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Advanced Research in Mathematical Physics Models with Painlevé Property and...
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Advanced Research in Mathematical Physics Models with Painlevé Property and Affine Weyl Group Symmetry

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: 31 December 2025 | Viewed by 4990

Special Issue Editor

Prof. Dr. Henrik Aratyn

E-Mail Website
Guest Editor
Department of Physics, University of Illinois at Chicago, 845 W. Taylor St., Chicago, IL 60607-7059, USA
Interests: theoretical particle physics; mathematical physics

Special Issue Information

Dear Colleagues,

We are pleased to announce a call for papers for a Special Issue dedicated to advanced research in mathematical physics models with the Painlevé property and affine Weyl group symmetry. Infinite-dimensional Lie algebras play a key role in many developments in integrable models that give rise to Painlevé equations in their self-similarity limits. These equations are invariant under symmetry groups of the Bäcklund transformations that form the affine Weyl groups. This connection relates the Painlevé property to the concept of integrability for non-linear differential equations.

This Special Issue seeks to exploit a link between infinite-dimensional Lie algebras and integrable systems and affine Weyl groups and Painlevé equations to form a unifying perspective that binds together a wide array of the latest developments on a wide variety of topics that include, but are not limited to:

  • A remarkable connection between Painlevé equations and a wide range of problems of mathematical physics, random matrix theory, etc., where Painlevé equations find active applications.
  • Novel descriptions of solutions of Painlevé equations, including rational solutions, the classical special functions such as hypergeometric-type solutions, transcendental functions and Umemura polynomials aided by various algebraic and group theoretical methods and techniques.
  • Discrete Painlevé equations and their relations to the symmetry structure of Painlevé equations and orthogonal polynomials.
  • Studies of the symmetry structure of Painlevé equations and their generalizations in terms of root systems associated with affine Weyl groups and their applications to solutions, including study of zeros and poles of rational solutions.
  • Origins of Painlevé equations as self-similarity limits of integrable models as studied by Lax, Hamiltonian and zero curvature approaches. Connection to various integrable models, e.g., Calogero–Moser models, Toda models, Volterra models and dressing chain equations.

We invite original research articles, review articles and short communications that address the above topics or related areas. All manuscripts will be subject to a rigorous peer review process, and accepted papers will be published in this Special Issue of the journal. We welcome submissions from researchers in mathematics, physics and related fields who are active in the study of integrable models and Painlevé equations in mathematical physics settings and interested in advances achieved through applications of symmetry considerations.

We look forward to your participation and contributions to this Special Issue. 

Prof. Dr. Henrik Aratyn
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematical Physics
  • integrable models
  • Painlevé equations
  • applications of symmetry

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Published Papers (4 papers)

