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.
Abstract : This talk will provide an overview of recent developments in integrable systems, orthogonal polynomials and asymptotics and the topics underlying the talks in this minisymposium.
[02937] Lagrangian multiform structure of discrete and semi-discrete KP typequations
Format : Talk at Waseda University
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
Format : Talk at Waseda University
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.
[04990] Deformed orthogonal functions and integrable lattices
Format : Talk at Waseda University
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.
[05494] Orthogonal polynomials on elliptic curves and Painlevé VI equation.
Author(s) :
Harini Desiraju (University of Sydney)
Pieter Roffelsen (University of Sydney)
Tomas Latimer (University of Sydney)
Abstract : Elliptic orthogonal polynomials are a family of special functions that satisfy certain orthogonality condition with respect to a weight function on an elliptic curve. Building up on several recent works on the topic, we establish a framework using Riemann-Hilbert problems to study such polynomials. When the weight function is constant, these polynomials relate to the elliptic form of the sixth Painleve equation. This talk is based on a recent work with Tomas Latimer and Pieter Roffelsen (arXiv: 2305.04404).
[04872] On q-Painlevé VI and the geometry of affine Segre surfaces
Format : Talk at Waseda University
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.
[05564] Riemann-HIlbert problem on the q-Painleve equations
Format : Talk at Waseda University
Author(s) :
Yousuke Ohyama (Tokushima University)
Abstract : We study monodromy spaces of $q$-Painleve equations. We apply the Riemann-Hilbert correspondence to analytic studies on $q$-Painleve equations.
[05591] A 3×3 Lax form for the q-P(E_6^{(1)})
Format : Talk at Waseda University
Author(s) :
Kanam Park (Toba college)
Abstract : For the q-Painlevé equation with the affine Weyl group symmetry of type E_6^{(1)}, a 2×2 matrix Lax form and a second order scalar lax form were known.
In this talk, we give a 3×3 matrix Lax form and a third order scalar equation related to it. We also give its continuous limit.
These Lax form and a scalar equation seems to be new.
[03983] Stokes' phenomenon, discretization, and discrete integrability
Format : Talk at Waseda University
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?
[05269] Borel analysis for the first difference q-Painlevé equation
Format : Talk at Waseda University
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
Format : Talk at Waseda University
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.
[05496] Asymptotic prediction of tau-function zeros of Painlevé equations
Author(s) :
Ines Varela Aniceto (University of Southampton)
Abstract : Transseries solutions of Painlevé I and II equations include both algebraic asymptotic expansions and exponentially small corrections, valid in pole-free regions. In this talk I will show how summing all exponential terms at each algebraic order provides an analytic continuation into the pole-filled regions of the solutions, where exponentials are no longer suppressed. The same can done for the respective tau-functions to obtain asymptotic predictions for all the arrays of the tau-function zeros.
