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Elliptic asymptotics of the first and the second Painlevé transcendents. (English) Zbl 0829.34040

This expository paper is concerned with asymptotic description of the general solutions of the first Painlevé equation (P 1 ) d 2 y/dx 2 =6y 2 +x, x, and of the second Painlevé equation (P 2 ) d 2 y/dx 2 =2y 3 +xy-ν 2 , x,ν 2 , in the neighborhood of the essential singularity, i.e. for |x|. The paper aims only to tackle the asymptotics in generic position which usually are constructed in terms of elliptic functions. Starting from one of the basic papers of Boutroux (1913) the elliptic asymptotics are followed in some fundamental papers and subsequent discrepancies in asymptotics are revealed.

Next isomonodromic deformations and asymptotics of the second Painlevé transcendent are studied, uniformization by elliptic Θ functions, degeneration of elliptic asymptotics into trigonometric ones and finally the distribution of zeros and poles of the first and second Painlevé transcendents are considered. The paper ends with a supplement where some recent papers published after the present paper had been written are commented.

MSC:
34E05Asymptotic expansions (ODE)
35Q53KdV-like (Korteweg-de Vries) equations
34M55Painlevé and other special equations; classification, hierarchies