scholarly journals Studies on the Painlevé equations II. Fifth Painlevé equation PV

1987 ◽  
Vol 13 (1) ◽  
pp. 47-76 ◽  
Author(s):  
Kazuo OKAMOTO
2000 ◽  
Vol 159 ◽  
pp. 87-111 ◽  
Author(s):  
Makoto Taneda

We study the Yablonskii-Vorob’ev polynomial associated with the second Painlevé equation. To study other special polynomials (Okamoto polynomials, Umemura polynomials) associated with the Painlevé equations, our purely algebraic approach is useful.


Author(s):  
Nalini Joshi ◽  
Yang Shi

In this paper, we present a new method of deducing infinite sequences of exact solutions of q -discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q -discrete version of the second Painlevé equation ( q -P II ) with affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . We show, for the first time, how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q -P II here, is also applicable to other discrete Painlevé equations.


2012 ◽  
Vol 22 (09) ◽  
pp. 1250211
Author(s):  
ATHANASSIOS S. FOKAS ◽  
DI YANG

One of the authors recently introduced the concept of conjugate Hamiltonian systems: the solution of the equation h = H(p, q, t), where H is a given Hamiltonian containing t explicitly, yields the function t = T(p, q, h), which defines a new Hamiltonian system with Hamiltonian T and independent variable h. By employing this construction and by using the fact that the classical Painlevé equations are Hamiltonian systems, it is straightforward to associate with each Painlevé equation two new integrable ODEs. Here, we investigate the conjugate Painlevé II equations. In particular, for these novel integrable ODEs, we present a Lax pair formulation, as well as a class of implicit solutions. We also construct conjugate equations associated with Painlevé I and Painlevé IV equations.


1995 ◽  
Vol 139 ◽  
pp. 37-65 ◽  
Author(s):  
Yoshihiro Murata

The big problem “Do Painlevé equations define new functions?”, what is called the problem of irreducibilities of Painlevé equations, was essentially solved by H. Umemura [16], [17] and K. Nishioka [9].


2019 ◽  
Vol 2020 (24) ◽  
pp. 9797-9843 ◽  
Author(s):  
Anton Dzhamay ◽  
Alisa Knizel

Abstract The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlevé equations. Namely, we consider the $q$-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete $q$-P$\left (E_7^{(1)}/A_{1}^{(1)}\right )$ equation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.


2019 ◽  
Vol 17 (1) ◽  
pp. 1014-1024
Author(s):  
Hong Yan Xu ◽  
Xiu Min Zheng

Abstract The purpose of this manuscript is to study some properties on meromorphic solutions for several types of q-difference equations. Some exponents of convergence of zeros, poles and fixed points related to meromorphic solutions for some q-difference equations are obtained. Our theorems are some extension and improvements to those results given by Qi, Peng, Chen, and Zhang.


Author(s):  
Giulio Bonelli ◽  
Fabrizio Del Monte ◽  
Alessandro Tanzini

AbstractWe study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $$\tau $$ τ -functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU(2) pure super Yang–Mills and $$N_f=2$$ N f = 2 on a circle.


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