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Special classes of solutions of Painlevé’s equations. (English) Zbl 0526.34002
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.
MSC:
34-02Research monographs (ordinary differential equations)
34M55Painlevé and other special equations; classification, hierarchies
35Q53KdV-like (Korteweg-de Vries) equations
35Q55NLS-like (nonlinear Schrödinger) equations
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
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34C11Qualitative theory of solutions of ODE: growth, boundedness