The Problem of Integrable Discretization: Hamiltonian Approach

ISBN-13: 9783034894043 / Angielski / Miękka / 2012 / 21 str.

Yuri B. Suris

Opis

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.

Spis treści

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 € ? ? 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(?; €).- 6.15 Different forms of MRTL(?; €).- 6.15.1 Change of variables corresponding to M) (?; €).- 6.15.2 Change of variables corresponding to MZ+) (?; €).- 6.16 Lax representations of MRTL(?; €).- 6.16.1 Lax representation corresponding to M) (?; €).- 6.16.2 Lax representation corresponding to MZ+) (?; €).- 6.17 r-matrix interpretation of MRTL(?; €).- 6.18 Discretization of MRTL+(?; €).- 6.18.1 Discretization based on the first Lax representation.- 6.18.2 Discretization based on the second Lax representation.- 6.19 Localizing change of variables for dMRTL+(?; €).- 6.20 Discretization of MRTL_ (?; €).- 6.21 Bibliographical remarks.- 7 Relativistic Volterra Lattice.- 7.1 Introduction.- 7.2 Quadratic invariant Poisson bracket of RVL(?).- 7.3 Cubic invariant Poisson bracket of RVL(?).- 7.4 Auto-transformation of RVL(?).- 7.5 The first Lax representation of RVL(?).- 7.6 Quadratic r-matrix for the first Lax representation.- 7.7 The second Lax representation of RVL(?).- 7.8 The third Lax representation of RVL(?).- 7.9 Quadratic r-matrix for third Lax representation.- 7.10 Discretization of RVL+(?).- 7.11 Localizing change of variables for dRVL+(?).- 7.12 Discretization of RVL-(?).- 7.13 Localizing change of variables for dRVL-(?).- 7.14 Modified relativistic Volterra lattice.- 7.15 Discretization of MRVL+(?; €).- 7.16 Appendix: selected results for M1-version of RVL(?).- 7.16.1 The flow RVL+(?) and its discretization.- 7.16.2 The flow RVL-(?) and its discretization.- 7.16.3 The flow MRVL+(?; €) and its discretization.- 7.17 Bibliographical remarks.- 8 Newtonian Equations of the Relativistic Toda Type.- 8.1 Introduction.- 8.2 Parametrizing the linear-quadratic bracket.- 8.2.1 Systems RTL+(?), dRTL+(?).- 8.2.2 Systems RTL-(?), dRTL-(?).- 8.3 Parametrizing the linear bracket.- 8.3.1 Systems RTL+(?), dRTL+(?).- 8.3.2 Systems RTL-(?), dRTL-(?).- 8.4 Dual linear parametrization.- 8.4.1 Systems RTL+(?), dRTL+(?).- 8.4.2 Systems RTL-(?), dRTL-(?).- 8.5 Parametrizing the quadratic bracket.- 8.5.1 Systems RVL+(?), dRVL+(?).- 8.5.2 Systems RVL-(?), dRVL-(?).- 8.6 Parametrizing the linear-quadratic bracket II.- 8.6.1 Systems MRTL+(?; €), dMRTL+(?; €).- 8.6.2 Systems MRTL-(?; €), dMRTL-(?; €).- 8.7 Parametrizing the cubic-quadratic bracket.- 8.8 Parametrizing the cubic bracket I.- 8.9 Parametrizing the cubic bracket II.- 8.10 Parametrizing the cubic bracket III.- 8.11 Bibliographical remarks.- 9 Explicit Discretizations for Toda Systems.- 9.1 Introduction.- 9.2 Explicit discretization for TL.- 9.3 Explicit discretization for MTL(€).- 9.4 Explicit discretization for VL.- 9.5 Explicit discretization for MVL(€).- 9.6 Explicit dRTL+(h) from implicit dTL.- 9.7 Explicit dRVL+(h) from implicit dVL.- 9.8 Bibliographical remarks.- 10 Explicit Discretizations of Newtonian Toda Systems.- 10.1 Introduction.- 10.2 Parametrizing special linear-quadratic bracket.- 10.3 Parametrizing the linear bracket.- 10.4 Dual parametrization of the linear bracket.- 10.5 Parametrizing the quadratic bracket.- 10.6 Parametrizing general linear-quadratic bracket.- 10.7 Parametrizing the cubic-quadratic bracket.- 10.8 Parametrizing the cubic bracket. I.