Symmetries in the fourth Painlevé equation and Okamoto polynomials

Abstract

Abstract. The fourth Painlevé equation PIV is known to have symmetry of the affine Weyl group of type A (1) 2 with respect to the Bäcklund transformations. We introduce a new representation of PIV , called the symmetric form, by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of PIV is given in terms of this representation. Through the symmetric form, it turns out that PIV is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions PIV , called Okamoto polynomials, are expressible in terms of the 3-reduced Schur functions.It is known by K. Okamoto [7] that the fourth Painlevé equation has symmetries under the affine Weyl group of type A 2 -symmetries become clearly visible. By means of this representation, we clarify the internal relation between the fourth Painlevé equation and the modified KP hierarchy. We obtain in particular a complete description of the rational solutions of the fourth Painlevé equation in terms of Schur functions. This implies that the so-called Okamoto polynomials, which arise from the τ -functions for rational solutions, are in fact expressible by the 3-reduced Schur functions. 1 §1. A symmetric form of the fourth Painlevé equationThe fourth Painlevé equation P IV is the following second order ordinary

Painlevé IV and degenerate Gaussian unitary ensembles

“…The solution of (4.17) may then be expressed in terms of τ -functions τ 0 (x), τ 1 (x), τ 2 (x) as 8) where the functions τ 0 and τ 2 will be defined later. The generalised Hermite polynomials [22] are defined as H m,n (x) = det (P n−i+j (x)) m i,j=1 Proof. For γ = 2K with K ∈ N the orthogonal polynomials with the weight w(x, t) can be expressed in terms of Hermite polynomials by the Christoffel formula ( [27], pg.…”

“…For the integer powers this can be seen from the original Kajiwara-Ohta determinant formula for the rational solutions of PIV ( [16], c.f. [22]) and it was later explored by Forrester and Witte [13].…”

“…In this case the recurrence coefficients are rational as functions of t . The theory of rational solutions to PIV [22] results in the expression of the recurrence coefficients through the generalized Hermite polynomials.…”

Solutions of a Derivative Nonlinear Schrödinger Hierarchy and Its Similarity Reduction

“…The Weyl group symmetry introduced in this article is isomorphic toW (A (1) 1 ), which does not seems to be a subgroup of theW (A (1) 2 )-symmetry discussed in [13,14]. To understand the relationship of ourW (A (1) 1 )-symmetry to the whole symmetry of the Painlevé IV, it seems that we need to consider a larger group that contain bothW (A (1) 1 ) andW (A (1) 2 ) as individual subgroups.…”

On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

“…Because of that result, it is easy to understand that Fuchs-Garnier pairs with the "defining equation" (1.9) in 3 × 3 matrices should exist for all Painlevé equations. Actually, M. Noumi and Y. Yamada [32,33] found such a pair for the symmetric version of P 4 . The latter pair was further studied by A. Sen, A. Hone, and P.A.…”