AN AUGMENTATION OF THE PHASE SPACE OF THE SYSTEM OF TYPE <i>A</i>₄⁽¹⁾

Abstract

Abstract. We investigate the differential system with affine Weyl group symmetry of type A (1) 4 and construct a space which parametrizes all meromorphic solutions of it. To demonstrate our method based on singularity analysis and affine Weyl group symmetry, we first study the system of type A (1) 2 , which is the equivalent of the fourth Painlevé equation, and obtain the space which augments the original phase space of the system by adding spaces of codimension 1. For the system of type A (1) 4 , codimension 2 spaces should be added to the phase space of the system in addition to codimension 1 spaces.

Coupled Painleve V Systems in Dimension 4

“…We call these spaces augmented phase spaces, in accordance with [9]. For these di¤erential equations in (1), we can construct the following phase spaces.…”

“…When we construct the phase spaces of the higher order Painlevé equations, an object that we call the local index is the key for determining when we need to make a blowing-up of an accessible singularity or a blowing-down to a minimal phase space. In the case of equations of higher order with favorable properties, for example the systems of type A 4 ð1Þ , the local index at the accessible singular point corresponds to the set of orders that appears in the free parameters of formal solutions passing through that point [9]. Definition 1.2.…”

Resolution of Singularities for a Family of Third-Order Differential Systems With Small Meromorphic Solution Spaces

“…(1) [2], the local index at the accessible singular point corresponds to the set of orders that appears in the free parameters of formal solutions passing through that point [5]. …”

“…For Painlevé equations, the dimension of the space of meromorphic solutions through any singular point is always codimension 1. However, in the case of higher-order Painlevé equations, the space of meromorphic solutions through a singular point may be of codimension greater than or equal to 2 [5]. In this paper, we will give an explicit resolution of singularities for a 3-parameter family of third-order differential systems with meromorphic solution spaces of codimension 2.…”

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

“…Other recent studies of the sP IV system include [52,79,82,83,[89][90][91][92][93][94][95]. EXAMPLE 2.…”