painleve%C3%A9%20transcendents

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1: 32.2 Differential Equations

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§32.2(i) Introduction

►The six Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are as follows: … ►The solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are called the Painlevé transcendents. The six equations are sometimes referred to as the Painlevé transcendents, but in this chapter this term will be used only for their solutions. … ► …

2: 32.16 Physical Applications

§32.16 Physical Applications

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Statistical Physics

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Integrable Continuous Dynamical Systems

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Other Applications

►For the Ising model see Barouch et al. (1973), Wu et al. (1976), and McCoy et al. (1977). …

3: 32.12 Asymptotic Approximations for Complex Variables

§32.12 Asymptotic Approximations for Complex Variables

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§32.12(i) First Painlevé Equation

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§32.12(ii) Second Painlevé Equation

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§32.12(iii) Third Painlevé Equation

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4: 32 Painlevé Transcendents

Chapter 32 Painlevé Transcendents

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5: 32.13 Reductions of Partial Differential Equations

§32.13 Reductions of Partial Differential Equations

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§32.13(i) Korteweg–de Vries and Modified Korteweg–de Vries Equations

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§32.13(ii) Sine-Gordon Equation

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§32.13(iii) Boussinesq Equation

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6: 32.17 Methods of Computation

§32.17 Methods of Computation

►The Painlevé equations can be integrated by Runge–Kutta methods for ordinary differential equations; see §3.7(v), Hairer et al. (2000), and Butcher (2003). …

7: Mark J. Ablowitz

… ►Their similarity solutions lead to special ODEs which have the Painlevé property; i. …ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. Some of the relationships between IST and Painlevé equations are discussed in two books: Solitons and the Inverse Scattering Transform and Solitons, Nonlinear Evolution Equations and Inverse Scattering. Widespread interest in Painlevé equations re-emerged in the 1970s and thereafter partially due to the connection with IST and integrable systems. …

8: Alexander A. Its

… ►Books by Its are The Isomonodromic Deformation Method in the Theory of Painlevé Equations (with V. … Matveev), published by Springer in 1994, and Painlevé Transcendents: The Riemann-Hilbert Approach (with A. …

9: 32.14 Combinatorics

§32.14 Combinatorics

… ►where the distribution function $F(s)$ is defined here by … ►The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994). ►See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.

10: 32.15 Orthogonal Polynomials

§32.15 Orthogonal Polynomials

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