scholarly journals Classical solutions of the third Painlevé equation

1995 ◽  
Vol 139 ◽  
pp. 37-65 ◽  
Author(s):  
Yoshihiro Murata
Keyword(s):  
Painlevé Equations ◽  
Painlevé Equation ◽  
Classical Solutions ◽  
Painleve Equations ◽  
The Third ◽  
Painleve Equation

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].

scholarly journals Remarks on the Yablonskii-Vorob’ev polynomials

2000 ◽  
Vol 159 ◽  
pp. 87-111 ◽  
Author(s):  
Makoto Taneda
Keyword(s):  
Algebraic Approach ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Special Polynomials ◽  
Painleve Equation ◽  
Second Painlevé Equation

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.

scholarly journals Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

2011 ◽  
Vol 467 (2136) ◽  
pp. 3443-3468 ◽  
Author(s):  
Nalini Joshi ◽  
Yang Shi
Keyword(s):  
Exact Solutions ◽  
Linear Problem ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Group Symmetry ◽  
Rational Solutions ◽  
Painleve Equations ◽  
Discrete Painlevé Equations ◽  
Painleve Equation ◽  
Second Painlevé Equation

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.

scholarly journals ON A NOVEL CLASS OF INTEGRABLE ODEs RELATED TO THE PAINLEVÉ EQUATIONS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250211
Author(s):  
ATHANASSIOS S. FOKAS ◽  
DI YANG
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Lax Pair ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Independent Variable ◽  
Painlevé Ii ◽  
Painleve Equation

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.

scholarly journals Solutions of the third Painlevé equation I

1998 ◽  
Vol 151 ◽  
pp. 1-24 ◽  
Author(s):  
Hiroshi Umemura ◽  
Humihiko Watanabe
Keyword(s):  
Painlevé Equation ◽  
Classical Solutions ◽  
The Third ◽  
Painleve Equation ◽  
Algebraic Solutions

Abstract.We classify transcendental classical solutions of the third Painlevé equation. This result combined with the list of algebraic solutions in [11] gives a complete table of classical solutions of the third Painlevé equation.

scholarly journals $q$ -Racah Ensemble and Discrete Painlevé Equation

2019 ◽  
Vol 2020 (24) ◽  
pp. 9797-9843 ◽  
Author(s):  
Anton Dzhamay ◽  
Alisa Knizel
Keyword(s):  
Orthogonal Polynomial ◽  
Lax Pair ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Symmetry Structure ◽  
The One ◽  
The Relationship ◽  
Gap Probabilities ◽  
Discrete Painlevé Equation

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.

scholarly journals Solutions of the second and fourth Painlevé equations, I

1997 ◽  
Vol 148 ◽  
pp. 151-198 ◽  
Author(s):  
Hiroshi Umemura ◽  
Humihiko Watanabe
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Painlevé Equations ◽  
Necessary Condition ◽  
Hamiltonian Vector Field ◽  
Rigorous Proof ◽  
Painleve Equations ◽  
Algebraic Differential Equations ◽  
The Third ◽  
Principal Ideals

AbstractA rigorous proof of the irreducibility of the second and fourth Painlevé equations is given by applying Umemura’s theory on algebraic differential equations ([26], [27], [28]) to the two equations. The proof consists of two parts: to determine a necessary condition for the parameters of the existence of principal ideals invariant under the Hamiltonian vector field; to determine the principal invariant ideals for a parameter where the principal invariant ideals exist. Our method is released from complicated calculation, and applicable to the proof of the irreducibility of the third, fifth and sixth equation (e.g. [32]).

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