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Airy Solutions of Painlevé’s Second Equation

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Abstract

We prove that any transcendental solution of Painlevé’s second equation w″ = α + zw + 2w 3, which has the form w = R(z,u), with R rational in both variables and non-linear with respect to u, is obtained by repeated application of the Bäcklund transformation to some solution of the Riccati equation U′ = ±(z/2 + U 2). In particular, \(\alpha = n+1/2, n\in {\rm Z}\), and w has order of growth ϱ 3/2. Moreover it is shown that u satisfies some Riccati differential equation u’ = a(z) + b(z)u + c(z)u with rational coefficients.

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Correspondence to Norbert Steinmetz.

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Dedicated to the memory of Professor Dieter Gaier

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Steinmetz, N. Airy Solutions of Painlevé’s Second Equation. Comput. Methods Funct. Theory 3, 117–126 (2004). https://doi.org/10.1007/BF03321029

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  • DOI: https://doi.org/10.1007/BF03321029

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