Symmetries and Integrability of Difference Equations.

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.

3.8 Soliton solutions.

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