Yuri B. Suris

The Problem ofIntegrable Discretization:Hamiltonian Approach

Birkhauser VerlagBasel • Boston • Berlin

Contents

Preface xvii

I General Theory

1 Hamiltonian Mechanics1.1 The problem of integrable discretization 31.2 Poisson brackets and Hamiltonian flows 41.3 Symplectic manifolds 71.4 Poisson submanifolds and symplectic leaves 111.5 Dirac bracket 121.6 Poisson reduction 141.7 Complete integrability 15

Bi-Hamiltonian systems 18Lagrangian mechanics on RN 20

1.10 Lagrangian mechanics on TV and on V x V 211.11 Lagrangian mechanics on Lie groups 25

1.11.1 Continuous time case 281.11.2 Discrete time case 30

1.12 Invariant Lagrangians and Lie-Poisson bracket 351.12.1 Continuous time case 351.12.2 Discrete time case 37

1.13 Lagrangian reduction and Euler-Poincare equations 401.13.1 Continuous time case 401.13.2 Discrete time case 44

A Appendix: Gradients, vector fields, and other notation 47B Appendix: Lie groups and Lie algebras 481.14 Bibliographical remarks 49

R-matrix Hierarchies2.1 Introduction 512.2 Lie-Poisson brackets 53

2.2.1 General construction 532.2.2 Tensor notation 55

vi Contents

2.2.3 Examples 572.3 Linear r-matrix structure 58

2.3.1 General construction 582.3.2 Tensor notation 602.3.3 Examples of .R-operators and r-matrices 64

2.4 Generalized linear r-matrix structure 712.5 Quadratic r-matrix structure 72

2.5.1 General construction 722.5.2 Tensor notation 762.5.3 Example 78

2.6 Poisson brackets on direct products 792.6.1 General construction 792.6.2 Tensor notation 812.6.3 Poisson properties of the monodromy map 82

2.7 i?-operators from splitting g = g+ (B g~ 852.8 Backlund transformations 892.9 Recipe for integrable discretization 92A Appendix: Backlund-Darboux transformation for KdV 942.10 Bibliographical remarks 97

II Lattice Systems

3 Toda Lattice3.1 Introduction 1033.2 Tri-Hamiltonian structure 1063.3 Basic algebras and operators 108

3.3.1 Open-end case 1083.3.2 Periodic case 109

3.4 Lax representation 1103.5 Linear r-matrix structure 1123.6 Quadratic r-matrix structure 1173.7 2x 2 Lax representation 1203.8 Discretization of the Toda lattice 1223.9 Localizing changes of variables 1253.10 Local equations of motion for dTL 1273.11 Second Toda flow and its discretization 1313.12 Local equations of motion for dTL2 1343.13 Third Toda flow and its discretization 1363.14 Local equations of motion for dTL3 1393.15 Modified Toda lattice 1413.16 Discretization of MTL 1463.17 Local equations of motion for dMTL 1493.18 Second modification of TL 1513.19 Discretization of M2TL 155

Contents vii

3.20 Local equations of motion for dM2TL 1593.21 Third modification of TL 162A Appendix: Miura transformations for KdV 1633.22 Bibliographical remarks 169

4 Volterra Lattice4.1 Introduction 1734.2 Bi-Hamiltonian structure 1764.3 Lax representation 1764.4 r-matrix structure 1784.5 Discretization 1794.6 Local equations of motion for dVL 1814.7 Second flow of the Volterra hierarchy 1844.8 Local discretization of VL2 1874.9 Local discretization of KdV 1884.10 Modified Volterra lattice 1894.11 Discretization of MVL 1904.12 Local equations of motion for dMVL 1924.13 Different forms of MVL and dMVL 1934.14 Particular case e —> ooofMVL 1954.15 Second modification of VL 1974.16 Factorizations and the two-field form of VL 1984.17 Lax representation in g <g> g 2034.18 Quadratic r-matrix structure in g <g> g 2054.19 Discretization of the two-field VL 2064.20 Local equations for the two-field dVL 2094.21 Two-field versions of VL2 and dVL2 2114.22 Two-field modified Volterra lattice 2154.23 Discretization of the two-field MVL 2174.24 Local equations for the two-field dMVL 221A Appendix: Tower of modifications of VL a la Yamilov 2234.26 Bibliographical remarks 227

