painleve%C3%A9%20equations

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11: Mark J. Ablowitz

… ►Certain nonlinear equations are special; e. …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. …

12: 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). …

13: 32.1 Special Notation

… ►The functions treated in this chapter are the solutions of the Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$.

14: 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. …

15: 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.

16: 32.5 Integral Equations

§32.5 Integral Equations

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17: 32.4 Isomonodromy Problems

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§32.4(ii) First Painlevé Equation

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§32.4(iii) Second Painlevé Equation

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§32.4(iv) Third Painlevé Equation

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§32.4(v) Other Painlevé Equations

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18: Peter A. Clarkson

… ►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. … Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …His well-known book Solitons, Nonlinear Evolution Equations and Inverse Scattering (with M. … ►

32 Painlevé Transcendents

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19: 15.17 Mathematical Applications

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§15.17(i) Differential Equations

►This topic is treated in §§15.10 and 15.11. ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … … ►These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

20: 32 Painlevé Transcendents

Chapter 32 Painlevé Transcendents

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