- 10.9 Parametrizing the cubic bracket. II.- 10.10 Parametrizing the cubic bracket. III.- 10.11 Bibliographical remarks.- 11 Bruschi-Ragnisco Lattice.- 11.1 Introduction.- 11.2 Bi-Hamiltonian structure.- 11.3 General construction.- 11.4 Orbit interpretation.- 11.5 Discretization.- 11.6 Newtonian equations of motion.- 11.7 Bibliographical remarks.- 12 Multi-field Toda-like Systems.- 12.1 Introduction.- 12.2 Multi-field analog of the Toda lattice.- 12.3 Linear r-matrix structure for TLm+1.- 12.4 Quadratic r-matrix structure for TLm+1.- 12.5 Discretization of TLm+1.- 12.6 Localizing change of variables for dTLm+1.- 12.7 Example: TL3.- 12.8 Multi-field analog of the modified Toda lattice.- 12.9 Quadratic r-matrix structure for MTLm+1(€).- 12.10 Discretization of MTLm+i (€).- 12.11 Localizing change of variables for dMTLm+i (E).- 12.12 Bibliographical remarks.- 13 Multi-field Relativistic Toda Systems.- 13.1 Introduction.- 13.2 Multi-field analog of RTL: first version.- 13.3 Linear r-matrix structure for RTLm+1(?).- 13.4 Quadratic bracket for RTLm+1(?).- 13.5 Introducing the gauge transformed hierarchy.- 13.6 Multi-field RTL: second version.- 13.7 Quadratic r-matrix structure for RTLm+1(?).- 13.8 Example: RTL3(?).- 13.9 Discretization of RTLm++1(?).- 13.10 Localizing change of variables for dRTL~++l (?).- 13.11 Local discretization for RTL, +1(?).- 13.12 Bibliographical remarks.- 14 Belov-Chaltikian Lattices.- 14.1 Introduction.- 14.2 Bi-Hamiltonian structure and Lax representation.- 14.3 Discretization of BCLm.- 14.4 Modified BCLm.- 14.5 Discretization of MBCLm(€).- 14.6 Localizing change of variables for dMBCLm( €).- 14.7 Relativistic deformation of BCLm.- 14.8 A gauge connection between MBCLm( €) and RBCLm(?).- 14.9 Discretization of RBCLm(?).- 14.10 Example: Volterra lattice as BCL1.- 14.11 Example: BCL2.- 14.12 Bibliographical remarks.- 15 Multi-field Volterra-like Systems.- 15.1 Introduction.- 15.2 Multi-field analog of the Volterra lattice.- 15.3 Quadratic r-matrix structure for VLm.- 15.4 Discretization of VLm.- 15.5 Localizing change of variables for dVLm.- 15.6 Example 1: VL3, three-field analog of Volterra lattice.- 15.7 A further generalization of VLm.- 15.8 Quadratic r-matrix structure for VLm(?).- 15.9 Discretization of VLm(?).- 15.10 Localizing change of variables for dVLm(?).- 15.11 The case of the signature ? = (+1, -1,…, -1).- 15.11.1 Lax representation and Hamiltonian structure.- 15.11.2 Discretization.- 15.11.3 Localizing change of variables.- 15.11.4 Miura relation to the Belov-Chaltikian lattices.- 15.12 Example 2: ? = (+1, -1, -1).- 15.13 Example 3: ? = (+1, +1, -1).- 15.14 Bibliographical remarks.- 16 Multi-field Relativistic Volterra Systems.- 16.1 Introduction.- 16.2 The RVLm(?; ?) hierarchy: first construction.- 16.3 Introducing the gauge transformed hierarchy.- 16.4 The RVLm( ?; ?) hierarchy: second construction.- 16.5 Discretization of RVI4,+) (?; ?).- 16.6 Localizing change of variables for dRVL,~,,+) (?; ?).- 16.7 Particular case h = a: explicit discretizations.- 16.8 The case of the signature v = (+1, +1,¡­, +1).- 16.8.1 Equations of motion and Hamiltonian structure.- 16.8.2 Discretization.- 16.8.3 Explicit discretization.- 16.8.4 Example: RVL3+“ (?), the three-field analog of the relativistic Volterra lattice.- 16.9 The case of the signature ? = (+1, ¡ª1,¡­, ¡ª1).- 16.9.1 Equations of motion and Hamiltonian structure.- 16.9.2 Discretization.- 16.9.3 Example: RVL3+“ (?; ?) with ? = (+1, ¡ª1, ¡ª1).- 16.10 Explicit dVL.m from the RVLmhierarchy.