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 $q$ -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 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 POLYNOMIAL HAMILTONIANS FOR QUANTUM PAINLEVÉ EQUATIONS

2009 ◽  
Vol 20 (11) ◽  
pp. 1335-1345 ◽  
Author(s):  
YUICHI UENO
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Point Of View ◽  
Painlevé Equations ◽  
Canonical Transformations ◽  
Painleve Equations ◽  
Quantum Version ◽  
Noncommutative Polynomial

Recently, a quantum version of Painlevé equations from the point of view of their symmetries was proposed by Nagoya. These quantum Painlevé equations can be written as Hamiltonian systems with a (noncommutative) polynomial Hamiltonian H J . We give a characterization of the quantum Painlevé equations by certain holomorphic properties. Namely, we introduce canonical transformations such that the Painlevé Hamiltonian system is again transformed into a polynomial Hamiltonian system, and we show that the Hamiltonian can be uniquely characterized through this holomorphic property.

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 V. On periodic solutions of Hamiltonian systems differential equations

1928 ◽  
Vol 227 (647-658) ◽  
pp. 137-221 ◽  
Keyword(s):  
Hamiltonian System ◽  
Differential Equations ◽  
Ab Initio ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Celestial Mechanics ◽  
Divergent Series ◽  
Hamiltonian Form ◽  
Systems Of Differential Equations ◽  
Independent Variable

In the following pages it is proposed to develop ab initio a theory of periodic solutions of Hamiltonian systems of differential equations. Such solutions are of theoretical importance for the following reason: that whereas the attempt to obtain, for a real Hamiltonian system, solutions valid for all real values of the independent variable leads in general to divergent series, for certain solutions which are formally periodic the series can be proved convergent. In the words of Poincare, “ce qui nous rend ces solutions périodiques si précieuses, c’est qu’elles sont, pour ainsi dire, la seule breche paroù nous puissions essayer de pénétrer dans une place jusqu'ici reputee inabordable. The existing theory of periodic solutions of differential equations was developed by Poincare mainly with reference to the equations of Celestial Mechanics. With a suitable choice of co-ordinates these are of the Hamiltonian form.

scholarly journals On the asymptotic representation of the solutions to the fourth general Painlevé equation

2003 ◽  
Vol 2003 (13) ◽  
pp. 845-851
Author(s):  
Youmin Lu
Keyword(s):  
Asymptotic Representation ◽  
Painlevé Equation ◽  
Painlevé Transcendents ◽  
Painlevé Transcendent ◽  
Straight Line ◽  
Painleve Transcendents ◽  
Independent Variable ◽  
Painleve Equation

There have been many results on the asymptotics of the Painlevé transcendents in recent years, but the asymptotics of the fourth Painlevé transcendent has not been studied much. In this note, we study the general fourth Painlevé equation and develop an asymptotic representation of a group of its solutions as the independent variable approaches infinity along a straight line.

Bäcklund transformation of Frobenius Painlevé equations

2018 ◽  
Vol 32 (17) ◽  
pp. 1850181 ◽  
Author(s):  
Haifeng Wang ◽  
Chuanzhong Li
Keyword(s):  
Lax Pair ◽  
Painlevé Equations ◽  
Bäcklund Transformations ◽  
Kp Hierarchy ◽  
Painleve Equations ◽  
Virasoro Constraint ◽  
Backlund Transformations ◽  
Hirota Bilinear Equation ◽  
Hirota Bilinear ◽  
Bilinear Equation

In this paper, in order to generalize the Painlevé equations, we give a [Formula: see text]-Painlevé IV equation which can apply Bäcklund transformations to explore. And these Bäcklund transformations can generate new solutions from seed solutions. Similarly, we also introduce a Frobenius Painlevé I equation and Frobenius Painlevé III equation. Then, we find the connection between the Frobenius KP hierarchy and Frobenius Painlevé I equation by the Virasoro constraint. Further, in order to seek different aspects of Painlevé equations, we introduce the Lax pair, Hirota bilinear equation and [Formula: see text] functions. Moreover, some Frobenius Okamoto-like equations and Frobenius Toda-like equations can also help us to explore these equations.

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