Discrete integrable systems

443 Pages·2004·2.792 MB·English

by  M.D. Kruskal, B. Grammaticos, T. Tamizhmani (auth.), Basil Grammaticos, Thamizharasi Tamizhmani, Yvette Kosmann-Schwarzbach (eds.)

B. Grammaticos Y. Kosmann-Schwarzbach
T. Tamizhmani (Eds.)
Discrete Integrable Systems
1 3
Editors
BasilGrammaticos ThamizharasiTamizhmani
GMPIB,Universite´ParisVII DepartmentofMathematics
e
Tour24-14,5 étage,case7021 KanchiMamunivarCentre
2placeJussieu forPostgraduateStudies
75251ParisCedex05,France Pondicherry,India
YvetteKosmann-Schwarzbach
CentredeMathématiques
ÉcolePolytechnique
91128Palaiseau,France
B.Grammaticos,Y.Kosmann-Schwarzbach,T.Tamizhmani(Eds.),DiscreteIntegrableSys-
tems,Lect.NotesPhys.644(Springer,BerlinHeidelberg2004),DOI10.1007/b94662
LibraryofCongressControlNumber:2004102969
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Table of Contents
Three Lessons on the Painlev´e Property
and the Painlev´e Equations
M.D. Kruskal, B. Grammaticos, T. Tamizhmani................... 1
1 Introduction .............................................. 1
2 The Painlev´e Property and the Naive Painlev´e Test............ 2
3 From the Naive to the Poly-Painlev´e Test .................... 7
4 The Painlev´e Property for the Painlev´e Equations ............. 11
Sato Theory and Transformation Groups.
A Unified Approach to Integrable Systems
R. Willox, J. Satsuma .......................................... 17
1 The Universal Grassmann Manifold.......................... 17
1.1 The KP Equation .................................... 18
1.2 Plu¨cker Relations..................................... 20
1.3 The KP Equation as a Dynamical System
on a Grassmannian ................................... 22
1.4 Generalization to the KP Hierarchy..................... 23
2 Wave Functions, τ-Functions and the Bilinear Identity ......... 24
2.1 Pseudo-differential Operators .......................... 24
2.2 The Sato Equation and the Bilinear Identity............. 25
2.3 τ-Functions and the Bilinear Identity ................... 28
3 Transformation Groups .................................... 31
3.1 The Boson-Fermion Correspondence .................... 31
3.2 Transformation Groups and τ-Functions................. 34
3.3 B¨acklund Transformations for the KP Hierarchy.......... 36
4 Extensions and Reductions ................................. 41
4.1 Extensions of the KP Hierarchy ........................ 42
4.2 Reductions of the KP Hierarchy........................ 46
Special Solutions of Discrete Integrable Systems
Y. Ohta ...................................................... 57
1 Introduction .............................................. 57
2 Determinant and Pfaffian................................... 58
2.1 Definition ........................................... 58
2.2 Linearity and Alternativity ............................ 62
XII Table of Contents
2.3 Cofactor and Expansion Formula....................... 71
2.4 Algebraic Identities................................... 72
2.5 Golden Theorem ..................................... 74
2.6 Differential Formula .................................. 76
3 Difference Formulas........................................ 77
3.1 Discrete Wronski Pfaffians............................. 77
3.2 Discrete Gram Pfaffians ............................... 78
4 Discrete Bilinear Equations................................. 80
4.1 Discrete Wronski Pfaffian.............................. 80
4.2 Discrete Gram Pfaffian................................ 80
5 Concluding Remarks....................................... 81
Discrete Differential Geometry. Integrability as Consistency
A.I. Bobenko.................................................. 85
1 Introduction .............................................. 85
2 Origin and Motivation: Differential Geometry ................. 85
3 Equations on Quad-Graphs. Integrability as Consistency ....... 88
3.1 Discrete Flat Connections on Graphs ................... 89
3.2 Quad-Graphs ........................................ 90
3.3 3D-Consistency ...................................... 92
3.4 Zero-Curvature Representation from the 3D-Consistency .. 94
4 Classification ............................................. 96
5 Generalizations: Multidimensional
and Non-commutative (Quantum) Cases ..................... 100
5.1 Yang-Baxter Maps ................................... 100
5.2 Four-Dimensional Consistency
of Three-Dimensional Systems ......................... 101
5.3 Noncommutative (Quantum) Cases ..................... 103
6 Smooth Theory from the Discrete One ....................... 105
Discrete Lagrangian Models
Yu.B. Suris ................................................... 111
1 Introduction .............................................. 111
2 Poisson Brackets and Hamiltonian Flows ..................... 112
3 Symplectic Manifolds ...................................... 115
4 Poisson Reduction......................................... 118
5 Complete Integrability ..................................... 118
6 Lax Representations ....................................... 119
7 Lagrangian Mechanics on RN ............................... 121
8 Lagrangian Mechanics on TP and on P ×P .................. 123
