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arXiv · 0805.2444

Studies on the second member of the second Painlevé hierarchy

Yusuke Sasano
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Abstract

In this paper, we study the second member of the second Painlevé hierarchy $P_{II}^{(2)}$. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four, and this Hamiltonian system can be considered as a 1-parameter family of coupled Painlevé systems. This Hamiltonian is new. We also show that this system admits extended affine Weyl group symmetry of type $A_1^{(1)}$, and can be recovered by its holomorphy conditions. We also study a fifth-order ordinary differential equation satisfied by this Hamiltonian. After we transform this equation into a system of the first-order ordinary differential equations of polynomial type in dimension five by birational transformations, we give its symmetry and holomorphy conditions.

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Yusuke Sasano. 2009-11-14. Studies on the second member of the second Painlevé hierarchy. https://arxiv.org/abs/0805.2444

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