CRM Series in Mathematical Physics

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CRM Series in Mathematical Physics

Conte, The Painleve Property: One Century Later MacKenzie, Paranjape, and Zakrzewski, Solitons: Properties, Dynamics,

Interactions, Applications Semenoff and Vinet, Particles and Fields

Robert Conte Editor

The Painleve Property

One Century Later

With 11 Illustrations

i Springer

Robert Conte Service de Physique de l'etat Condense CEA SacIay F91191 Gif-sur-Yvette Cedex France

Editorial Board

Joel S. Feldman Department of Mathematics University of British Columbia Vancouver, BC V6T lZ2 Canada feldman@math.ubc.ca

Yvan Saint-Aubin Departement de Mathematiques

et Statistique Universite de Montreal c.P. 6128, Succursale Centre-ville Montreal, Quebec H3C 317 Canada saint@dms.umontreal.ca

Duong H. Phong Department of Mathematics Columbia University New York, NY 10027-0029 USA phong@math.columbia.edu

Luc Vinet McGill University James Administration Building, Room 504 Montreal, Quebec H3A 2T5 Canada vinet@vpa.mcgill.ca

Library of Congress Cataloging-in-Publication Data The Painleve property: one century later / editor, Robert Conte.

p. cm. - (CRM series in mathematical physics) Includes bibliographical references and index. ISBN 0-387-98888-2 (alk. paper) I. Painleve equations. 2. Mathematical physics. I. Conte,

Robert, 1943- . II. Series: CRM series in mathematical physics. QC20.7.D5P34 1999 530.1 '535---dc21 99-16039

Printed on acid-free paper.

© 1999 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by MaryAnn Cottone; manufacturing supervised by Jeffrey Taub. Photocomposed copy prepared from CRM's LaTeX files.

9 876 5 432 1

ISBN 0-387-98888-2 Springer-Verlag New York Berlin Heidelberg SPIN 10731750

To Florent Bureau (1906-1999)

1 A

lexandr Orlov

10 N

ikolai A. K

udryashov 19

Hiroyuki K

awam

uko 28

Micheline M

usette 2

Monica U

gaglia 11

Milena A

. Khlabystova

20 K

azuo Okam

oto 29

Sim

onetta Abenda

3 T

amara G

rava 12

Brenton L

eMesurier

21 T

hierry Lehner

30 V

angelis Marinakis

4 A

ndrei K. S

vinin 13

Marina C

ooks 22

Martin D

. Kruskal

31 C

hristian Scheen 5

Marta M

azzocco 14

Philippe D

i Francesco

23 V

ladimir V

. Sokolov 32

Maciej D

unajski 6

Serguei A

. Zykov

15 G

ilbert Mahoux

24 A

drian-S C

arstea 33

Jarmo H

ietarinta 7

Vladim

ir V. T

segel'nik 16

Zora T

homova

25 L

uciano Seta

34 X

ing-biao Hu

8 Isidore N

dayirinde 17

Pavel W

internitz 26

Valerii I. G

romak

35 V

aleria Ricci

9 A

nnie Touchant

18 R

obert Conte

27 M

etin Unal

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Xing-biao H

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c. Sinica, Institute of C

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alini Joshi V

. of New

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Hiroyuki K

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Milena A

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Milena@

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Martin D

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Rutgers V

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Hill C

enter, Busch C

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ikolai A. K

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oscow E

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Stephane L

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. de Montreal, C

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hierry Lehner

Ecole poly technique, P

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Shanna L

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CE

A Saclay, S

ervice de physique theorique F

Mahoux@

spht.saclay.cea.fr 91191 G

if/Yvette C

edex V

angelis Marinakis

U. of P

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agelis@m

ath.upatras.gr 26110 P

atras M

arta Afazzocco

Scuola intern. sup. di studi avanzati

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ann.sissa.it V

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rieste V

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R, B

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icheline Musette

Vrije U

. Brussel, T

heoretische Natu

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MM

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Pleinlaan 2-1050 B

russel Isidore N

dayirinde U

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de

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Universiteitsplein 1-B

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Frank W

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GB

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rank@am

sta.leeds.ac.uk L

eeds LS2 9

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Kazuo O

kamoto

U. of T

okyo, Dept. of M

ath. Sciences

JPN

O

kamoto@

Poisson.m

s.u-tokyo.ac.jp 3-8-1 K

omaba, M

eguroku-Tokyo 153-8914

Alexandr O

rlov U

. de Montreal, C

entre de rech. math.