5 Newtonian Equations of the Toda Type5.1 Introduction 2315.2 Exponential form of the Toda lattice 2345.3 Dual Toda lattice 2385.4 Modified exponential Toda lattice 2405.5 Parametrizing the linear-quadratic bracket 2435.6 Parametrizing the cubic-quadratic bracket 2475.7 Parametrizing the cubic bracket I 2515.8 Parametrizing the cubic bracket II 2545.9 Parametrizing the cubic bracket III 2565.10 Newtonian equations for TL2 2585.11 Bibliographical remarks 261

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6 Relativistic Toda Lattice6.1 Introduction 2636.2 The first Lax representation of RTL(a) 2676.3 Linear r-matrix for the first Lax representation 2696.4 Quadratic r-matrix for the first Lax representation 2746.5 Tri-Hamiltonian structure of RTL(a) 2766.6 The second Lax representation of RTL (a) 2796.7 Linear r-matrix for the second Lax representation 2816.8 Quadratic r*'matrix for the second Lax representation 2846.9 2x2 Lax representations 2866.10 Discretization of the flow RTL+(a) 2866.11 Localizing change of variables for dRTL+(a) 2906.12 Discretization of the flow RTL- (a) 2926.13 Localizing change of variables for dRTL_ (a) 2966.14 Modified relativistic Toda lattice MRTL(a;e) 2996.15 Different forms of MRTL(a;e) 301

6.15.1 Change of variables corresponding to M\+' (a; e) 302

6.15.2 Change of variables corresponding to Mj (Q; e) 3036.16 Lax representations of M R T L ( Q ; e) 304

6.16.1 Lax representation corresponding to M[+^ (a; e) 304

6.16.2 Lax representation corresponding to M ^ (a; e) 3066.17 r-matrix interpretation of MRTL(a;e) 3096.18 Discretization of MRTL+(a;e) 311

6.18.1 Discretization based on the first Lax representation 3116.18.2 Discretization based on the second Lax representation 313

6.19 Localizing change of variables for dMRTL+(ct;e) 3156.20 Discretization of MRTL_ (a; e) 3186.21 Bibliographical remarks 320

7 Relativistic Volterra Lattice7.1 Introduction 3217.2 Quadratic invariant Poisson bracket of RVL(a) 3227.3 Cubic invariant Poisson bracket of RVL(a) 3247.4 Auto-transformation of RVL(a) 3257.5 The first Lax representation of RVL(ct) 3267.6 Quadratic r-matrix for the first Lax representation 3287.7 The second Lax representation of RVL(a) 3317.8 The third Lax representation of RVL(Q) 3337.9 Quadratic r-matrix for third Lax representation 3357.10 Discretization of RVL+(a) 3367.11 Localizing change of variables for dRVL+(a) 3377.12 Discretization of RVL_ (a) 3407.13 Localizing change of variables for dRVL_(a) 341

Contents ix

7.14 Modified relativistic Volterra lattice 3437.15 Discretization of MRVL+(a; e) 3477.16 Appendix: selected results for Mi-version of RVL(a) 349

7.16.1 The flow RVL+ (a) and its discretization 3497.16.2 The flow RVL_ (a) and its discretization 3517.16.3 The flow MRVL+(a; e) and its discretization 352

7.17 Bibliographical remarks 353

8 Newtonian Equations of the Relativistic Toda Type8.1 Introduction 3558.2 Parametrizing the linear-quadratic bracket 361

8.2.1 Systems RTL+ (a), dRTL+(a) 3628.2.2 Systems R T L _ ( Q ) , dRTL_ (a) 365

8.3 Parametrizing the linear bracket 3688.3.1 SystemsRTL+(a),dRTL+(a) 3698.3.2 Systems RTL_(c*),dRTL_ (a) 371

8.4 Dual linear parametrization 3748.4.1 SystemsRTL+(a),dRTL+(a) 3758.4.2 Systems RTL_ (a), dRTL_(a) 377

8.5 Parametrizing the quadratic bracket 3808.5.1 SystemsRVL+(a),dRVL+(Q:) 3818.5.2 Systems RVL_ (a), dRVL-(a) 384

8.6 Parametrizing the linear-quadratic bracket II 3878.6.1 SystemsMRTL+(a;e),dMRTL+(a;e) 3888.6.2 Systems MRTL_(a;e),dMRTL_ (a; e) 391

8.7 Parametrizing the cubic-quadratic bracket 3948.8 Parametrizing the cubic bracket I 3998.9 Parametrizing the cubic bracket II 4018.10 Parametrizing the cubic bracket III 4048.11 Bibliographical remarks 407

