Painlev%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: 18.42 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …

3: 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). …

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: 32.12 Asymptotic Approximations for Complex Variables

§32.12 Asymptotic Approximations for Complex Variables

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7: Bibliography I

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207

External Links:

ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§32.11(iii).

Encodings:

BibTeX

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Quasi-linear Stokes phenomenon for the second Painlevé transcendent. Nonlinearity 16 (1), pp. 363–386.

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External Links:

ISSN 0951-7715, Document, MathReview (Youmin Lu), ZentralBlatt

Referenced by:

§32.12(ii).

Encodings:

BibTeX

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Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.

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External Links:

ISSN 0305-4470, Document, MathReview (Yoshitsugu Takei), ZentralBlatt

Referenced by:

§32.11(v).

Encodings:

BibTeX

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8: 20 Theta Functions

Chapter 20 Theta Functions

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9: 32.17 Methods of Computation

§32.17 Methods of Computation

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10: 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). …