CD

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OrlovS

@w

ave.sio.rssi.ru C

P 6128, succ. C

entre-ville-Montreal H

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Andrew

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U. of K

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Alfred R

amani

Ecole poly technique, C

entre de phys. theor. F

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alaiseau Cedex

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Valeria R

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. di Rom

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Colin R

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. of New

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P. C

hristian Scheen

U. de L

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Constance Schober

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pus box 526-Boulder, C

o 80309 L

uciano Seta

U. di P

alermo, D

ipart. di Matem

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LS

eta@grem

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Via A

rchirafi 34-90123 Palerm

o V

ladimir V

. Sokolov R

ussian Ac. of Sciences, Inst. of M

ath. R

U

Sokolov@

nkc.bashkiria.su C

hernyshevsky ul. 112-Ufa

Andrei K

. Svinin

Inst. of System

Dynam

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RU

Svinin@

k517.icc.ru L

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O B

ox 1233-664033 Irkutsk Z

ora Thom

ova U

. de Montreal, C

entre de rech. math.

CD

N

Thom

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sunyit.edu C

P 6128, Succ. C

entre-ville-Montreal H

3C 3J7

Vladim

ir V. T

segel'nik B

ielorussian State U

., Mathem

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Y

math@

micro.rei.m

insk.by P. B

rovka ul. 6-220027 Minsk

Monica U

gaglia S

cuola intern. sup. di studi avanzati I

U gaglia@

neumann.sissa.it

Via B

eirut 4-34014 Trieste

Metin U

nal U

. of Glasgow

, Dept. of M

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GB

M

etin@m

aths.gla.ac.uk U

niversity gardens-Glasgow

G12 8Q

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Luc V

inet U

. de Montreal, C

entre de rech. math.

CD

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Vinet@

crm.um

ontreal.ca C

P 6128, succ. C

entre-ville-Montreal H

3C 3J7

Pavel W

internitz U

. de Montreal, C

entre de rech. math.

CD

N

Wintern@

crm.um

ontreal.ca C

P 6128, succ. C

entre-ville-Montreal H

3C 3J7

Serguei A

. Zykov

Inst. of Metal P

hysics R

U

Serg@

imp.uran.ru

Kovalevskaya ul. 18, G

SP

-170-620219 Ekaterinburg

Series Preface

The Centre de recherches mathCmatiques (CRM) was created in 1968 by the Universite de Montreal to promote research in the mathematical sci­ences. It is now a national institute that hosts several groups, holds special theme years, summer schools, workshops, postdoctoral program. The focus of its scientific activities ranges from pure to applied mathematics, and includes satistics, theoretical computer science, mathematical methods in biology and life sciences, and mathematical and theoretical physics. The CRM also promotes collaboration between mathematicians and industry. It is subsidized by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR od the Province of Quebec, the Canadian Institute for Advanced Research and has private endowments. Current ac­tivities, fellowships, and annual reports can be found on the CRM web page at http://www . CRM. UMontreal. CAl.

The CRM Series in Mathematical Physics will publish monographs, lec­ture notes, and proceedings base on research pursued and events held at the Centre de recherches mathematiques.

Yvan Saint-Aubin Montreal

Preface

The subject of this three-week school was the explicit integration, that is, analytical as opposed to numerical, of all kinds of nonlinear differential equations (ordinary differential, partial differential, finite difference). The result of such integration is ideally the "general solution," but there are numerous physical systems for which only a particular solution is accessible, for instance the solitary wave of the equation of Kuramoto and Sivashinsky in turbulence.

