Computational Methods and Function Theory 3 (2003), No. 1, 117--126
Copyright Heldermann Verlag 2003
Norbert Steinmetz
stein@math.uni-dortmund.de, Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany.
We prove that any transcendental solution of Painlev\'e's second equation $w''=\alpha+zw+2w^3$, 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 B\"acklund transformation to some solution of the Riccati equation $U'=\pm (z/2+U^2)$. In particular, $\alpha=n+1/2$, $n\in \mathbb{Z}$, and $w$ has order of growth $\varrho=3/2$. Moreover it is shown that $u$ satisfies some Riccati differential equation $u'=a(z)+b(z)u+c(z)u^2$ with rational coefficients.
Keywords: Painlevé, Airy and Riccati differential equation, Bäcklund transformation.
MSC 2000: 34M55, 30D35.
[FullText-pdf (320 K)] [FullText-ps (256 K)]