# Registered Data

## [02025] Recent Advances on the Analysis and Applications of Continuous and Discrete Integrable Systems

**Session Date & Time**:- 02025 (1/3) : 3C (Aug.23, 13:20-15:00)
- 02025 (2/3) : 3D (Aug.23, 15:30-17:10)
- 02025 (3/3) : 3E (Aug.23, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: Research in integrable systems has led to numerous important new concepts, ideas and techniques of mathematics and physics in the last few decades. The very concept of integrability has been found in deep connections with a large spectrum of mathematics such as algebraic geometry, differential geometry, representation theory, random matrix theory, nonlinear waves, etc. In this minisymposium, we will focus on the recent developments of both continuous and discrete integrable systems. Specially, it will cover discrete and ultradiscrete integrable systems, noncommutative integrable systems and rogue waves. This is a unique platform for the interaction of international researchers in relevant fields.**Organizer(s)**: Kenichi Maruno, Linyu Peng, Cheng Zhang**Classification**:__39A36__,__37K10__,__35Q51__,__37J70__,__35C08__**Speakers Info**:- Cheng Zhang (Shanghai University )
- Claire Gilson (University of Glasgow)
- Kazushige Endo (National Institute of Technology, Gunma College)
- Andy Hone (University of Kent)
- Rod Halburd (University College London)
- Yaqing Liu (Beijing Information Science and Technology University)
- Hidetomo Nagai (Tokai University)
- Irina Kogan (North Carolina State University)
- Zhiwei Wu (Sun Yat-Sen University)
- Sikarin Yoo-Kong (Naresuan University)
- Chengfa Wu (Shenzhen University)
- Ting-Jung Kuo (National Taiwan Normal University)

**Talks in Minisymposium**:**[03194] A Generalization of an Integrability Theorem of Darboux and the Stable Configuration Condition.****Author(s)**:**Irina A. Kogan**(North Carolina State University)

**Abstract**: In his monograph "Systemes Orthogonaux" (1910), Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of PDEs, where for each unknown function a certain subset of partial derivatives is prescribed and the values of the unknown functions are prescribed along transversal coordinate affine subspaces. The more general of the theorems, Theorem III, was proved by Darboux only for the cases of 2 and 3 independent variables. We formulate and prove a generalization of Theorem III. Instead of partial derivatives, we prescribe derivatives of the unknown functions along vector fields. The values of the unknown functions are prescribed along arbitrary transversal submanifolds. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the set of vector fields and on the initial manifolds. SCC is automatically met in the case considered by Darboux. Assuming the SCC and the relevant integrability conditions are satisfied, we establish local existence and uniqueness of a $C^1$-smooth solution via Picard iteration for any number of independent variables. If the SCC is not satisfied, we show on a concrete example that the uniqueness can fail the following strong sense: for the same initial data, there are two solutions that differ on any open subset of their domains. This talk is based on joint publications with Michael Benfield and Kris Jenssen.

**[03221] Geometric Aspects of Miura Transformations****Author(s)**:- Changzheng Qu (Ningbo University)
**Zhiwei Wu**(Sun Yat-sen University)

**Abstract**: The Miura transformation and its extensions play a crucial role in the study of integrable systems, which have been used to relate different kinds of integrable equations and to classify the bi-Hamiltonian structures. In this talk, we will discuss the geometric aspects of the Miura transformation. The generalized Miura transformations from the mKdV-type hierarchies to the KdV-type hierarchies are constructed under both algebraic and geometric settings. We show that the Miura transformations not only relate integrable curve flows in different geometries but also induce the transition between different moving frames. Moreover, the Miura transformation gives the factorization of generating operators of constraint Gelfand-Dickey hierarchy. This talk is based on a recent joint work with Prof. Qu, Changzheng.

**[03369] Quantum variational principle for Lagrangian 1-forms****Author(s)**:**Sikarin Yoo-Kong**(Naresuan University)

**Abstract**: In this talk, we will present a new type of the propagator associated with the Lagrangian 1-forms called the (continuous) multi-time propagator. With this new type of the propagator, a new paradigm on summing over possible paths arises since one needs to take into account not only summing over possible spatial paths but also summing over possible temporal paths. The quantum intragrability (a.k.a multi-dimensional consistency), which mainly relies on the classical Lagrangian 1-form closure relation, will be captured in the language of Feynman path integration.

**[03558] Integrable boundary conditions for quad-graph systems: classification and applications****Author(s)**:**Cheng Zhang**(Shanghai University)

**Abstract**: The notion of boundary conditions for quad-graph systems will first be introduced. The boundary conditions are naturally defined on triangles that arise as dualization of given quad-graphs with boundary. For three-dimensionally consistent quad-graph systems, the so-called integrable boundary conditions will be characterized as boundary conditions satisfying the boundary consistency condition that is a consistency condition defined on a half of a rhombic-dodecahedron. Based on these notions, three main results will then be presented: a classification of integrable boundary conditions for quad-equations of the ABS classification; Lax formulations of integrable boundary conditions; and the so-called open boundary reduction technique as systematic a means to construct integrable mappings from integrable initial-boundary value problems for quad-graph systems.

