scholarly journals From Heun Class Equations to Painlevé Equations

2021 ◽  
Author(s):  
Jan Dereziński ◽  
◽  
Artur Ishkhanyan ◽  
Adam Latosiński ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Direct Relationship ◽  
Time Variable ◽  
Painlevé Equations ◽  
Complex Domain ◽  
Heun Equation ◽  
Painleve Equations ◽  
Time Variables

In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonlogarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can be also divided into 5 supertypes, and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the ''time variable'' may depend on the type. We describe unified treatments for several possible ''time variables''.

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]).

New exact solutions of the Boussinesq equation

1990 ◽  
Vol 1 (3) ◽  
pp. 279-300 ◽  
Author(s):  
Peter A. Clarkson
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Boussinesq Equation ◽  
Painlevé Equations ◽  
Painleve Equations ◽  
New Exact Solutions ◽  
New Space ◽  
Manipulation Language ◽  
Similarity Reductions

In this paper new exact solutions are derived for the physically and mathematically significant Boussinesq equation. These are obtained in two different ways: first, by generating exact solutions to the ordinary differential equations which arise from (classical and nonclassical) similarity reductions of the Boussinesq equation (these ordinary differential equations are solvable in terms of the first, second and fourth Painlevé equations); and second, by deriving new space-independent similarity reductions of the Boussinesq equation. Extensive sets of exact solutions for both the second and fourth Painlevé equations are also generated. The symbolic manipulation language MACSYMA is employed to facilitate the calculations involved.

scholarly journals The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles

2019 ◽  
Vol 17 (1) ◽  
pp. 1014-1024
Author(s):  
Hong Yan Xu ◽  
Xiu Min Zheng
Keyword(s):  
Fixed Points ◽  
Difference Equations ◽  
Painlevé Equations ◽  
Meromorphic Solutions ◽  
Painleve Equations ◽  
Zeros And Poles

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.

scholarly journals BPS Quivers of Five-Dimensional SCFTs, Topological Strings and q-Painlevé Equations

2021 ◽  
Author(s):  
Giulio Bonelli ◽  
Fabrizio Del Monte ◽  
Alessandro Tanzini
Keyword(s):  
Topological String ◽  
Cluster Algebra ◽  
Painlevé Equations ◽  
Partition Functions ◽  
Quantum Field Theories ◽  
Particle Spectrum ◽  
Yang Mills ◽  
Painleve Equations ◽  
Rank One ◽  
Bilinear Equations

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.

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.

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