9 Explicit Discretizations for Toda Systems9.1 Introduction 4099.2 Explicit discretization for TL 4119.3 Explicit discretization for MTL(e) 4139.4 Explicit discretization for VL 4169.5 Explicit discretization for MVL(e) 4189.6 Explicit dRTL+(/i) from implicit dTL 4209.7 Explicit dRVL+(ft) from implicit dVL 4239.8 Bibliographical remarks 427

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10 Explicit Discretizations of Newtonian Toda Systems10.1 Introduction 42910.2 Parametrizing special linear-quadratic bracket 43110.3 Parametrizing the linear bracket 43310.4 Dual parametrization of the linear bracket 43410.5 Parametrizing the quadratic bracket 43510.6 Parametrizing general linear-quadratic bracket 43610.7 Parametrizing the cubic-quadratic bracket 43810.8 Parametrizing the cubic bracket. I 43910.9 Parametrizingthecubicbracket.il 44010.10Parametrizingthecubicbracket.III 44210.11 Bibliographical remarks 443

11 Bruschi-Ragnisco Lattice11.1 Introduction 44511.2 Bi-Hamiltonian structure 44611.3 General construction 44711.4 Orbit interpretation 44811.5 Discretization 45111.6 Newtonian equations of motion 45211.7 Bibliographical remarks 454

12 Multi-field Toda-like Systems12.1 Introduction 45512.2 Multi-field analog of the Toda lattice 45612.3 Linear r-matrix structure for T L m + ] 45812.4 Quadratic r-matrix structure for T L m + i 45912.5 Discretization of T L m + i 46212.6 Localizing change of variables for dTL m + 1 46512.7 Example: TL3 46712.8 Multi-field analog of the modified Toda lattice 46912.9 Quadratic r-matrix structure for MTLm+1(e) 47112.10 Discretization of MTLm + ](e) 47512.11 Localizing change of variables for dMTLm+i(e) 47812.12 Bibliographical remarks 481

13 Multi-field Relativistic Toda Systems13.1 Introduction 48313.2 Multi-field analog of RTL: first version 48513.3 Linear r-matrix structure for RTL m + i (a ) 48613.4 Quadratic bracket for RTLTO+] (a) 48913.5 Introducing the gauge transformed hierarchy 49013.6 Multi-field RTL: second version 49313.7 Quadratic r-matrix structure for RTLTO+i (a) 49513.8 Example: RTL3(a) 499

Contents xi

13.9 Discretization of R T L ^ I J (a) 500

13.10 Localizing change of variables for d R T L ^ j ^ a ) 503

13.11 Local discretization for R T L I " ^ (a) 50513.12 Bibliographical remarks 506

14 Belov-Chaltikian Lattices14.1 Introduction 50714.2 Bi-Hamiltonian structure and Lax representation 50814.3 Discretization of BCLm 51014.4 Modified BCLm 51314.5 Discretization of MBCLm(e) 51514.6 Localizing change of variables for dMBCLTO(e) 51814.7 Relativistic deformation of BCLm 52014.8 A gauge connection between MBCLm(e) and RBCLm(a) 52214.9 Discretization of RBCLm (a) 52614.10 Example: Volterra lattice as BCLi 52814.11 Example: BCL2 52914.12 Bibliographical remarks 532

15 Multi-field Volterra-like Systems15.1 Introduction 53315.2 Multi-field analog of the Volterra lattice 53415.3 Quadratic r-matrix structure for VLm 53615.4 Discretization of VLm 53815.5 Localizing change of variables for dVLTO 54215.6 Example 1: VL.i, three-field analog of Volterra lattice 54315.7 A further generalization of VLm 54515.8 Quadratic r-matrix structure for VLm(<x) 54715.9 Discretization of VLm(<r) 54915.10 Localizing change of variables for dVLm(<r) 55115.11 The case of the signature o = ( + 1 , - 1 , . . . , -1 ) 553

15.11.1 Lax representation and Hamiltonian structure 55315.11.2 Discretization 55415.11.3 Localizing change of variables 55715.11.4 Miura relation to the Belov-Chaltikian lattices 558

15.12 Example 2: a = ( + 1 , - 1 , - 1 ) 55915.13 Example 3: a = (+1, + 1 , - 1 ) 56215.14 Bibliographical remarks 565

16 Multi-field Relativistic Volterra Systems16.1 Introduction 56716.2 The RVLm(<r; a) hierarchy: first construction 57016.3 Introducing the gauge transformed hierarchy 57416.4 The RVLTO(o-; a) hierarchy: second construction 57816.5 Discretization of R V L ^ V ; a) 582

