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Table of Contents
                            Frontmatter Pages
	1 Analytic Methods
	2 Painlev´e Analysis
	3 τ -functions, Bilinear and Trilinear Forms
	4 Lie-Algebraic and Group-Theoretical Methods
	5 Bihamiltonian Structures
	1 Fundamentals of Waves
	2 IST for Nonlinear Equations in 1+1 Dimensions
	3 Scattering and the Inverse Scattering Transform
	4 IST for 2+1 Equations
Grammaticos Ramani
	1 General Introduction: Who Cares about Integrability?
	2 Historical Presentation: From Newton to Kruskal
	3 Towards a Working De.nition of Integrability
		3.1 Complete Integrability
		3.2 Partial and Constrained Integrability
	4 Integrability and How to Detect It
		4.1 Fixed and Movable Singularities
		4.2 The Ablowitz-Ramani-Segur Algorithm
	5 Implementing Singularity Analysis: From Painlev´e to ARS and Beyond
	6 Applications to Finite and In.nite Dimensional Systems
		6.1 Integrable Di.erential Systems
		6.2 Integrable Two-Dimensional Hamiltonian Systems
		6.3 In.nite-Dimensional Systems
	7 Integrable Discrete Systems Do Exist!
	8 Singularity Con.nement: The Discrete Painlev´e Property
	9 Applying the Con.nement Method: Discrete Painlev´e Equations and Other Systems
		9.1 The Discrete Painlev´e Equations
		9.2 Multidimensional Lattices and Their Similarity Reductions
		9.3 Linearizable Mappings
	10 Discrete/Continuous Systems: Blending Con.nement with Singularity Analysis
		10.1 Integrodi.erential Equations of the Benjamin-Ono Type
		10.2 Multidimensional Discrete/Continuous Systems
		10.3 Delay-Di.erential Equations
	11 Conclusion
	1 Why the Bilinear Form?
	2 From Nonlinear to Bilinear
		2.1 Bilinearization of the KdV Equation
		2.2 Another Example: The Sasa-Satsuma Equation
	3 Constructing Multi-soliton Solutions
		3.1 The Vacuum, and the One-Soliton Solution
		3.2 The Two-Soliton Solution
		3.3 Multi-soliton Solutions
	4 Searching for Integrable Evolution Equations
		4.1 KdV
		4.2 mKdV and sG
		4.3 nlS
	1 Lie Bialgebras
		1.1 An Example: sl(2, C)
		1.2 Lie-Algebra Cohomology
		1.3 De.nition of Lie Bialgebras
		1.4 The Coadjoint Representation
		1.5 The Dual of a Lie Bialgebra
		1.6 The Double of a Lie Bialgebra. Manin Triples
		1.7 Examples
		1.8 Bibliographical Note
	2 Classical Yang-Baxter Equation and r-Matrices
		2.1 When Does δr a Lie-Bialgebra Structure on g?
		2.2 The Classical Yang-Baxter Equation
		2.3 Tensor Notation
		2.4 R-Matrices and Double Lie Algebras
		2.5 The Double of a Lie Bialgebra Is a Factorizable Lie Bialgebra
		2.6 Bibliographical Note
	3 Poisson Manifolds. The Dual of a Lie Algebra. Lax Equations
		3.1 Poisson Manifolds
		3.2 The Dual of a Lie Algebra
		3.3 The First Russian Formula
		3.4 The Traces of Powers of Lax Matrices Are in Involution
		3.5 Symplectic Leaves and Coadjoint Orbits
		3.6 Double Lie Algebras and Lax Equations
		3.7 Solution by Factorization
		3.8 Bibliographical Note
	4 Poisson Lie Groups
		4.1 Multiplicative Tensor Fields on Lie Groups
		4.2 Poisson Lie Groups and Lie Bialgebras
		4.3 The Second Russian Formula (Quadratic Brackets)
		4.5 The Dual of a Poisson Lie Group
		4.6 The Double of a Poisson Lie Group
		4.7 Poisson Actions
		4.8 Momentum Mapping
		4.9 Dressing Transformations
		4.10 Bibliographical Note
	Appendix 1 The ‘Big Bracket’ and Its Applications
	Appendix 2 The Poisson Calculus and Its Applications
	Background on Manifolds, Lie Algebras and Lie Groups, and Hamiltonian Systems
	A. Fundamental Articles on Poisson Lie Groups
	B. Books and Lectures on Poisson Manifolds, Lie Bialgebras, r-Matrices, and Poisson Lie Groups
	C. Further Developments on Lie Bialgebras, r-Matrices and Poisson Lie Groups
	D. Bibliographical Note Added in the Second Edition
Kruskal Joshi Halburd
	1 Introduction
	2 Nonlinear-Regular-Singular Analysis
		2.1 The Painlev´e Property
		2.2 The α-Method
		2.3 The Painlev´e Test
		2.4 Necessary versus Su.cient Conditions for the Painlev´e Property
		2.5 A Direct Proof of the Painlev´e Property for ODEs
		2.6 Rigorous Results for PDEs
	3 Nonlinear-Irregular-Singular Point Analysis
		3.1 The Chazy Equation
		3.2 The Bureau Equation
	4 Coalescence Limits
Magri Casati Falqui Pedroni
	1st Lecture: Bihamiltonian Manifolds
	2nd Lecture: Marsden–Ratiu Reduction
	3rd Lecture: Generalized Casimir Functions
	4th Lecture: Gel’fand–Dickey Manifolds
	5th Lecture: Gel’fand–Dickey Equations
		5.1. The Spectral Analysis of V (λ)
		5.2. The Auxiliary Eigenvalue Problem
		5.3 Dressing Transformations
	6th Lecture: KP Equations
	7th Lecture: Poisson–Nijenhuis Manifolds
	8th Lecture: The Calogero System
	1 Introduction
	2 Hirota’s Method
	3 Algebraic Inentities
	4 Extensions
		4.1 q-Discrete Toda Equation
		4.2 Trilinear Formalism
		4.3 Ultra-discrete Systems
	5 Concluding Remarks
	1 Introduction
	2 Generalities
		2.1 Basic Theorem: Linear Case
		2.2 Two Examples
	3 Quadratic Case
		3.1 Abstract Case: Poisson Lie Groups and Factorizable Lie Bialgebras
		3.2 Duality Theory for Poisson Lie Groups and Twisted Spectral Invariants
		3.3 Sklyanin Bracket on G(z)
	4 Quantization
		4.1 Linear Case
		4.2 Quadratic Case. Quasitriangular Hopf Algebras
Document Text Contents
Page 1

