Integrability of Nonlinear Systems

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X Contents Eight Lectures on Integrable Systems F. Magri, P. Casati, G. Falqui, M. Pedroni ...........................209 1st Lecture: Bihamiltonian Manifolds .................................210 2nd Lecture: Marsden–Ratiu Reduction ...............................215 3rd Lecture: Generalized Casimir Functions ...........................220 4th Lecture: Gel’fand–Dickey Manifolds ...............................225 5th Lecture: Gel’fand–Dickey Equations ..............................229 6th Lecture: KP Equations ..........................................235 7th Lecture: Poisson–Nijenhuis Manifolds .............................241 8th Lecture: The Calogero System ...................................247 Bilinear Formalism in Soliton Theory J. Satsuma ........................................................251 1 Introduction ...................................................251 2 Hirota’s Method ...............................................252 3 Algebraic Inentities .............................................255 4 Extensions ....................................................259 4.1 q-Discrete Toda Equation ...................................259 4.2 Trilinear Formalism ........................................260 4.3 Ultra-discrete Systems ......................................263 5 Concluding Remarks ............................................266 Quantum and Classical Integrable Systems M.A. Semenov-Tian-Shansky ........................................269 1 Introduction ...................................................269 2 Generalities ....................................................271 2.1 Basic Theorem: Linear Case .................................273 2.2 Two Examples ............................................283 3 Quadratic Case ................................................291 3.1 Abstract Case: Poisson Lie Groups and Factorizable Lie Bialgebras ..............................293 3.2 Duality Theory for Poisson Lie Groups and Twisted Spectral Invariants .............................296 3.3 Sklyanin Bracket on G (z)...................................304 4 Quantization...................................................305 4.1 Linear Case ...............................................305 4.2 Quadratic Case. Quasitriangular Hopf Algebras ................319
List of Contributors Mark J. Ablowitz Department of Applied Mathematics, Campus Box 526, University of Colorado at Boulder, Boulder Colorado 80309-0526, USA markjab@newton.colorado.edu Paolo Casati Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via degli Arcimboldi 8, 20126 Milano, Italy casati@matapp.unimib.it Gregorio Falqui SISSA, Via Beirut 2/4, 34014 Trieste, Italy falqui@sissa.it Basil Grammaticos GMPIB, Université Paris VII, Tour 24-14, 5 e étage, case 7021, 75251 Paris, France grammati@paris7.jussieu.fr Rod Halburd Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom r.g.halburd@lboro.ac.uk Jarmo Hietarinta Department of Physics, University of Turku, 20014 Turku, Finland hietarin@utu.fi Nalini Joshi School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia nalini@maths.usyd.edu.au Yvette Kosmann-Schwarzbach Centre de Mathématiques, École Polytechnique, 91128 Palaiseau, France yks@math.polytechnique.fr Martin D. Kruskal Department of Mathematics, Rutgers University, New Brunswick NJ 08903, USA kruskal@math.rutgers.edu Franco Magri Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via degli Arcimboldi 8, 20126 Milano, Italy magri@matapp.unimib.it
Page 1 and 2: Y. Kosmann-Schwarzbach B. Grammatic
Page 3 and 4: Lecture Notes in Physics Editorial
Page 5: Preface This second edition of Inte
Page 8 and 9: VIII Contents 9 Applying the Confin
Page 12 and 13: XII List of Contributors Marco Pedr
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Page 18 and 19: 6 M.J. Ablowitz ∂φ + gη =0 on z
Page 20 and 21: 8 M.J. Ablowitz The quasilinear equ
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Integrability - and How to Detect I
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8 Singularity Confinement: The Disc
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and we obtain, Integrability - and
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Introduction to the Hirota Bilinear
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3.3 Multi-soliton Solutions Introdu
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108 Y. Kosmann-Schwarzbach vector s
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110 Y. Kosmann-Schwarzbach volume,
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112 Y. Kosmann-Schwarzbach Since, w
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114 Y. Kosmann-Schwarzbach An alter
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116 Y. Kosmann-Schwarzbach define a
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118 Y. Kosmann-Schwarzbach e 1 = (
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120 Y. Kosmann-Schwarzbach r(ξ)(η
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122 Y. Kosmann-Schwarzbach The last
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124 Y. Kosmann-Schwarzbach Now 〈s
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126 Y. Kosmann-Schwarzbach 2.3 Tens
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128 Y. Kosmann-Schwarzbach or, in t
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130 Y. Kosmann-Schwarzbach Thus, by
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132 Y. Kosmann-Schwarzbach Note tha
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134 Y. Kosmann-Schwarzbach Definiti
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136 Y. Kosmann-Schwarzbach If, in p
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138 Y. Kosmann-Schwarzbach we obtai
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140 Y. Kosmann-Schwarzbach Let G be
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142 Y. Kosmann-Schwarzbach for all
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144 Y. Kosmann-Schwarzbach parallel
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146 Y. Kosmann-Schwarzbach = d ds f
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148 Y. Kosmann-Schwarzbach Taking i
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150 Y. Kosmann-Schwarzbach By the t
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152 Y. Kosmann-Schwarzbach where
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154 Y. Kosmann-Schwarzbach Example
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156 Y. Kosmann-Schwarzbach Definiti
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158 Y. Kosmann-Schwarzbach 4.8 Mome
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160 Y. Kosmann-Schwarzbach Let g be
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162 Y. Kosmann-Schwarzbach where th
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164 Y. Kosmann-Schwarzbach defining
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166 Y. Kosmann-Schwarzbach For any
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168 Y. Kosmann-Schwarzbach Proposit
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170 Y. Kosmann-Schwarzbach 3. O. Ba
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172 Y. Kosmann-Schwarzbach 39. Y. K
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Analytic and Asymptotic Methods for
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This gives Analytic and Asymptotic
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Eight Lectures on Integrable System
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• they preserve the Poisson brack
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The Poisson tensors P 0 and P 1 are
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4th Lecture: Gel’fand-Dickey Mani
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After an integration by parts, we f
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starting from h (0) = 1. A simple c
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Bilinear Formalism in Soliton Theor
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Quantum and Classical Integrable Sy
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g ++ D Quantum and Classical Integr
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Let ( V k (λ,0) Quantum and Classi
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Integrability of Nonlinear Systems

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