9 Lagrangian Mechanics on Lie Groups ........................ 125
10 Invariant Lagrangians and the Lie–Poisson Bracket ............ 128
10.1 Continuous–Time Case................................ 129
10.2 Discrete–Time Case .................................. 131
Table of Contents XIII
11 Lagrangian Reduction and Euler–Poincar´e
Equations on Semidirect Products ........................... 134
11.1 Continuous–Time Case................................ 135
11.2 Discrete–Time Case .................................. 138
12 Neumann System.......................................... 141
12.1 Continuous–Time Dynamics ........................... 141
12.2 B¨acklund Transformation for the Neumann System ....... 144
12.3 Ragnisco’s Discretization of the Neumann System ........ 147
12.4 Adler’s Discretization of the Neumann System ........... 149
13 Garnier System ........................................... 150
13.1 Continuous–Time Dynamics ........................... 150
13.2 B¨acklund Transformation for the Garnier System......... 151
13.3 Explicit Discretization of the Garnier System ............ 152
14 Multi–dimensional Euler Top ............................... 153
14.1 Continuous–Time Dynamics ........................... 153
14.2 Discrete–Time Euler Top.............................. 156
15 Rigid Body in a Quadratic Potential......................... 159
15.1 Continuous–Time Dynamics ........................... 159
15.2 Discrete–Time Top in a Quadratic Potential ............. 161
16 Multi–dimensional Lagrange Top ............................ 164
16.1 Body Frame Formulation.............................. 164
16.2 Rest Frame Formulation............................... 166
16.3 Discrete–Time Analogue of the Lagrange Top:
Rest Frame Formulation............................... 168
16.4 Discrete–Time Analogue of the Lagrange Top:
Moving Frame Formulation ............................ 169
17 Rigid Body Motion in an Ideal Fluid:
The Clebsch Case ......................................... 171
17.1 Continuous–Time Dynamics ........................... 171
17.2 Discretization of the Clebsch Problem, Case A=B2 ...... 173
17.3 Discretization of the Clebsch Problem, Case A=B ....... 174
18 Systems of the Toda Type.................................. 175
18.1 Toda Type System ................................... 175
18.2 Relativistic Toda Type System......................... 177
19 Bibliographical Remarks ................................... 179
Symmetries of Discrete Systems
P. Winternitz ................................................. 185
1 Introduction .............................................. 185
1.1 Symmetries of Differential Equations.................... 185
1.2 Comments on Symmetries of Difference Equations ........ 191
2 Ordinary Difference Schemes and Their Point Symmetries ...... 192
2.1 Ordinary Difference Schemes........................... 192
2.2 Point Symmetries of Ordinary Difference Schemes ........ 194
2.3 Examples of Symmetry Algebras of O∆S ................ 199
XIV Table of Contents
3 Lie Point Symmetries of Partial Difference Schemes............ 203
3.1 Partial Difference Schemes............................. 203
3.2 Symmetries of Partial Difference Schemes ............... 206
3.3 The Discrete Heat Equation ........................... 208
3.4 Lorentz Invariant Difference Schemes ................... 211
4 Symmetries of Discrete Dynamical Systems ................... 213
4.1 General Formalism ................................... 213
4.2 One-Dimensional Symmetry Algebras ................... 217
4.3 Abelian Lie Algebras of Dimension N ≥2 ............... 218
4.4 Some Results on the Structure of Lie Algebras ........... 220
4.5 Nilpotent Non-Abelian Symmetry Algebras .............. 222
4.6 Solvable Symmetry Algebras
with Non-Abelian Nilradicals .......................... 222
4.7 Solvable Symmetry Algebras with Abelian Nilradicals..... 224
4.8 Nonsolvable Symmetry Algebras ....................... 224
4.9 Final Comments on the Classification ................... 225
5 Generalized Point Symmetries
of Linear and Linearizable Systems .......................... 225
5.1 Umbral Calculus ..................................... 225
5.2 Umbral Calculus and Linear Difference Equations ........ 227
5.3 Symmetries of Linear Umbral Equations................. 232
5.4 The Discrete Heat Equation ........................... 234
5.5 The Discrete Burgers Equation and Its Symmetries ....... 235
Discrete Painlev´e Equations: A Review
B. Grammaticos, A. Ramani .................................... 245
1 The (Incomplete) History of Discrete Painlev´e Equations ....... 247
2 Detectors, Predictors, and Prognosticators (of Integrability) .... 253
3 Discrete P’s Galore ........................................ 262
4 Introducing Some Order into the d-P Chaos .................. 268
5 What Makes Discrete Painlev´e Equations Special? ............ 274
6 Putting Some Real Order to the d-P Chaos ................... 282
7 More Nice Results on d-P’s ................................. 300
8 Epilogue ................................................. 317
Special Solutions for Discrete Painlev´e Equations
K.M. Tamizhmani, T. Tamizhmani, B. Grammaticos, A. Ramani.... 323