Y. Kosmann-Schwarzbach B. Grammaticos
K.M. Tamizhmani (Eds.)

of Nonlinear Systems

1 3

Page 2


Yvette Kosmann-Schwarzbach
Centre de Mathématiques
École Polytechnique Palaiseau
91128 Palaiseau, France

Basil Grammaticos
GMPIB, Université Paris VII
Tour 24-14, 5e étage, case 7021
2 place Jussieu
75251 Paris, France

K. M. Tamizhmani
Department of Mathematics
Pondicherry University
Pondicherry 605 014, India

Y. Kosmann-Schwarzbach, B. Grammaticos, K. M. Tamizhmani (eds.), Integrability of
Nonlinear Systems, Lect. Notes Phys. 638 (Springer-Verlag Berlin Heidelberg 2004), DOI

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ISSN 0075-8450
ISBN 3-540-20630-2 Springer-Verlag Berlin Heidelberg New York

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Page 171

Lie Bialgebras, Poisson Lie Groups, and Dressing Transformations 161

for x ∈ g, ξ ∈ g∗ (see Sect. 1.6). In the linearization of the left (resp., right)
action of G on G∗, the image of (x, ξ) ∈ g ⊕ g∗ is the projection onto g∗ of
[x, ξ]d (resp., [ξ, x]d), i.e., the linearized action is the coadjoint action (resp., the
opposite of the coadjoint action) of g on g∗.

Similarily the linearized action of the left (resp., right) action of G∗ on G is
the coadjoint (resp., the opposite of the coadjoint action) of g∗ on g.

From relation (4.22) we deduce that the vector fields of the left infinitesimal
action of g on G∗ satisfy the twisted multiplicativity property (4.17). This fact
and the fact that the linearized action is the coadjoint action of g on g∗ permit
identifying this infinitesimal action with the left infinitesimal dressing action of
g on G∗ (see Lu and Weinstein [45]).

The proofs for the right action of G on G∗, and for the left and right actions
of G∗ on G are similar.

