About Painlevé equations
نویسنده
چکیده
In this paper we study the equivalence problem with the first Painlevé equation y = 6y + x (resp. the second Painlevé equation y = 2y + yx+ α) under the action of fiber-preserving and point transformations. More specifically, we explicitly compute the change of coordinates that maps the generic second order differential equation to the Painlevé first equation (resp. the Painlevé second equation). The main innovation of this work lies in the exploitation of discrete symmetries for solving the equivalence problem.
منابع مشابه
Second-order second degree Painlevé equations related with Painlevé I, II, III equations
The algorithmic method introduced by Fokas and Ablowitz to investigate the transformation properties of Painlevé equations is used to obtain a one-to-one correspondence between the Painlevé I, II and III equations and certain second-order second degree equations of Painlevé type.
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Starting from the first Painlevé equation, Painlevé type equations of higher order are obtained by using the singular point analysis.
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Based on the works by Kajiwara, Noumi and Yamada, we propose a canonically quantized version of the rational Weyl group representation which originally arose as “symmetries” or the Bäcklund transformations in Painlevé equations. We thereby propose a quantization of discrete Painlevé VI equation as a discrete Hamiltonian flow commuting with the action of W (D (1) 4 ). 1
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We use the middle convolution to obtain some old and new algebraic solutions of the Painlevé VI equations.
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Starting from the second Painlevé equation, we obtain Painlevé type equations of higher order by using the singular point analysis.
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