Nonlinear differential equations describe many physical phenomena whose behavior can be either integrable (such as solitonic equations), par­tially integrable (the complex Ginzburg--Landau equation of nonlinear op­tics), or chaotic (dynamical systems such as the Lorenz model). In statisti­cal physics, the models that are defined on a lattice are described by finite difference equations, in short, discrete equations.

This subject is quite important for applications, because any analytic result (particular solution, first integral, etc.) is preferable to a numerical computation, which by definition is long, costly, and, worse, intrinsically subject to numerical error. But the distinction is even stronger. Indeed, the analytic approach, which was the subject of this school, often provides global knowledge of the solution, while the numerical approach is always local and hence insufficient: This is not because some initial state generates a "regular" regime that there do not exist other initial conditions with a "chaotic" regime. This question can then be settled only by a global study.

This explicit integration is not based on recipes, as we learned, but on quite powerful methods, most often not taught in universities, that were developed by Henri Poincare and mainly Paul Prudent Painleve in his fa­mous Ler;ons de Stockholm of 1895. The guideline is the in-depth study of singularities, because "les fonctions, comme les ctres vivants, sont car­acterisees par leurs singularites" (Montel). The landmark is the discovery, by Painleve and his student Bertrand Gambier, of six new functions, each defined by a second-order nonlinear differential equation, the first of which, (PI), is as simple as d2ujdx2 = 6u2 + x.

Initially qualified by Poincare as "lIe originale et splendide dans l'ocean voisin [du continent des mathematiques]," i.e., without any link with the exterior world, this discovery for a long time excited the curiosity only of mathematicians (mainly from Belgium, Japan, and Russia). Its second youth dates from about thirty years ago, and this renewal of interest has

xvi Preface

never decreased from then on. It arises from three, apparently disjoint, fields: statistical physics and field theory (Ising model, Wu and McCoy, Brczin and Kazakov), solitonic evolution equations (Kruskal, Ablowitz, Ra­mani, and Segur), and dynamical systems (Lorenz model, etc.). In all these fields were encountered the six Painlevc functions, whether in integrable or partially integrable cases; for instance, the self-dual Yang~Mills equa­tions, so fundamental in field theory, admit reductions to the six functions (Pl)~(P6). The physicists then became strongly interested in understand­ing the reason for that, and they took over from the mathematicians to make progress in the theory: They are responsible for extension of the the­ory to partial differential equations, in which the integrability manifests itself as solitons, and more recently to discrete equations.

The different courses presented at Cargcse intentionally alternated be­tween mathematics and physics. These courses were aimed at gradually bringing the audience (thirty participants were near the doctorate) to the level of current research. In this book as well as during the school, the se­quence of contributions appears in the following logical order (linear before nonlinear, elementary before advanced, etc.) In the following we abbrevi­ate ordinary differential equation as ODE, partial differential equation as PDE.

~ A reminder of more or less well known results on singularities (Fuch­sian, non-Fuchsian) of linear ODEs (Reignier)

~ A long course on the theory and practice of explicit integration of nonlinear ODEs by the study of only their singularities in the complex plane ("Painlevc analysis," properly said) (Conte)

~ Two successive long courses on the questions of monodromy and the connection of linear, then nonlinear, ODEs, in the modern formalism of the Japanese school (Mahoux, Joshi).

~ A long course on statistical physics, on the approach to bidimensional quantum gravity with the random matrices formalism, and how one naturally encounters the Painlevc functions (Di Francesco)

~ Another long course of "physics" on bidimensional topological field theory and its link with the Painlevc functions (Dubrovin)

~ A long course on the discretization of the equations of Painlevc, a blooming subject under the impulse of lattice models of statistical physics (Nijhoff and Ramani)

~ A medium length course on the extension of Painlevc methods to PDEs, whether integrable or not (Musette)

~ An introduction to the method of the inverse spectral transform, which extends to the nonlinear case the well-known Fourier transform,

Preface xvii

followed by a generalization of the famous system of Darboux and Halphen (Ablowitz)

- A long course on the method of symmetry and reduction, very useful to derive particular analytic solutions to PDEs by reducing them to ODEs (Winternitz, Clarkson)

