy
pp 197-205
- By Clio Cresswell, School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia, Nalini Joshi, Department of Pure Mathematics, University of Adelaide, Adelaide SA 5005, Australia
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Rational solutions to dPIV
pp 206-216
- By Jarmo Hietarinta, Department of Physics, University of Turku FIN-20014 Turku, Finland, Kenji Kajiwara, Department of Electrical Engineering Doshisha University, Tanabe, Kyoto 610-03, Japan
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The discrete Painlevé II equation and the classical special functions
pp 217-227
- By Kenji Kajiwara, Department of Electrical Engineering, Doshisha University, Tanabe, Kyoto 610-03, JAPAN
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Freud's equations for orthogonal polynomials as discrete Painlevé equations
pp 228-244
- By Alphonse P. Magnus, Institut de Mathématique Pure et Appliquée, Université Catholique de Louvain, Chemin du Cyclotron,2, B-1348 Louvain-la-Neuve, Belgium
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6 - Symmetries of Difference Equations
pp 245-246
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An approach to master symmetries of lattice equations
pp 247-260
- By Benno Fuchssteiner, Universität-GH Paderborn, D-33098 Paderborn, Germany, Wen-Xiu Ma, Department of Mathematics, University of Manchester Institute of Science and Technology, Manchester M60 1QD, UK
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Symmetries and generalized symmetries for discrete dynamical systems
pp 261-274
- By D. Levi, Dipartimento di Fisica – Universitá di Roma 3 and INFN-Sezione di Roma I, Via della Vasca Navale 84, 00146 Roma, Italy
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