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Elliptic asymptotic forms of solutions to the Painlev'e equations

A. D. Bruno, I. V. Goryuchkina


Abstract: Here we set out algorithm based on the three-dimensional power geometry, which allows to find elliptic asymptotic forms of solutions to a wide class of differential equations, including Painlev'e equations. We use this algorithm and methods of the plane and three-dimensional power geometry to calculate power, complicated, exotic and elliptic asymptotic forms of the solutions to the first four Painlevé equations and also exponentially small additions to these asymptotic forms. Asymptotic forms of the solutions to the first Painlev'e equations, expressed in elliptic functions, were found by Boutroux in 1913.

UDC: 517.91