Title Symmetries and Integrability of Difference Equations

Published Cambridge : Cambridge University Press, 2011

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Description 1 online resource (362 pages) Series London Mathematical Society Lecture Note Series, 381 ; v. 381 London Mathematical Society Lecture Note Series, 381 Contents Cover; Title; Copyright; Contents; List of figures; List of contributors; Preface; Introduction; 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals V. orodnitsyn and R. Kozlov; Abstract; 1.1 Introduction; 1.2 Invariance of Euler-Lagrange equations; 1.3 Lagrangian formalism for second-order difference equations; 1.4 Hamiltonian formalism for differential equations; 1.4.1 Canonical Hamiltonian equations; 1.4.2 The Legendre transformation; 1.4.3 Invariance of canonical Hamiltonian equations; 1.5 Discrete Hamiltonian formalism 1.5.1 Discrete Legendre transform1.5.2 Variational formulation of the discrete Hamiltonian equations; 1.5.3 Symplecticity of the discrete Hamiltonian equations; 1.5.4 Invariance of the Hamiltonian action; 1.5.5 Discrete Hamiltonian identity and discrete Noether theorem; 1.5.6 Invariance of the discrete Hamiltonian equations; 1.6 Examples; 1.6.1 Nonlinear motion; 1.6.2 A nonlinear ODE; 1.6.3 Discrete harmonic oscillator; 1.6.4 Modified discrete harmonic oscillator (exact scheme); 1.7 Conclusion; Acknowledgments; References 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete B. Grammaticos and A. RamaniAbstract; 2.1 Introduction; 2.2 A rough sketch of the top-down description of the Painlevé equations; The Hamiltonian formulation of Painlevé equations; 2.3 A succinct presentation of the bottom-up description of the Painlevé equations; Derivation of continuous Painlevé equations; 2.4 Properties of the, continuous and discrete, Painlevé equations: a parallel presentation; 2.4.1 Degeneration cascade; 2.4.2 Lax pairs; 2.4.3 Miura and Bäcklund relations; 2.4.4 Particular solutions; 2.4.5 Contiguity relations 2.5 The ultradiscrete Painlevé equations2.5.1 Degeneration cascade; 2.5.2 Lax pairs; 2.5.3 Miura and Bäcklund relations; 2.5.4 Particular solutions; 2.5.5 Contiguity relations; 2.6 Conclusion; References; 3 Definitions and Predictions of Integrability for Difference Equations J. Hietarinta; Abstract; 3.1 Preliminaries; 3.1.1 Points of view on integrability; 3.1.2 Preliminaries on discreteness and discrete integrability; 3.2 Conserved quantities; 3.2.1 Constants of motion for continuous ODE; 3.2.2 The standard discrete case; 3.2.3 The Hirota-Kimura-Yahagi (HKY) generalization 3.3 Singularity confinement and algebraic entropy3.3.1 Singularity analysis for difference equations; 3.3.2 Singularity confinement in projective space; 3.3.3 Singularity confinement is not sufficient; 3.4 Integrability in 2D; 3.4.1 Definitions and examples; 3.4.2 Quadrilateral lattices; 3.4.3 Continuum limit; 3.4.4 Conservation laws; 3.5 Singularity confinement in 2D; 3.6 Algebraic entropy for 2D lattices; 3.6.1 Default growth of degree and factorization; 3.6.2 Search based on factorization; 3.7 Consistency around a cube; 3.7.1 Definition; 3.7.2 Lax pair; 3.7.3 CAC as a search method Summary A comprehensive introduction to and survey of the state of the art, suitable for graduate students and researchers alike Notes 3.8 Soliton solutions Bibliography Includes bibliographical references Notes English Print version record Subject Difference equations. Symmetry (Mathematics) Integrals. MATHEMATICS -- Calculus. MATHEMATICS -- Mathematical Analysis. Difference equations Integrals Symmetry (Mathematics) Form Electronic book Author Olver, Peter Thomova, Zora Winternitz, Pavel LC no. 2011006852 ISBN 9781139117098 1139117092 9781139127752 1139127756 9780511997136 0511997132 1139114921 9781139114929 1280776099 9781280776090 1139122835 9781139122832 9786613686480 6613686484 1139112732 9781139112734

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