- Three more specialized courses dealing with the six equations of Painleve: one on their Hamiltonian formalism (Okamoto), one on their groups of transformations (transformations of Schlesinger) (Gro­mak), one on their Hirota bilinear form (Hietarinta)

- Finally, a brief contribution aimed at revising some conventional ideas and proposing new ones (Kruskal)

We deeply regret the cancellation of the course to be given by Florent Bureau, one of the last true disciples of the Painleve school, due to health problems. This was also the reason for the absence of Peter Clarkson, whose joint course with Pavel Winternitz was given by the latter.

This school would not have been possible without the generous support of the following organizations, to which we express our deepest gratitude: Centre national de la recherche scientifique; Commission europeenne; Col­lectivite territoriale de Corse; Commissariat it l'energie atomique; Direction des recherches, etudes et techniques (DRET); ministere de la Recherche (programme ACCES); ministcre des Affaires etrangcres (bureau des con­grcs); Logovaz foundation (International Science Foundation).

Robert Conte Gif-sur-Yvette

Contents

Participants ix

Series Preface xiii

Preface xv

Contributors xxv

1 Singularities of Ordinary Linear Differential Equations and Integrability 1 Jean Reignier 1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Structure of the Solutions of the Homogeneous Equation

Around an Isolated Singular Point . 6 3 Weakly Singular Equations (Fuchs) 12 4 Thome's Equations . . 22 5 Global Considerations 27 6 References . . . . . . . 32

2 Introduction to the Theory of Isomonodromic Deformations of Linear Ordinary Differential Equations with Rational Coefficients 35 Gilbert M ahoux 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35 2 Isomonodromic Deformations of Linear ODEs with Fuchsian

Singularities . . . . . . . . . . . . . . . . . . . . . . . . . .. 38 3 The Isomonodromic Deformation Problem for Painleve VI. 46 4 Isomonodromic Deformations of Linear ODEs with Thome

Singularities . . . . . . . . . . . . . . . . . . . . . . . . . 52 5 An Isomonodromic Deformation Problem for Painleve I. 63 6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . 68 7 Appendix A: Matrix Versus Scalar Formalisms and Fuchs's

Theorem. . . . . . . . . . . . . . . . . 70 8 Appendix B: Asymptotic Power Series 73 9 References . . . . . . . . . . . . . . . . 74

3 The Painleve Approach to Nonlinear Ordinary Differential Equations 77

xx Contents

Robert Conte 1 Introduction . . . . . . . . . . 77 2 The Meromorphy Assumption 83 3 The True Problems . . . . . . 96 4 The Classical Results (L. Fuchs, Poincare, Painleve) 105 5 Construction of Necessary Conditions. The Theory . 108 6 Construction of Necessary Conditions. The Painleve Test 153 7 Sufficiency: Explicit Integration Methods 164 8 Conclusion 171 9 References . . . . . . . . . . . . . . . . . 172

4 Asymptotic Studies of the Painleve Equations 181 Nalini Joshi 1 Introduction . . . . . . . . . . . . . . . . . 181 2 Linear and Nonlinear Asymptotic Models. 192 3 The First and Second Painleve Equations 205 4 Global Extensions . 217 5 Conclusion 223 6 References . . . . . 224

5 2-D Quantum and Topological Gravities, Matrix Models, and Integrable Differential Systems 229 Philippe Di Francesco Part A 2-D Quantum Gravity. . . . . . . 229 1 Introduction . . . . . . . . . . . . . . . 229 2 The One-Matrix Model: Large N Limit. 238 3 The One-Matrix Model: Exact Solution. 246 4 The Double-Scaling Limit 251 5 Multimatrix Models. . . . . . 261 6 Conclusion . . . . . . . . . . . 269 Part B 2-D Topological Gravity 270 7 Introduction . . . . . . . . . . 270 8 Computing the Kontsevich Integral 273 9 The Kontsevich Integral as r-Function of the KdV Hierarchy. 278 10 Main Equivalence Theorem Between Topological and

