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Rational solutions of the second and the fourth Painlevé equations. (English) Zbl 0597.34004

This paper establishes by group theory methods the conditions under which the second and fourth Painlevé equations

P 2 (α):d 2 λ dt 2 =2λ 3 +tλ+α
P(α,θ):d 2 λ dt 2 =1 2λ(dλ dt) 2 + 32λ 3 +4tλ 2 +2(t 2 -α)λ-2θ 2 λ

have rational solutions for λ (t). Specifically P 2 (α) has a unique rational solution if and only is α is an integer and P 4 (α,θ) likewise for specifically characterised values of α and θ. In each case explicit solutions are given for small values of α and θ.

Reviewer: G.G.Wake
MSC:
34M55Painlevé and other special equations; classification, hierarchies
34A34Nonlinear ODE and systems, general
34A05Methods of solution of ODE