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Research

28 pages, 518 KiB  
Article
What Is the Symmetry Group of a d-PII Discrete Painlevé Equation?
by Anton Dzhamay, Yang Shi, Alexander Stokes and Ralph Willox
Mathematics 2025, 13(7), 1123; https://doi.org/10.3390/math13071123 - 28 Mar 2025
Viewed by 184
Abstract
The symmetry group of a (discrete) Painlevé equation provides crucial information on the properties of the equation. In this paper, we argue against the commonly held belief that the symmetry group of a given equation is solely determined by its surface type as [...] Read more.
The symmetry group of a (discrete) Painlevé equation provides crucial information on the properties of the equation. In this paper, we argue against the commonly held belief that the symmetry group of a given equation is solely determined by its surface type as given in the famous Sakai classification. We will dispel this misconception by using a specific example of a d-PII equation, which corresponds to a half-translation on the root lattice dual to its surface-type root lattice but becomes a genuine translation on a sub-lattice thereof that corresponds to its real symmetry group. The latter fact is shown in two different ways, first by a brute force calculation, and then through the use of normalizer theory, which we believe to be an extremely useful tool for this purpose. We finish the paper with the analysis of a sub-case of our main example, which arises in the study of gap probabilities for Freud unitary ensembles, and the symmetry group of which is even further restricted due to the appearance of a nodal curve on the surface on which the equation is regularized. Full article
(This article belongs to the Special Issue Advanced Research in Mathematical Physics Models with Painlevé Property and Affine Weyl Group Symmetry)
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25 pages, 408 KiB  
Article
Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions
by Henrik Aratyn, José Francisco Gomes, Gabriel Vieira Lobo and Abraham Hirsz Zimerman
Mathematics 2024, 12(23), 3701; https://doi.org/10.3390/math12233701 - 26 Nov 2024
Viewed by 621
Abstract
The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N, the symmetry group A^N−1(1) contains [...] Read more.
The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N, the symmetry group A^N−1(1) contains the conventional Bäcklund transformations sj,j=1,…,N, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of N points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to the existence of degenerated solutions, and for N=4, we explicitly show how even reflection automorphisms cause degeneracy of a class of rational solutions obtained on the orbit of the translation operators of A^3(1). We obtain the closed expressions for the solutions and their degenerated counterparts in terms of the determinants of the Kummer polynomials. Full article
(This article belongs to the Special Issue Advanced Research in Mathematical Physics Models with Painlevé Property and Affine Weyl Group Symmetry)
13 pages, 278 KiB  
Article
Polynomial Tau-Functions of the n-th Sawada–Kotera Hierarchy
by Victor Kac and Johan van de Leur
Mathematics 2024, 12(5), 681; https://doi.org/10.3390/math12050681 - 26 Feb 2024
Cited by 2 | Viewed by 2427
Abstract
We give a review of the B-type Kadomtsev–Petviashvili (BKP) hierarchy and find all polynomial tau-functions of the n-th reduced BKP hierarchy (=n-th Sawada–Kotera hierarchy). The name comes from the fact that, for n=3, the simplest equation of [...] Read more.
We give a review of the B-type Kadomtsev–Petviashvili (BKP) hierarchy and find all polynomial tau-functions of the n-th reduced BKP hierarchy (=n-th Sawada–Kotera hierarchy). The name comes from the fact that, for n=3, the simplest equation of the hierarchy is the famous Sawada–Kotera equation. Full article
(This article belongs to the Special Issue Advanced Research in Mathematical Physics Models with Painlevé Property and Affine Weyl Group Symmetry)
21 pages, 1097 KiB  
Article
Classification of Real Solutions of the Fourth Painlevé Equation
by Jeremy Schiff and Michael Twiton
Mathematics 2024, 12(3), 463; https://doi.org/10.3390/math12030463 - 31 Jan 2024
Viewed by 1018
Abstract
Painlevé transcendents are usually considered as complex functions of a complex variable, but, in applications, it is often the real cases that are of interest. We propose a classification of the real solutions of Painlevé IV (or, more precisely, symmetric Painlevé IV) using [...] Read more.
Painlevé transcendents are usually considered as complex functions of a complex variable, but, in applications, it is often the real cases that are of interest. We propose a classification of the real solutions of Painlevé IV (or, more precisely, symmetric Painlevé IV) using a sequence of symbols describing their asymptotic behavior and singularities. We give rules for the sequences that are allowed (depending on the parameter values). We look in detail at the existence of globally nonsingular real solutions and determine the dimensions of the spaces of such solutions with different asymptotic behaviors. We show that for a generic choice of the parameters, there exists a unique finite sequence of singularities for which Painlevé IV has a two-parameter family of solutions with this singularity sequence. There also exist solutions with singly infinite and doubly infinite sequences of singularities, and we identify which such sequences are possible. We look at the singularity sequences (and asymptotics) of special solutions of Painlevé IV. We also show two other results concerning special solutions: rational solutions can be obtained as the solution of a set of polynomial equations, and other special solutions can be obtained as the solution of a first-order differential equation. Full article
(This article belongs to the Special Issue Advanced Research in Mathematical Physics Models with Painlevé Property and Affine Weyl Group Symmetry)
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