Quantum Gravities 280 11 Conclusion 283 12 References . . . . . 283

6 Painleve Transcendents in Two-Dimensional Topological Field Theory 287 Boris Dubrovin 1 Algebraic Properties of Correlators in 2-D Topological Field

Theories. Moduli of a 2-D TFT and WDVV Equations of Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

Contents xxi

2 Equations of Associativity and Frobenius Manifolds. Deformed Flat Connection and Its Monodromy at the Origin . . . . . . 300

3 Semisimplicity and Canonical Coordinates . . . . . . . . . . . 333 4 Stokes Matrices and Classification of Semisimple Frobenius

Manifolds ............................. 344 5 Monodromy Group and Mirror Construction for Semisimple

Frobenius Manifolds 379 6 References . . . . . . . . . . . 406

7 Discrete Painleve Equations 413 Basile Gmmmaticos, Prank W. NijhojJ, and Alfred Ramani Introduction. . . . . . . . . . . . . . . . . . . . . 413 1 Integrable Discrete Systems . . . . . . . . . . 416 2 Similarity Reduction and Direct Linearization 432 3 The Painleve Property for Discrete Systems . 455 4 Properties of the Discrete Painleve Equations 476 5 Monodromy Problems and q-Difference Equations 486 6 References . . . . . . . . . . . . . . . . . . . . . . 505

8 Painleve Analysis for Nonlinear Partial Differential Equations 517 Micheline Musette 1 Introduction . . . . . . . . . 517 2 Integrable Equations . . . . 519 3 Painleve Analysis for PDEs 535 4 Partially Integrable and Nonintegrable Equations 558 5 References . . . . . . . . . . . . . . . . . . . . . . 562

9 On Painleve and Darboux-Halphen-Type Equations 573 Mark J. Ablowitz, Barby Chakmvarty, and Rod Halburd 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 573 2 Painleve Equations and 1ST . . . . . . . . . . . . . . . . 574 3 Darboux-Halphen Systems and Their Linear Problems as

Reductions of SDYM . . . . . . . . . . . . . . . . . . . . . 578 4 The Monodromy Evolving System and the Solution of the

Generalized DH System 585 5 Discussion 587 6 References . . . . . . . . 588

10 Symmetry Reduction and Exact Solutions of Nonlinear Partial Differential Equations 591 Peter A. Clarkson and Pavel Winternitz 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 2 Algorithm for Calculating the Symmetry Group of a

Differential System . . . . . . . . . . . . . . . . . . . . . . . . 592

xxii Contents

3 Examples of Symmetry Groups . . . . . . . . . . . . . . . 596 4 Symmetry Reduction, Group Invariant Solutions, Partially

Invariant Solutions . . . . . . . . . . . . . . . . . . . . . . 604 5 Classification of the Subalgebras of a Finite-Dimensional Lie

Algebra. . . . . . . . . . . . . . . . . . . . . . . 615 6 Direct Reductions and Conditional Symmetries 623 7 Conclusions 645 8 References . . . . . . . . . . . . . . . . . . . . . 646

11 Painleve Equations in Terms of Entire Functions 661 Jarmo Hietarinta 1 Introduction. . . . . . . . . . . . . . . . . . . . 661 2 Hirota's Bilinear Method for Soliton Equations 663 3 Bilinear Forms and Similarity Reduction 667 4 Solutions in Terms of Entire Functions 672 5 Discrete Painleve 681 6 References . . . . . . . . . . . . . . . . 685

12 Backlund Transformations of Painleve Equations and Their Applications 687 Valerii I. Gromak 1 Introduction. . . . . . . . . . . . . . . . 688 2 The Second Painleve Equation . . . . . . 695 3 Rational Solutions of (P2) (O-Solutions) . 700 4 One-Parameter Families of Classical Solutions (I-Solutions) 702 5 Algebraic Nonintegrability of (P2 ) 703 6 Higher Analogue of (P2 ) . . . . 705 7 The Fourth Painleve Equation . 708 8 Classical Solutions of (P4 ) . . 712 9 Rational Solutions of (P4 ) . . . 716 10 The Third Painleve Equation . 719 11 Equation (P3 ) for, = 0, aD -=1= 0 722 12 Equation (P3 ) for ,D -=1= 0 . . . . 724 13 Rational and Classical Solutions of (P3 ) for ,D -=1= 0 727 14 The Fifth Painleve Equation . 729 15 The Sixth Painleve Equation . 732 16 References . . . . . . . . . . . 734

