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Painlevé equations. --- Hamiltonian systems. --- Backlund transformations

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Lie algebras --- Differential equations --- Geometry, Algebraic --- Painlevé equations

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There are a number of intriguing connections between Painlevé equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these. Researchers in integrable systems and non-linear equations will find the many explicit examples where Painlevé equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painlevé transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painlevé equations are worked out in detail, with a survey of recent results and an outline of their close relationship with orthogonal polynomials. Exercises throughout the book help the reader to get to grips with the material. The author is a leading authority on orthogonal polynomials, giving this work a unique perspective on Painlevé equations.

Orthogonal polynomials. --- Polynomials. --- Painlevé equations. --- Differential equations, Nonlinear.

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Painleve equations --- Riemann-Hilbert problems --- Differential equations, Nonlinear --- Asymptotic theory --- Painlevé equations. --- Riemann-Hilbert problems. --- 517.9 --- Asymptotic theory in nonlinear differential equations --- Asymptotic expansions --- Hilbert-Riemann problems --- Riemann problems --- Boundary value problems --- Equations, Painlevé --- Functions, Painlevé --- Painlevé functions --- Painlevé transcendents --- Transcendents, Painlevé --- Asymptotic theory. --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Painlevé equations --- Differential equations, Nonlinear - Asymptotic theory

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This is a proceedings of the international conference "Painlevé Equations and Related Topics" which was taking place at the Euler International Mathematical Institute, a branch of the Saint Petersburg Department of the Steklov Institute of Mathematics of the Russian Academy of Sciences, in Saint Petersburg on June 17 to 23, 2011. The survey articles discuss the following topics: General ordinary differential equations Painlevé equations and their generalizations Painlevé property Discrete Painlevé equations Properties of solutions of all mentioned above equations:- Asymptotic forms and asymptotic expansions- Connections of asymptotic forms of a solution near different points- Convergency and asymptotic character of a formal solution- New types of asymptotic forms and asymptotic expansions- Riemann-Hilbert problems- Isomonodromic deformations of linear systems- Symmetries and transformations of solutions- Algebraic solutions Reductions of PDE to Painlevé equations and their generalizations Ordinary Differential Equations systems equivalent to Painlevé equations and their generalizations Applications of the equations and the solutions

Painlevé equations --- Equations, Painlevé --- Functions, Painlevé --- Painlevé functions --- Painlevé transcendents --- Transcendents, Painlevé --- Differential equations, Nonlinear --- Ordinary Differential Equation. --- Painlevé Property. --- Panilevé Equation.