Bilinear Discrete Painleve-II and its Particular Solutions

Abstract

By analogy to the continuous Painlevé II equation, we present particular solutions of the discrete Painlevé II (d-P${}_{\mathrm{II}}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the bilinear formalism that allows us to obtain the $\tau$ function for d-P${}_{\mathrm{II}}$. Two different forms of bilinear d-P${}_{\mathrm{II}}$ are obtained and we show that they can be related by a simple gauge transformation.

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References in corpus (2)

• Linearization And Solutions Of The Discrete Painlevé-III Equation
• Casorati Determinant Solutions for the Discrete Painlevé-II Equation

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• Rational Solutions for the Discrete Painlevé II Equation
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• Discrete equations from Bäcklund transformations of the fifth Painlevé equation