To conclude, we wish to relate the dressing transformations defined above
with the formulæ expressing the dressing of G-valued fields satisfying a zero-
curvature equation. (See Faddeev and Takhtajan [29], Babelon and Bernard
[34].) The field equation expresses a compatibility condition for a linear system,
the Lax representation, equivalent to a nonlinear soliton equation. This nonlinear
equation admits a Hamiltonian formulation such that the Poisson brackets of
the g-valued Lax matrix are expressed in terms of a factorizable r-matrix. The
dressing transformations act on the G-valued fields and preserve the solutions of
the field equation. This action is in fact a Poisson action of the dual group, G∗.

If the Poisson Lie structure of G is defined by a factorizable r-matrix, the
double, D, is isomorphic to G × G, with, as subgroups, the diagonal subgroup
{(g, g)|g ∈ G} ≈ G, and {(g+, g−)|g± = R±h, h ∈ G} ≈ G∗, with Lie algebras g
and {(r+x, r−x)|x ∈ g} ≈ g∗, respectively (see Sect. 2.5 and 4.6).

In this case, the factorization problems consist in finding group elements
g′ ∈ G and g′± = R±h′, h′ ∈ G, satisfying

(g+, g−)(g, g) = (g
′, g′)(g′+, g

−) , (4.23)

or g′ ∈ G and g′± = R±h′, h′ ∈ G, satisfying
(g, g)(g+, g−) = (g

+, g


′, g′) . (4.24)

Let us write the left action of G∗ on G in this case. From ug = ug ug, we
obtain from (4.23),

g+g = g
′g′+, g−g = g


Eliminating g′, we find that g′+
g′− is obtained from g

+ g− by conjugation by

g−1, and that g′+, g

− solve the factorization equation

g′− = g

−1g−)g . (4.25)

It follows that the action of the element (g+, g−) ∈ G∗ on g ∈ G is given by
g′ = g+gg


= g−gg

, (4.26)

Page 172

162 Y. Kosmann-Schwarzbach

where the group elements g′+ and g

− solve the factorization equation (4.25).

Similarly for the right action of G∗ on G, we obtain the factorization equation



= g(g+g−
−1)g−1 , (4.27)

and the action of (g+, g−) ∈ G∗ on g ∈ G is given by

g′ = g′+
gg+ = g

gg− , (4.28)

where the group elements g′+ and g

− solve the factorization equation (4.27).

Thus we recover the formula of the dressing transformation in Faddeev and
Takhtajan [29] and in Babelon and Bernard [34]. (In the convention of [29],
the g′− considered here is replaced by its inverse, while in [34], the factorization
equation is g′−

g′+ = g(g−


4.10 Bibliographical Note

Multiplicative fields of tensors, were introduced by Lu and Weinstein [45]. See
Kosmann-Schwarzbach [39], Vaisman [30], Dazord and Sondaz [37]. The results
of Sect. 4.2 are due to Drinfeld [15], and are further developed in Kosmann-
Schwarzbach [38] [39], Verdier [31], and Lu and Weinstein [45]. For formula
(4.11), see e.g., Takhtajan [29]. For the examples of Sect. 4.5, see [45] and Majid
[46][24]. Poisson actions were introduced by Semenov-Tian-Shansky [19], who
showed that they were needed to explain the properties of the dressing trans-
formations in field theory. Their infinitesimal characterization is due to Lu and
Weinstein [45]. The generalization of the momentum mapping to the case of
Poisson actions is due to Lu [44], while Babelon and Bernard [34], who call it
the “non-Abelian Hamiltonian”, have shown that in the case of the dressing
transformations of G-valued fields the momentum mapping is given by the mon-
odromy matrix of the associated linear equation. For the properties of the dres-
sing transformations, see Semenov-Tian-Shansky [19], Lu and Weinstein [45],
Vaisman [30], Alekseev and Malkin [32] (also, Kosmann-Schwarzbach and Magri
[40] for the infinitesimal dressing transformations). A comprehensive survey of
results, including examples and further topics, such as affine Poisson Lie groups,
is given by Reyman [26].

Appendix 1
The ‘Big Bracket’ and Its Applications

Let F be a finite-dimensional (complex or real) vector space, and let F ∗ be its
dual vector space. We consider the exterior algebra of the direct sum of F and

F ∗,

(F ⊕ F ∗) =




F ∗ ⊗∧p+1 F )


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