The book explores the theory of discrete integrable systems, with an emphasis on the following general problem: how to discretize one or several of independent variables in a given integrable system of differential equations, maintaining the integrability property? This question (related in spirit to such a modern branch of numerical analysis as geometric integration) is treated in the book as an immanent part of the theory of integrable systems, also commonly termed as the theory of solitons.

Among several possible approaches to this theory, the Hamiltonian one is chosen as the guiding principle. A self-contained exposition of the Hamiltonian (r-matrix, or "Leningrad") approach to integrable systems is given, culminating in the formulation of a general recipe for integrable discretization of r-matrix hierarchies. After that, a detailed systematic study is carried out for the majority of known discrete integrable systems which can be considered as discretizations of integrable ordinary differential or differential-difference (lattice) equations. This study includes, in all cases, a unified treatment of the correspondent continuous integrable systems as well. The list of systems treated in the book includes, among others: Toda and Volterra lattices along with their numerous generalizations (relativistic, multi-field, Lie-algebraic, etc.), Ablowitz-Ladik hierarchy, peakons of the Camassa-Holm equation, Garnier and Neumann systems with their various relatives, many-body systems of the Calogero-Moser and Ruijsenaars-Schneider type, various integrable cases of the rigid body dynamics. Most of the results are only available from recent journal publications, many of them are new.

Thus, the book is a kind of encyclopedia on discrete integrable systems. It unifies the features of a research monograph and a handbook. It is supplied with an extensive bibliography and detailed bibliographic remarks at the end of each chapter. Largely self-contained, it will beaccessible to graduate and post-graduate students as well as to researchers in the area of integrable dynamical systems. Also those involved in real numerical calculations or modelling with integrable systems will find it very helpful.

Klappentext

An exploration of the theory of discrete integrable systems, with an emphasis on the following general problem: how to discretize one or several of independent variables in a given integrable system of differential equations, maintaining the integrability property? This question (related in spirit to such a modern branch of numerical analysis as geometric integration) is treated in the book as an immanent part of the theory of integrable systems, also commonly termed as the theory of solitons. Most of the results are only available from recent journal publications, many of them are new.

Thus, the book is a kind of encyclopedia on discrete integrable systems. It unifies the features of a research monograph and a handbook. It is supplied with an extensive bibliography and detailed bibliographic remarks at the end of each chapter. Largely self-contained, it will be accessible to graduate and post-graduate students as well as to researchers in the area of integrable dynamical systems.

