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Studies on the Painlevé equations. IV: Third Painlevé equation P III . (English) Zbl 0639.58013

Summary: [For part III see Math. Ann. 275, 221-255 (1986; Zbl 0589.58008).]

The present article deals with the third Painlevé equation P III ; we consider instead the equation P III ' , equivalent to the former. By defining the Painlevé system , we consider the group G * of birational canonical transformations of ; G * is isomorphic to the affine Weyl group of the root system of the type B 2 . A sequence of solutions of is obtained from that of τ-functions, satisfying the Toda equation and vice versa. We consider also particular solutions of written in terms of the cylinder function.

MSC:
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
14E05Rational and birational maps