13 The Hamiltonians Associated to the Painleve Equations 735 Kazuo Okamoto 1 Introduction. . . . . . . . . . . . . . 735 2 Hamiltonians and Painleve Analysis . 736 3 The Space of Initial Conditions . . . 744 4 The Irreducibility of Pn ...... . 753 5 The T-Functions of the Second Painleve System 762

6 The Painleve System of Two Variables 7 References . . . . . . . . . . . . . . . .

14 "Completeness" of the Painleve Test-General Considerations-Open Problems Martin D. Kruskal 1 Cultures in Mathematics 2 The Painleve Test . . . . 3 The PolyPainleve Test . 4 Asymptotic Expansions . 5 References . . . . . . . .

Index

Contents xxiii

771 784

789

789 792 797 800 803

805

Contributors

Mark J. Ablowitz Program in Applied Mathematics, University of Colorado, Campus box 526, Boulder, CO 80309, USA MarkjAb@newton.colorado.edu

Barby Chakravarty Program in Applied Mathematics, University of Colorado, Campus box 526, Boulder, Co 80309, USA Chuck@newton.colorado.edu

Peter A. Clarkson Institute of Mathematics and Statistics, University of Kent at Canterbury, Cornwallis bldg, Canterbury CT2 7NF, UK P.A.Clarkson@ukc.ac.uk

Robert Conte Service de physique de l'etat condense, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France Conte@spec.saclay.cea.fr

Philippe Di Francesco Service de physique theorique, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France Philippe@spht.saclay.cea.fr

Boris Dubrovin SISSA, Via Beirut 2, 1-34013 Trieste, Italy Dubrovin@tsmi19.sissa.it

Basile Grammaticos GMPIB, Universite Paris VII Denis Diderot, 2, place Jussieu, F-75251 Paris Cedex 05, France Grammati@paris7.jussieu.fr

Valerii 1. Gromak Department of mechanics and mathematics, Prospekt F. Skarina 4, Bielorussian State University of Informatics and Radioelectronics, 220050 Minsk, Bielorussia Grom@mmf.bsu.unibel.by

Rod Halburd Program in Applied Mathematics, University of Colorado, Campus box 526, Boulder, Co 80309, USA Rod@newton.colorado.edu

Jarmo Hietarinta Department of physics, University of Turku, FIN-20014 Turku, Finland Hietarin@utu.fi

xxvi Contributors

Nalini Joshi Department of Pure Mathematics, University of Adelaide, Adelaide 5005, Australia NJoshi~maths.adelaide.edu.au

Martin D. Kruskal Department of Mathematics, Hill Center, Busch Campus, Rutgers University, New Brunswick, NJ 08903, USA Kruskal~math.rutgers.edu

Gilbert Mahoux Service de physique thCorique, CEA Saclay, F-91191 G if-sur-Yvette Cedex, France Mahoux~spht.saclay.cea.fr

Micheline Musette Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium MMusette~vub.ac.be

Frank W. NijhofJ Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK Frank~amsta.leeds.ac.uk

Kazuo Okamoto Department of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguroku, Tokyo 153~8914, Japan Okamoto~Poisson.ms.u-tokyo.ac.jp

Alfred Ramani Centre de physique thCorique , Ecole poly technique, F-91128 Palaiseau Cedex, France Ramani~orphee.polytechnique.fr

Jean Reignier Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium JReignie~vub.ac.be

Pavel Winternitz Centre de recherches mathematiques, Universite de Montreal, C.P. 6128. Succ. Centre ville, Montreal, Quebec H3C 3J7, Canada Wintern~CRM.UMontreal.CA