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arXiv · 2205.12800

Exponentially-improved asymptotics and numerics for the (un)perturbed first Painlev\'e equation

Adri B. Olde Daalhuis
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Abstract

The solutions of the perturbed first Painlev\'e equation $y"=6y^2-x^\mu$, $\mu>-4$, are uniquely determined by the free constant $C$ multiplying the exponentially small terms in the complete large $x$ asymptotic expansions. Full details are given, including the nonlinear Stokes phenomenon, and the computation of the relevant Stokes multipliers. We derive asymptotic approximations, depending on $C$, for the locations of the singularities that appear on the boundary of the sectors of validity of these exponentially-improved asymptotic expansions. Several numerical examples illustrate the power of the approximations. For the tri-tronqu\'ee solution of the unperturbed first Painlev\'e equation we give highly accurate numerics for the values at the origin and the locations of the zeros and poles.

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Adri B. Olde Daalhuis. 2022-05-25. Exponentially-improved asymptotics and numerics for the (un)perturbed first Painlev\'e equation. https://doi.org/10.1088/1751-8121%2Fac7bbb

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