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16.6 Localizing change of variables for dRVL^(cr; a) 58416.7 Particular case h = a: explicit discretizations 58616.8 The case of the signature a = ( + 1 , + 1 , . . . , + 1 ) 588

16.8.1 Equations of motion and Hamiltonian structure 58816.8.2 Discretization 58916.8.3 Explicit discretization 59016.8.4 Example: RVL^+) (a), the three-field analog

of the relativistic Volterra lattice 59116.9 The case of the signature a = ( + 1 , - 1 , . . . , - 1 ) 592

16.9.1 Equations of motion and Hamiltonian structure 59216.9.2 Discretization 59416.9.3 Example: RVL4+)(cr;a) wither = ( + 1 , - 1 , - 1 ) 595

16.10 Explicit dVLm from the RVLm hierarchy 59716.11 Bibliographical remarks 603

17 Bogoyavlensky Lattices17.1 Introduction 60517.2 Lax representations 60617.3 Quadratic r-matrix structure of BLl(m) 60817.4 Quadratic r-matrix structure of BL2(p) and BL3(p) 60917.5 Examples of Hamiltonian structures 613

17.5.1 Lattice BL2(p),p> 1 61417.5.2 Lattice BL3(p),p> 2 615

17.6 Discretization of the lattice BLl(m) 61617.7 Discretization of the lattice BL2(p) 62117.8 Discretization of the lattice BL3(p) 62617.9 Modified Volterra lattice 63117.10 Alternative approach to BLl(m) 63417.11 Alternative approach to BL2(p) 63617.12 Alternative approach to BL3(p) 63817.13 Bibliographical remarks 640

18 Ablowitz-Ladik Hierarchy18.1 Introduction 64318.2 AKNS hierarchy 64518.3 Ablowitz-Ladik hierarchy 64818.4 Non-local difference schemes 654

18.4.1 Difference schemes for NLS 65418.4.2 Difference schemes for MKdV 657

18.5 Elementary flows of the AL hierarchy 65818.6 Local discretizations for T±i 66018.7 Symplectic properties 66418.8 Local discretizations for NLS 66518.9 Local discretizations for T±i 670

Contents xiii

18.10 Local discretizations for MKdV 67518.11 Connection with relativistic Toda lattice 67818.12 Bibliographical remarks 685

III Systems of Classical Mechanics19 Peakons System

19.1 Introduction 68919.2 Lax representation and r-matrix 69019.3 Discretization 69319.4 Lagrangian interpretation 69719.5 Bibliographical remarks 699

20 Standard-like Discretizations20.1 Introduction 70120.2 Integrable scalar equations: Examples 70220.3 Integrable scalar equations: Classification 70520.4 Bibliographical remarks 709

21 Lie-algebraic Toda Systems21.1 Introduction 71121.2 Lie-algebraic open-end Toda lattices 71421.3 Lie-algebraic periodic Toda lattices 72021.4 Toda lattices AN_X andA<^)_1 724

21.5 Discrete time lattices ^TV-I and AJY^J 72721.6 List of generalized Toda lattices 73221.7 Discretization of lattices BN,CN, C^\ A{^, and D^+1 741

21.8 Discretizationoflattices£» iv,£ )Jv )"BN ) 'and^2iv-i 7 4 6

21.9 2 x 2 Lax representations: continuous time case 75221.10 2 x 2 Lax representations: discrete time case 75621.11 Lattice G? as a reduction of the lattice B3 76021.12 Lattice G^ as a reduction of the lattice B^ 76421.13 Toda lattice D^ , continuous and discrete 76921.14 Bibliographical remarks 775

22 Gamier System22.1 Introduction 77722.2 Gamier system 779

22.2.1 Equations of motion and Hamiltonian structure 77922.2.2 Integrals of motion 77922.2.3 "Big" Lax representation 78122.2.4 "Small" Lax representation 783

22.3 Anharmonic oscillator 78522.3.1 Equations of motion and Hamiltonian structure 78522.3.2 Integrals of motion 785

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22.3.3 "Big" Lax representation 78622.3.4 "Small" Lax representation 787

22.4 Wojciechowski system 78822.4.1 Equations of motion and Hamiltonian structure 78822.4.2 Integrals of motion 79022.4.3 "Big" Lax representation 79022.4.4 "Small" Lax representation 792

22.5 Backlund transformation for the Gamier system 79222.6 Backlund transformation for anharmonic oscillator 79522.7 Backlund transformation for Wojciechowski system 79622.8 Explicit discretization of the Gamier system 799