- 16.11 Bibliographical remarks.- 17 Bogoyavlensky Lattices.- 17.1 Introduction.- 17.2 Lax representations.- 17.3 Quadratic r-matrix structure of BL1(m).- 17.4 Quadratic r-matrix structure of BL2(p) and BL3(p).- 17.5 Examples of Hamiltonian structures.- 17.5.1 Lattice BL2(p), p > 1.- 17.5.2 Lattice BL3(p), p ? 2.- 17.6 Discretization of the lattice BL1(m).- 17.7 Discretization of the lattice BL2(p).- 17.8 Discretization of the lattice BL3 (p).- 17.9 Modified Volterra lattice.- 17.10 Alternative approach to BL1(m).- 17.11 Alternative approach to BL2(p).- 17.12 Alternative approach to BL3 (p).- 17.13 Bibliographical remarks.- 18 Ablowitz-Ladik Hierarchy.- 18.1 Introduction.- 18.2 AKNS hierarchy.- 18.3 Ablowitz-Ladik hierarchy.- 18.4 Non-local difference schemes.- 18.4.1 Difference schemes for NLS.- 18.4.2 Difference schemes for MKdV.- 18.5 Elementary flows of the AL hierarchy.- 18.6 Local discretizations for.F1.- 18.7 Symplectic properties.- 18.8 Local discretizations for NLS.- 18.9 Local discretizations for F 2.- 18.10 Local discretizations for MKdV.- 18.11 Connection with relativistic Toda lattice.- 18.12 Bibliographical remarks.- III Systems of Classical Mechanics.- 19 Peakons System.- 19.1 Introduction.- 19.2 Lax representation and r-matrix.- 19.3 Discretization.- 19.4 Lagrangian interpretation.- 19.5 Bibliographical remarks.- 20 Standard-like Discretizations.- 20.1 Introduction.- 20.2 Integrable scalar equations: Examples.- 20.3 Integrable scalar equations: Classification.- 20.4 Bibliographical remarks.- 21 Lie-algebraic Toda Systems.- 21.1 Introduction.- 21.2 Lie-algebraic open-end Toda lattices.- 21.3 Lie-algebraic periodic Toda lattices.- 21.4 Toda lattices AN-1and A 1.- 21.5 Discrete time lattices AN-1and 1.- 21.6 List of generalized Toda lattices.- 21.7 Discretization of lattices BN, CN, CN1), AzN, and DNZ+1.- 21.8 Discretization of lattices DN, D(Air), BN1, and A2N_1.- 21.9 2 x 2 Lax representations: continuous time case.- 21.10 2 x 2 Lax representations: discrete time case.- 21.11 Lattice G2as a reduction of the lattice B3.- 21.12 Lattice G as a reduction of the lattice B1).- 21.13 Toda lattice D(43), continuous and discrete.- 21.14 Bibliographical remarks.- 22 Gamier System.- 22.1 Introduction.- 22.2 Gamier system.- 22.2.1 Equations of motion and Hamiltonian structure.- 22.2.2 Integrals of motion.- 22.2.3 “Big” Lax representation.- 22.2.4 “Small” Lax representation.- 22.3 Anharmonic oscillator.- 22.3.1 Equations of motion and Hamiltonian structure.- 22.3.2 Integrals of motion.- 22.3.3 “Big” Lax representation.- 22.3.4 “Small” Lax representation.- 22.4 Wojciechowski system.- 22.4.1 Equations of motion and Hamiltonian structure.- 22.4.2 Integrals of motion.- 22.4.3 “Big” Lax representation.- 22.4.4 “Small” Lax representation.- 22.5 Bäcklund transformation for the Gamier system.- 22.6 Bäcklund transformation for anharmonic oscillator.- 22.7 Bäcklund transformation for Wojciechowski system.- 22.8 Explicit discretization of the Gamier system.- 22.8.1 Equations of motion and symplectic properties.- 22.8.2 Integrals of motion.- 22.8.3 “Big” Lax representation.- 22.8.4 “Small” Lax representation.- 22.9 Explicit discretization of anharmonic oscillator.- 22.9.1 Equations of motion and symplectic properties.- 22.9.2 Integrals of motion.- 22.9.3 “Big” Lax representation.- 22.9.4 “Small” Lax representation.- 22.10 Explicit discretization of Wojciechowski system.- 22.10.1 Equations of motion and symplectic properties.