**[03560] Rogue waves and their patterns in the vector nonlinear Schrödinger equation****Author(s)**:- Baofeng Feng (University of Texas Rio Grande Valley )
- Peng Huang (Shenzhen University)
**Chengfa Wu**(Shenzhen University)- Guangxiong Zhang (Shenzhen University)

**Abstract**: This talk presents the general rogue wave solutions and their patterns in the vector (or M-component) nonlinear Schrödinger (NLS) equation. We derived the explicit solution for the rogue wave expressed by tau-functions that are determinants of K×K block matrices with an index jump of M+1. Patterns of the rogue waves for M=3,4 and K=1 are thoroughly investigated.

**[03656] Whitham modulation theory of Riemann problem for nonlinear integrable equations****Author(s)**:**Yaqing Liu**(Beijing Information Science and Technology University)

**Abstract**: The Riemann problem of the nonlinear integrable equation with step-like initial value is explored by Whitham modulation theory, which is a modified version of the well-known finite-gap integration method. Based on the reparameterization of the solution with the use of algebraic resolvent of the polynomial defining the solution, the periodic wave solutions of the nonlinear integrable equation are described by the elliptic function along with the Whitham modulation equations. Complete classification of possible wave structures is given for all possible jump conditions at the discontinuity initial value. The proposed analytic results are confirmed through direct numerical simulations.

**[03957] Addition formulae for ultradiscrete hafnians****Author(s)**:**Hidetomo Nagai**(Tokai University)

**Abstract**: Ultradiscrete hafnian is an ultradiscrete analogue of hafnian, which is signature free pfaffian. In this talk we propose some formulae for the ultradiscrete hafnians with some conditions, which are related to the ultradiscrete soliton solutions.

**[04193] Curvature equation with conic singularities and integrable system****Author(s)**:**Ting-Jung Kuo**(National Taiwan Normal University)

**Abstract**: Let $\left( E_{\tau },dz^{2}\right) $, $\tau \in \mathbb{H}$ be a flat torus. We consider the follwing PDE: \begin{equation} \Delta u+e^{u}=\sum_{j=1}^{N}4\pi \alpha _{j}\delta _{p_{j}}\text{ on } E_{\tau } ~~~~~(1) \end{equation} where $\delta _{p_{j}}$ is the dirac measure and $\alpha _{j}>-1$. In the literature, equation (1) arised from conformal geometry. Indeed, (1) is equivalent to saying that the conformal metric $ds^{2}=\frac{1}{2 }e^{u}\left\vert dz\right\vert ^{2}$ with conic singularties at $p_{j}$ has the Gaussian curvature $1$. By classical Liouville theorem, the curvature equation is also an integrable system which yields a complex ODE (a generalization of the classical Lame equation ) and the solvability of equation (1) is equivalent to saying that the corresponding complex ODE is always apparent and has unitary monodromy. The study of the monodromy of a general complex ODE is difficult in general. However, recently, we also discover its relation with KdV theory. In this talk, I will talk about this deep connection and focus on the study of the complex ODE from monodromy point of view.

**[05088] Beyond the Painlevé property****Author(s)**:**Rod Halburd**(University College London)

**Abstract**: The Painlevé property and associated tests have led to the identification of integrable cases of many families of equations. In this talk I will describe some generalisations of the Painlevé property that allow for the identification of a wider set of integrable equations. Various necessary conditions will be discussed and implemented. Examples from Newtonian and relativistic stellar models will be analysed

**[05092] Recent Advances on the Analysis and Applications of Continuous and Discrete Integrable Systems****Author(s)**:**Andrew Hone**(University of Kent)

**Abstract**: "New discrete integrable systems from deformed cluster mutations" We describe how to obtain integrable maps by deforming cluster algebra mutations that display Zamolodchikov periodicity. The simplest example is the general Lyness map in 2D, arising as a 2-parameter family of deformations of the cluster algebra of finite type A_2. Results will be presented on integrability of deformed mutations in type A_n, and other finite root systems. This is joint work with J.Grabowski, W.Kim, and T.Kouloukas.

**[05098] Three-dimensional fundamental diagram of stochastic cellular automata****Author(s)**:**Kazushige Endo**(Kindai University)

**Abstract**: Cellular automata including Burgers cellular automaton are not only examples of ultradiscrete analogue of integrable systems, but also mathematical models which show fundamental mechanisms of traffic flow. For example, a phase transition from free flow to congested flow in a traffic system is well-known and has been studied using fundamental diagram. Fundamental diagram is an object showing relation between the density of traffic (particles) and their mean momentum. However, the density of particles is not a unique parameter to determine the mean momentum. Several systems whose mean momentum is uniquely determined by a pair of the density and another conserved quantity have been discovered. In this talk, we show a three-dimensional framework of the fundamental diagram of stochastic cellular automata and its theoretical derivation.

**[05195] Constructing non-commutative systems with Pfaffian type solutions****Author(s)**:**Claire Gilson**(University of Glasgow)

**Abstract**: In this talk we look to construct non-commutative systems from quasi-determinants of Pfaffian type by considering quasi-determinant identities.