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Zbl 0526.34002
Gromak, V.I.; Lukashevich, N.A.
Special classes of solutions of Painlevé's equations. (English. Russian original)
[J] Differ. Equations 18, 317-326 (1982); translation from Differ. Uravn. 18, 419-429 (1982). ISSN 0012-2661

This is a survey paper on research works in the past 30 years about the six famous second order differential equations of Painlevé. it is concerned with conditions of integrability of Equations (II)--(VI), conditions for the existence of rational and classical transcendental solutions, relations between solutions of the same equation which correspond to different values of parameters, as well as solutions of Equations (III) and (V), methods for the construction of one-parameter family of solutions, and the connection between Painlevé equations and famous equations of mathematical physics, such as Korteweg-de Vries equation, Sine-Gordon and Schrödinger equations.

Display scanned Zentralblatt-MATH page with this review.
[Ye Yanqian (Nanjing)]

MSC 2000:: *34-02 Research monographs (ordinary differential equations)
34M55 Painlevé and other special equations
35Q53 KdV-like equations
35Q55 NLS-like (nonlinear Schroedinger) equations
34A12 Initial value problems for ODE
34B30 Special ODE
34A25 Analytical theory of ODE
34C11 Qualitative theory of solutions of ODE: Growth, etc.

Keywords: Painlevé equations; Korteweg-de Vries equation; Sine-Gordon equation; Schrödinger equation; transcendental solutions; one-parameter family of solutions

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