Abstract:We consider ordinary differential equations (ODE) of the form $u''u - (u')^2 = e^{-x}P(u) - 1$, where $P$ is a polynomial. In previous work, necessary conditions on $P$ have been established for certain families of solutions of these ODEs to have asymptotic expansions of the form $u(x) = \sum_{k=0}^{\infty} p_k(x+c)e^{-kx}$ for $Re\,x \to +\infty$, where $c \in \mathbb C$ is an arbitrary constant parameterizing the solution family, and $p_k$ are polynomials, with $p_0(x) = x$. These conditions amount to $P(0) = 0$ and $P'(0) = \frac12P''(0)$. Here we show that these two conditions are also sufficient. The results imply the existence of corresponding expansions for certain degenerate Painlevé III transcendents.
From: Roland Hildebrand [view email]
[v1]
Tue, 23 Jun 2026 06:05:57 UTC (10 KB)
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