22.8.1 Equations of motion and symplectic properties 79922.8.2 Integrals of motion 80022.8.3 "Big" Lax representation 80122.8.4 "Small" Lax representation 805

22.9 Explicit discretization of anharmonic oscillator 80622.9.1 Equations of motion and symplectic properties 80622.9.2 Integrals of motion 80722.9.3 "Big" Lax representation 80722.9.4 "Small" Lax representation 810

22.10 Explicit discretization of Wojciechowski system 81022.10.1 Equations of motion and symplectic properties 81022.10.2 Integrals of motion 81222.10.3 "Big" Lax representation 81322.10.4 "Small" Lax representation 814

22.11 Bibliographical remarks 815

23 Henon-Heiles System23.1 Introduction 81723.2 Lax representation 81823.3 Discretization of Henon-Heiles system 82023.4 Bibliographical remarks 822

24 Neumann System24.1 Introduction 82324.2 Double Neumann system 826

24.2.1 Equations of motion and Hamiltonian structure 82624.2.2 "Big" Lax representation 82824.2.3 "Small" Lax representation 83124.2.4 Unconstrained version 833

24.3 Neumann system 83524.3.1 Equations of motion and Hamiltonian structure 83524.3.2 "Big" Lax representation 83724.3.3 "Small" Lax representation 83924.3.4 Unconstrained version 840

Contents xv

24.4 Rosochatius system 84124.4.1 Equations of motion and Hamiltonian structure 84124.4.2 "Big" Lax representation 84424.4.3 "Small" Lax representation 84524.4.4 Unconstrained version 846

24.5 Backlund transformation for double Neumann system 84724.6 Backlund transformation for Neumann system 85324.7 Backlund transformation for Rosochatius system 85624.8 Ragnisco's discretization of Neumann system 85924.9 V. Adler's discretization of Neumann system 86224.10 Coupled Neumann system 86624.11 Discretizations of the coupled Neumann system 869

24.11.1 Discretization a la Ragnisco 86924.11.2DiscretizationalaV. Adler 871

24.12 Bibliographical remarks 874

25 Lie-algebraic Generalizations of the Gamier Systems25.1 Introduction 87725.2 Gamier systems related to symmetric spaces 87825.3 Gamier systems related to AIII 883

25.3.1 Equations of motion and Lax representation 88325.3.2 Example: M = 2 88525.3.3 Discretizations 88625.3.4 Example: discrete systems with M = 2 889

25.4 Gamier systems related to CI and Dili 89025.5 Gamier systems related to BDI 891

25.5.1 Equations of motion and Lax representation 89125.5.2 Discretization 893

25.6 Bibliographical remarks 896

26 Integrable Cases of Rigid Body Dynamics26.1 Introduction 89726.2 Multi-dimensional Euler top 90126.3 Discrete time Euler top 90526.4 Rigid body in a quadratic potential 90926.5 Discrete time top in a quadratic potential 91226.6 Multi-dimensional Lagrange top 916

26.6.1 Body frame formulation 91626.6.2 Rest frame formulation 919

26.7 Discrete time analog of the Lagrange top 92026.7.1 Rest frame formulation 92026.7.2 Moving frame formulation 924

26.8 Three-dimensional Lagrange top 92626.9 Discrete time three-dimensional Lagrange top 93126.10 Rigid body motion in an ideal fluid: Clebsch case 936

xvi Contents

26.11 Discretization of the Clebsch problem 93826.11.1 Case A = B2 of the Clebsch problem 93826.11.2 Case A = B of the Clebsch problem 939

A Appendix: Lagrange top and Heisenberg magnetic 94126.12 Bibliographical remarks 943

27 Systems of Calogero-Moser Type27.1 Introduction 94727.2 Lax representations: rational and hyperbolic cases 95127.3 Dynamical ^-matrix formulation 95327.4 Explicit solutions 957

27.4.1 Rational systems 95727.4.2 Hyperbolic systems 959

27.5 Discrete time evolution: rational systems 96127.5.1 Rational CM system 96227.5.2 Rational RS system 964

27.6 Discrete time evolution: hyperbolic systems 96727.6.1 Hyperbolic CM system 96827.6.2 Hyperbolic RS systems 971

27.7 Elliptic CM type models: Lax representations 97427.8 Elliptic CM type models: r-matrix structure 97627.9 Discretization of elliptic CM and RS models 98027.10 Strong coupling limit of RS models 984

27.10.1 Rational system 98627.10.2 Hyperbolic system 989

27.11 Bibliographical remarks 993

Bibliography 1001

List of Notations 1043

Index 1061