B. Grammaticos Y. Kosmann-Schwarzbach T. Tamizhmani (Eds.) D iscrete Integrable Systems 13 Editors Basil Grammat icos GMPIB, Universit ´ e Paris VII Tour 24-14, 5 e étage, case 7021 2 place Jussieu 75251 Paris Cedex 05, France Yvette Kosmann-Schwarzbach Centre de Mathématiques École Polytechnique 91128 Palaiseau, France Thamizharasi Tamizhmani Department of Mathematics Kanchi Mamunivar Centre for Postgraduate Studies Pondicherry, India B. Grammaticos, Y. Kosmann-Schwarzbach, T. Tamizhmani (Eds.), Discrete Integrable Sys- tems,Lect.NotesPhys.644 (Springer, Berlin Heidelberg 2004), DOI 10.1007/b94662 Library of Congress Control Number: 2004102969 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Biblio- thek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de> ISSN 0075-8450 ISBN 3-540- Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. 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LNP Homepage (springerlink.com) On the LNP homepage you will find: −The LNP online archive. It contains the full texts (PDF) of all volumes published since 2000. Abstracts, table of contents and prefaces are accessible free of charge to everyone. Information about the availability of printed volumes can be obtained. −The subscription information. T he online archive is free of charge to all subscribers of the printed volumes. −The editorial contacts, with respect to both scientific and technical matters. −Theauthor’s/editor’sinstructions. 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´eTest 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 Pl¨ucker 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 R N 121 8 Lagrangian Mechanics on T P 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 = B 2 173 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-B¨acklund 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) M Crystal Lattice 417 6.4 PBBS as A (1) N −1 Crystal Chains 419 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. Kruskal 1 , B. Grammaticos 2 , and T. Tamizhmani 3 1 Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA, kruskal@math.rutgers.edu 2 GMPIB, Universit´e Paris VII, Tour 24-14, 5 e ´etage, case 7021, 75251 Paris, France, grammati@paris7.jussieu.fr 3 Department of Mathematics, Kanchi Mamunivar Centre for Postgraduate Studies, Pondicherry 605008, India, arasi55@yahoo.com Abstract. While this school focuses on discrete integrable systems we feel it nec- essary, if only for reasons of comparison, to go back to fundamentals and introduce 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 we will focus on simpler systems and consider only ordinary differential equa- tions. Six such equations play a fundamental role in integrability theory, the six Painlev´e equations [1]: x =6x 2 + t P I x =2x 3 + tx + a P II x = x 2 x − x t + 1 t (ax 2 + b)+cx 3 + d x P III x = x 2 2x + 3x 3 2 +4tx 2 +2(t 2 − a)x − b 2 2x P IV x = x 2 1 2x + 1 x − 1 − x t + (x − 1) 2 t 2 ax + b x + c x t + dx(x +1) x − 1 P V x = x 2 2 1 x + 1 x − 1 + 1 x − t − x 1 t + 1 t − 1 + 1 x − t + x(x − 1)(x −t) 2t 2 (t − 1) 2 a − bt x 2 + c t − 1 (x − 1) 2 + (d − 1)t(t −1) (x − t) 2 P VI Here the dependent variable x is a function of the independent variable t, while a, b, c, and d are parameters (constants). These are second order equa- tions in normal form (solved for x ), rational in x and x. M.D. Kruskal, B. Grammaticos, and T. Tamizhmani, Three Lessons on the Painlev´e Property and the Painlev´e Equations, Lect. Notes Phys. 644, 1–15 (2004) http://www.springerlink.com/ c Springer-Verlag Berlin Heidelberg 2004 [...]