[00170] Integrable systems, orthogonal polynomials and asymptotics
Session Date & Time :
00170 (1/3) : 2C (Aug.22, 13:20-15:00)
00170 (2/3) : 2D (Aug.22, 15:30-17:10)
00170 (3/3) : 2E (Aug.22, 17:40-19:20)
Type : Proposal of Minisymposium
Abstract : Interest in nonlinear dynamical systems has grown dramatically over the past half century. Profound advances have been fueled by the discovery of integrable systems that are applicable in a wide range of applications. In particular, nonlinear ODEs called the Painlev\' equations model applications in many fields, in particular in random matrix theory and growth processes. Their appearance in quantum gravity and orthogonal polynomial theory has led to widening interest in integrable discrete versions of these equations. This minisymposium will bring together recent developments in integrable systems, orthogonal polynomials and asymptotics with a view to describing new special functions.
[02937] Lagrangian multiform structure of discrete and semi-discrete KP typequations
Author(s) :
Frank Willem Nijhoff (University of Leeds)
Abstract : A brief review of of Lagrangian multiform theory for integrable discrete and continuous equations will be presented. As specific examples I will discuss the recently established 3-form structure of the KP hierarchy, and its discrete and semi-discrete counterparts.
[03500] Charge-conserving solutions to the constant Yang-Baxter equations
Author(s) :
Jarmo Hietarinta (University of Turku)
Paul Martin (University of Leeds)
Eric C. Rowell (Texas A&M University)
Abstract :
The Yang-Baxter equation is difficult to solve even in the constant form $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$ and a complete solution is known only for rank two. For further progress it is important to make a meaningful ansatz. Recently Martin and Rowell proposed charge-conservation as an effective constraint (arXiv:2112.04533). We explore the results obtained by a slightly different charge-conservation rule.
[03983] Stokes' phenomenon, discretization, and discrete integrability
Author(s) :
Christopher Lustri (Macquarie University)
Abstract : This talk is concerned with integrability in discrete systems, and its relationship with Stokes' phenomenon. Discrete equations such as the discrete Painleve I equation can be written in terms of an infinite-order differential equation. We will consider a family of equations obtained by truncating this infinite-order differential equation at different orders. In this talk we will answer two questions: (1) How does discretization connect the Stokes' phenomenon in continuous and discrete Painleve I? (2) How does integrability emerge in this family of equations in the discrete limit?
[04872] On q-Painlevé VI and the geometry of affine Segre surfaces
Author(s) :
Pieter Roffelsen (University of Sydney)
Abstract : A famous result by M. Jimbo (1982) relates Painlevé VI to a family of affine cubic surfaces via the Riemann-Hilbert correspondence. In recent work with Nalini Joshi, a $q$-analog of this result was obtained, relating $q$-Painlevé VI to a family of affine Segre surfaces. I will explain this result and show how the geometry of these surfaces is reflected in the asymptotic expansions of solutions around the two critical points of $q$-Painlevé VI.
[04990] Deformed orthogonal functions and integrable lattices
Author(s) :
Xiangke Chang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Abstract : Since the 1990s, the theory of orthogonal polynomials has been increasingly playing an important role in the studies of Toda type lattices, peakon dynamical systems of the Camassa-Holm type, as well as specific Painlevè equations. These integrable lattices can be derived according to deformations of orthogonal functions, directly or indirectly. This talk is devoted to exploring some of related works with focus on our recent results for some new orthogonality.
[05269] Borel analysis for the first difference q-Painlevé equation
Author(s) :
Adri Olde Daalhuis (The University of Edinburgh)
Abstract : We discuss the asymptotics of solutions of the first -difference $q$-Painlevé equation $w(qt)w^2(t)w(t/q)=w(t)-t$. Via the $q$-Borel transform we obtain an interesting singularity distribution in the Borel plane.
[05326] Non-linear Stokes phenomenon for Painleve transcendents and topological recursion
Author(s) :
Kohei Iwaki (The University of Tokyo)
Abstract : I will propose a conjectural statement on the Stokes phenomenon for the topological recursion partition function. Our claim is based on a relation between the topological recursion and the Painleve tau-function through the exact WKB analysis.
