. Q A ] 3 A ug 1 99 8 HIGHER ORDER PAINLEVÉ
نویسنده
چکیده
A series of systems of nonlinear equations with affine Weyl group symmetry of type A (1) l is studied. This series gives a generalization of Painlevé equations P IV and PV to higher orders. Introduction In this paper we propose a series of systems of nonlinear differential equations which have symmetry under the affine Weyl groups of type A (1) l (l ≥ 2). These systems are thought of as higher order analogues of the Painlevé equations P IV and PV. We will study in particular their Hamiltonian structures and some basic properties of their τ -functions. For each l = 2, 3, . . . , we consider a differential system for (l + 1) unknown functions f0, . . . , fl, containing complex parameters α0, . . . , αl which correspond to the simple roots of the affine root system of type A (1) l . In what follows, we set α0 + · · · + αl = k. Our differential system is defined as follows according to the parity of l: For l = 2n (n = 1, 2, . . . ) A (1) 2n : f ′ j = fj ( ∑
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