- 22.10.2 Integrals of motion.- 22.10.3 “Big” Lax representation.- 22.10.4 “Small” Lax representation.- 22.11 Bibliographical remarks.- 23 Hénon-Heiles System.- 23.1 Introduction.- 23.2 Lax representation.- 23.3 Discretization of Hénon-Heiles system.- 23.4 Bibliographical remarks.- 24 Neumann System.- 24.1 Introduction.- 24.2 Double Neumann system.- 24.2.1 Equations of motion and Hamiltonian structure.- 24.2.2 “Big” Lax representation.- 24.2.3 “Small” Lax representation.- 24.2.4 Unconstrained version.- 24.3 Neumann system.- 24.3.1 Equations of motion and Hamiltonian structure.- 24.3.2 “Big” Lax representation.- 24.3.3 “Small” Lax representation.- 24.3.4 Unconstrained version.- 24.4 Rosochatius system.- 24.4.1 Equations of motion and Hamiltonian structure.- 24.4.2 “Big” Lax representation.- 24.4.3 “Small” Lax representation.- 24.4.4 Unconstrained version.- 24.5 Bäcklund transformation for double Neumann system.- 24.6 Bäcklund transformation for Neumann system.- 24.7 Bäcklund transformation for Rosochatius system.- 24.8 Ragnisco’s discretization of Neumann system.- 24.9 V. Adler’s discretization of Neumann system.- 24.10 Coupled Neumann system.- 24.11 Discretizations of the coupled Neumann system.- 24.11.1 Discretization à la Ragnisco.- 24.11.2 Discretization à la V. Adler.- 24.12 Bibliographical remarks.- 25 Lie-algebraic Generalizations of the Gamier Systems.- 25.1 Introduction.- 25.2 Gamier systems related to symmetric spaces.- 25.3 Gamier systems related to AIII.- 25.3.1 Equations of motion and Lax representation.- 25.3.2 Example: M = 2.- 25.3.3 Discretizations.- 25.3.4 Example: discrete systems with M = 2.- 25.4 Gamier systems related to CI and DIII.- 25.5 Gamier systems related to BDI.- 25.5.1 Equations of motion and Lax representation.- 25.5.2 Discretization.- 25.6 Bibliographical remarks.- 26 Integrable Cases of Rigid Body Dynamics.- 26.1 Introduction.- 26.2 Multi-dimensional Euler top.- 26.3 Discrete time Euler top.- 26.4 Rigid body in a quadratic potential.- 26.5 Discrete time top in a quadratic potential.- 26.6 Multi-dimensional Lagrange top.- 26.6.1 Body frame formulation.- 26.6.2 Rest frame formulation.- 26.7 Discrete time analog of the Lagrange top.- 26.7.1 Rest frame formulation.- 26.7.2 Moving frame formulation.- 26.8 Three-dimensional Lagrange top.- 26.9 Discrete time three-dimensional Lagrange top.- 26.10 Rigid body motion in an ideal fluid: Clebsch case.- 26.11 Discretization of the Clebsch problem.- 26.11.1 Case A = B2of the Clebsch problem.- 26.11.2 Case A = B of the Clebsch problem.- A Appendix: Lagrange top and Heisenberg magnetic.- 26.12 Bibliographical remarks.- 27 Systems of Calogero-Moser Type.- 27.1 Introduction.- 27.2 Lax representations: rational and hyperbolic cases.- 27.3 Dynamical r-matrix formulation.- 27.4 Explicit solutions.- 27.4.1 Rational systems.- 27.4.2 Hyperbolic systems.- 27.5 Discrete time evolution: rational systems.- 27.5.1 Rational CM system.- 27.5.2 Rational RS system.- 27.6 Discrete time evolution: hyperbolic systems.- 27.6.1 Hyperbolic CM system.- 27.6.2 Hyperbolic RS systems.- 27.7 Elliptic CM type models: Lax representations.- 27.8 Elliptic CM type models: r-matrix structure.- 27.9 Discretization of elliptic CM and RS models.- 27.10 Strong coupling limit of RS models.- 27.10.1 Rational system.- 27.10.2 Hyperbolic system.- 27.11 Bibliographical remarks.- List of Notations.

Blurb

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 be accessible 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.