1 What Is a Discrete Painlev´e Equation?....................... 324
2 Finding Special-Function Solutions .......................... 328
2.1 The Continuous Painlev´e Equations
and Their Special Solutions............................ 328
2.2 Special Function Solutions
for Symmetric Discrete Painlev´e Equations .............. 332
2.3 The Case of Asymmetric Discrete Painlev´e Equations ..... 338
Table of Contents XV
3 Solutions by Direct Linearisation ............................ 345
3.1 Continuous Painlev´e Equations......................... 346
3.2 Symmetric Discrete Painlev´e Equations ................. 349
3.3 Asymmetric Discrete Painlev´e Equations ................ 356
3.4 Other Types of Solutions for d-P’s...................... 365
4 From Elementary to Higher-Order Solutions .................. 366
4.1 Auto-Ba¨cklund and Schlesinger Transformations.......... 366
4.2 The Bilinear Formalism for d-Ps........................ 368
4.3 The Casorati Determinant Solutions .................... 370
5 Bonus Track: Special Solutions
of Ultra-discrete Painlev´e Equations ......................... 377
Ultradiscrete Systems (Cellular Automata)
T. Tokihiro.................................................... 383
1 Introduction .............................................. 383
2 Box-Ball System .......................................... 385
3 Ultradiscretization......................................... 386
3.1 BBS as an Ultradiscrete Limit
of the Discrete KP Equation........................... 386
3.2 BBS as Ultradiscrete Limit
of the Discrete Toda Equation ......................... 391
4 Generalization of BBS ..................................... 395
4.1 BBS Scattering Rule and Yang-Baxter Relation .......... 395
4.2 Extensions of BBSs
and Non-autonomous Discrete KP Equation ............. 399
5 From Integrable Lattice Model to BBS ....................... 405
5.1 Two-Dimensional Integrable Lattice Models
and R-Matrices ...................................... 405
5.2 Crystallization and BBS............................... 408
6 Periodic BBS (PBBS)...................................... 412
6.1 Boolean Formulae for PBBS ........................... 414
6.2 PBBS and Numerical Algorithm ....................... 415
6.3 PBBS as Periodic A(1) Crystal Lattice .................. 417
M
6.4 PBBS as A(1) Crystal Chains......................... 419
N−1
6.5 Fundamental Cycle of PBBS........................... 421
7 Concluding Remarks....................................... 423
Time in Science: Reversibility vs. Irreversibility
Y. Pomeau.................................................... 425
1 Introduction .............................................. 425
2 On the Phenomenon of Irreversibility in Physical Systems ...... 426
3 Reversibility of Random Signals............................. 429
4 Conclusion and Perspectives ................................ 435
Index........................................................ 437
Three Lessons on the Painlev´e Property
and the Painlev´e Equations
M.D. Kruskal1, B. Grammaticos2, and T. Tamizhmani3
1 Department of Mathematics, Rutgers University, New Brunswick, NJ 08903,
USA, [email protected]
2 GMPIB, Universit´e Paris VII, Tour 24-14, 5e´etage, case 7021, 75251 Paris,
France, [email protected]
3 Department of Mathematics, Kanchi Mamunivar Centre for Postgraduate
Studies, Pondicherry 605008, India, [email protected]
Abstract. While this school focuses on discrete integrable systems we feel it nec-
essary,ifonlyforreasonsofcomparison,togobacktofundamentalsandintroduce
the basic notion of the Painlev´e property for continuous systems together with a
critical analysis of what is called the Painlev´e test. The extension of the latter to
what is called the poly-Painlev´e test is also introduced. Finally we devote a lesson
to the proof that the Painlev´e equations do have the Painlev´e property.
1 Introduction
A course on integrability often starts with introducing the notion of soliton
and how the latter emerges in integrable partial differential equations. Here
wewillfocusonsimplersystemsandconsideronlyordinarydifferentialequa-
tions. Six such equations play a fundamental role in integrability theory, the
six Painlev´e equations [1]:
x(cid:1)(cid:1) =6x2+t P
I
x(cid:1)(cid:1) =2x3+tx+a P
II
x(cid:1)2 x(cid:1) 1 d
x(cid:1)(cid:1) = − + (ax2+b)+cx3+ P
x t t x III
x(cid:1)2 3x3 b2
x(cid:1)(cid:1) = + +4tx2+2(t2−a)x− P
2x 2 2x IV
(cid:1) (cid:2) (cid:1) (cid:2)
x(cid:1)(cid:1) =x(cid:1)2 1 + 1 − x(cid:1) + (x−1)2 ax+ b +cx + dx(x+1) P
x(cid:1)2(cid:1)1 2x 1x−1 1 t(cid:2) (cid:1)t12 1 x 1 (cid:2)t x−1 V
x(cid:1)(cid:1) = + + −x(cid:1) + +
2 x x−1 x−t t t−1 x−t
(cid:3) (cid:4)
x(x−1)(x−t) bt t−1 (d−1)t(t−1)
+ a− +c + P
2t2(t−1)2 x2 (x−1)2 (x−t)2 VI
Here the dependent variable x is a function of the independent variable t,
whilea,b,c,anddareparameters(constants).Thesearesecondorderequa-
tions in normal form (solved for x(cid:1)(cid:1)), rational in x(cid:1) and x.
M.D. Kruskal, B. Grammaticos, and T. Tamizhmani, Three Lessons on the Painlev´e Property
andthePainlev´eEquations,Lect.NotesPhys.644,1–15(2004)
http://www.springerlink.com/ (cid:2)c Springer-VerlagBerlinHeidelberg2004