... understanding of integrable systems with infinitely many degrees of freedom Starting with an elementary introduction to Sato theory, followed by an expos´ of its interpretation in terms of infinite-dimensional Clifford e algebras and their representations, the scope of the theory is gradually extended to include multi-component systems, integrable lattice equations and fully discrete systems Special emphasis... theory [36], offers a deep algebraic and geometric understanding of integrable systems with infinitely many degrees of freedom and their solutions At the heart of the theory lies the idea that integrable systems are not isolated but should be thought of as belonging to infinite families, so-called hierarchies of mutually compatible systems, i.e., systems governed by an infinite set of evolution parameters in... proof thus appeared highly desirable 12 M.D Kruskal, B Grammaticos, T Tamizhmani In a series of papers [8,9] one of the authors (MDK) together with various collaborators has proposed a straightforward proof of the Painlev´ property e that can be applied to all six equations The latest version of this proof is that obtained in collaboration with K.M Tamizhmani [10] In what follows we shall outline this... lattice equations and fully discrete systems Special emphasis is placed on the symmetries of the integrable equations described by the theory and especially on the Darboux transformations and elementary B¨cklund transformations for these equations Finally, reductions to lower a dimensional systems and eventually to integrable ordinary differential equations are discussed As an example, the origins of the fourth... 10 M.D Kruskal, K.M Tamizhmani, N Joshi and O Costin, “The Painlev´ prope erty: a simple proof for Painlev´ equation III”, preprint (2004) e 11 M.D Kruskal, Asymptotology, in Mathematical Models in Physical Sciences (University of Notre Dame, 1962), S Drobot and P.A Viebrock, eds., PrenticeHall 1963, pp 17-48 Sato Theory and Transformation Groups A Unified Approach to Integrable Systems Ralph Willox1,2... which can be derived from W satisfy the linear systems, LΨ (t, λ) = λΨ (t, λ) , ∂tn Ψ (t, λ) = Bn Ψ (t, λ) L∗ Ψ ∗ (t, λ) = λΨ ∗ (t, λ) , ∗ ∂tn Ψ ∗ (t, λ) = −Bn Ψ ∗ (t, λ) (48) as well as the bilinear identity, Resλ=∞ [Ψ (t, λ)Ψ ∗ (t , λ)] = 0 In (49), Res denotes the operation A(λ) := i ∀t, t (49) ai λi , Resλ=∞ [A(λ)] := a−1 The Sato Approach to Integrable Systems 27 Several remarks are in order First...2 M.D Kruskal, B Grammaticos, T Tamizhmani These may look like more or less random equations, but that is not the case Apart from some simple transformations they cannot have a form other than shown above They are very special... we choose to work with the case of u small, but u large is entirely similar, mut mut To solve the equation by iteration, we note that the contribution of 1/u2 is more important 14 M.D Kruskal, B Grammaticos, T Tamizhmani than that of u2 The integral of z/u2 may be an important contribution but since it is taken over a short path it is much smaller than the z 2 /u2 term outside the integral in (4.2)... Ramani and B Grammaticos, NATO ASI Series C 310, Kluwer 1989, p 321 3 S Kovalevskaya, Acta Math 12 (1889) 177 4 M.J Ablowitz, A Ramani and H Segur, Lett Nuov Cim 23 (1978) 333 5 M.D Kruskal, NATO ASI B278, Plenum 1992, p 187 6 M.D Kruskal and P.A Clarkson, Stud Appl Math 86 (1992) 87 7 P Painlev´, Acta Math 25 (1902) 1 e 8 N Joshi and M.D Kruskal, in “Nonlinear evolution equations and dynamical systems ... other than poles (Poles can be viewed as nonsingular values of ∞ on the “complex sphere,” the compact closure of the complex plane obtained by adjoining the point at infinity.) 4 M.D Kruskal, B Grammaticos, T Tamizhmani Fixed singularities do not pose a major problem Linear equations can have only the singularities of their coefficients and thus these singularities are fixed The case of fixed singularities . gravity, (A, B, C) the moments of inertia, (x 0 ,y 0 ,z 0 ) the centre of mass of the system, M the mass of the top, and g the acceleration of gravity. Complete integrability of the system requires. property. Since t 0 is an arbitrary point this means that f (t) =0andf must be linear. (We can take f (t) =at + b but it is then straightforward to transform it to just f (t) =t. ) So the only equation of. Postgraduate Studies Pondicherry, India B. Grammaticos, Y. Kosmann-Schwarzbach, T. Tamizhmani (Eds. ), Discrete Integrable Sys- tems,Lect.NotesPhys.644 (Springer, Berlin Heidelberg 2004 ), DOI 10.1007 /b9 4662 Library