Inhalt

I General Theory.- 1 Hamiltonian Mechanics.- 1.1 The problem of integrable discretization.- 1.2 Poisson brackets and Hamiltonian flows.- 1.3 Symplectic manifolds.- 1.4 Poisson submanifolds and symplectic leaves.- 1.5 Dirac bracket.- 1.6 Poisson reduction.- 1.7 Complete integrability.- 1.8 Bi-Hamiltonian systems.- 1.9 Lagrangian mechanics on ?N.- 1.10 Lagrangian mechanics on TP and on P × P.- 1.11 Lagrangian mechanics on Lie groups.- 1.11.1 Continuous time case.- 1.11.2 Discrete time case.- 1.12 Invariant Lagrangians and Lie-Poisson bracket.- 1.12.1 Continuous time case.- 1.12.2 Discrete time case.- 1.13 Lagrangian reduction and Euler-Poincaré equations.- 1.13.1 Continuous time case.- 1.13.2 Discrete time case.- A Appendix: Gradients, vector fields, and other notation.- B Appendix: Lie groups and Lie algebras.- 1.14 Bibliographical remarks.- 2 R-matrix Hierarchies.- 2.1 Introduction.- 2.2 Lie-Poisson brackets.- 2.2.1 General construction.- 2.2.2 Tensor notation.- 2.2.3 Examples.- 2.3 Linear r-matrix structure.- 2.3.1 General construction.- 2.3.2 Tensor notation.- 2.3.3 Examples of R-operators and r-matrices.- 2.4 Generalized linear r-matrix structure.- 2.5 Quadratic r-matrix structure.- 2.5.1 General construction.- 2.5.2 Tensor notation.- 2.5.3 Example.- 2.6 Poisson brackets on direct products.- 2.6.1 General construction.- 2.6.2 Tensor notation.- 2.6.3 Poisson properties of the monodromy map.- 2.7 R-operators from splitting g = g+? g-.- 2.8 Bäcklund transformations.- 2.9 Recipe for integrable discretization.- A Appendix: Bäcklund-Darboux transformation for KdV.- 2.10 Bibliographical remarks.- II Lattice Systems.- 3 Toda Lattice.- 3.1 Introduction.- 3.2 Tri-Hamiltonian structure.- 3.3 Basic algebras and operators.- 3.3.1 Open-end case.- 3.3.2 Periodic case.- 3.4 Lax representation.- 3.5 Linear r-matrix structure.- 3.6 Quadratic r-matrix structure.- 3.7 2 × 2 Lax representation.- 3.8 Discretization of the Toda lattice.- 3.9 Localizing changes of variables.- 3.10 Local equations of motion for dTL.- 3.11 Second Toda flow and its discretization.- 3.12 Local equations of motion for dTL2.- 3.13 Third Toda flow and its discretization.- 3.14 Local equations of motion for dTL3.- 3.15 Modified Toda lattice.- 3.16 Discretization of MTL.- 3.17 Local equations of motion for dMTL.- 3.18 Second modification of TL.- 3.19 Discretization of M2TL.- 3.20 Local equations of motion for dM2TL.- 3.21 Third modification of TL.- A Appendix: Miura transformations for KdV.- 3.22 Bibliographical remarks.- 4 Volterra Lattice.- 4.1 Introduction.- 4.2 Bi-Hamiltonian structure.- 4.3 Lax representation.- 4.4 r-matrix structure.- 4.5 Discretization.- 4.6 Local equations of motion for dVL.- 4.7 Second flow of the Volterra hierarchy.- 4.8 Local discretization of VL2.- 4.9 Local discretization of KdV.- 4.10 Modified Volterra lattice.- 4.11 Discretization of MVL.- 4.12 Local equations of motion for dMVL.- 4.13 Different forms of MVL and dMVL.- 4.14 Particular case EUR ? ? of MVL.- 4.15 Second modification of VL.- 4.16 Factorizations and the two-field form of VL.- 4.17 Lax representation in g ? g.- 4.18 Quadratic r-matrix structure in g ? g.- 4.19 Discretization of the two-field VL.- 4.20 Local equations for the two-field dVL.- 4.21 Two-field versions of VL2 and dVL2.- 4.22 Two-field modified Volterra lattice.- 4.23 Discretization of the two-field MVL.- 4.24 Local equations for the two-field dMVL.- A Appendix: Tower of modifications of VL à la Yamilov.- 4.26 Bibliographical remarks.- 5 Newtonian Equations of the Toda Type.- 5.1 Introduction.- 5.2 Exponential form of the Toda lattice.- 5.3 Dual Toda lattice.- 5.4 Modified exponential Toda lattice.- 5.5 Parametrizing the linear-quadratic bracket.- 5.6 Parametrizing the cubic-quadratic bracket.- 5.7 Parametrizing the cubic bracket I.- 5.8 Parametrizing the cubic bracket II.- 5.9 Parametrizing the cubic bracket III.- 5.10 Newtonian equations for TL2.- 5.11 Bibliographical remarks.- 6 Relativistic Toda Lattice.- 6.1 Introduction.- 6.2 The first Lax representation of RTL(?).- 6.3 Linear r-matrix for the first Lax representation.- 6.4 Quadratic r-matrix for the first Lax representation.- 6.5 Tri-Hamiltonian structure of RTL(?).- 6.6 The second Lax representation of RVL(?).- 6.7 Linear r-matrix for the second Lax representation.- 6.8 Quadratic r-matrix for the second Lax representation.- 6.9 2 × 2 Lax representations.- 6.10 Discretization of the flow RTL+(?).- 6.11 Localizing change of variables for dRTL+(?).- 6.12 Discretization of the flow RTL-(?).- 6.13 Localizing change of variables for dRTL-(?).- 6.14 Modified relativistic Toda lattice MRTL(?; EUR).- 6.15 Different forms of MRTL(?; EUR).- 6.15.1 Change of variables corresponding to M)…