Sub-normal Solutions to Painleve's Second Differential Equation
Norbert Steinmetz
TL;DR
This paper completes the classification of transcendental solutions to Painleve's second differential equation by proving that no sub-normal solutions exist apart from known Airy solutions, refining the understanding of their growth properties.
Contribution
It proves that the only sub-normal solutions to Painleve's second equation are the Airy solutions, closing a gap in the classification of solutions based on growth order.
Findings
No sub-normal solutions other than Airy solutions exist
Transcendental solutions have growth order 3 or 3/2
Complete classification of solutions based on growth order
Abstract
In a recent paper, Aimo Hinkkanen and Ilpo Laine proved that the transcendental solutions to Painleve's second differential equation w"=a+zw+w^3 have either order of growth 3 or else 3/2. We complete this result by proving that there exist no sub-normal solutions (order of growth 3/2) other than the so-called Airy solutions.
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