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# On the Second Painlevé Transcendent

J. W. Miles
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 361, No. 1706 (Jun. 1, 1978), pp. 277-291
Stable URL: http://www.jstor.org/stable/79576
Page Count: 15
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## Abstract

Numerical and asymptotic approximations to the second Painlevé transcendent, F±(z;a), as determined by the solution of F′ ′-zF± 2F3=0 and F∼ aAi $(z)(z\uparrow \infty)$, are presented. The solution for F+ is finite for all real z and $01$. The asymptotic behaviour of F± in the oscillatory régime $(-z\pm 2F^{2}>0)$, which bears a qualitative resemblance to that of Ai (z), is determined for a2≪ 1 and for ± ln (1 ± a2) ≫ 1. The results are relevant for several recent investigations of nonlinear wave motion.

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