Abstract:We consider ordinary differential equations (ODE) of the form $u''u - (u')^2 = e^{-x}P(u) - 1$, where $P$ is a polynomial. For $P = u^k$, $k = 3,4,6$ this ODE is equivalent to certain degenerate Painlevé III equations. We study whether families of solutions of these ODEs 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, $p_k$ are polynomials, with $p_0(x) = x$. We find necessary conditions on $P$ for such expansions to exist. Numerical experiments suggest that these conditions are also sufficient, and the expansions are not only formal, but actually provide a series representation of the solutions. Numerical evidence also suggests a conjecture on the nonnegativity of coefficients of the $p_k$.
From: Roland Hildebrand [view email]
[v1]
Fri, 19 Jun 2026 06:36:58 UTC (3,716 KB)
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