Vincent Caudrelier 博士学术报告
Title: Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies
Speaker: Vincent Caudrelier
Affiliation: University of Leeds
Time: 15:30-16:30, Tuesday, 10th May, 2022 (UTC+8, Beijing Time)
Venue：Zoom Meeting
Inviter: Ryo SUZUKI
Abstract
The notion of integrability for classical (field) theories has been almost entirely studied from the Hamiltonian point of view since the early days of the modern theory of integrable systems. In 2009, the notion of Lagrangian multiform was first put forward by Lobb and Nijhoff as a purely Lagrangian framework to capture integrability. The main idea is to formulate a generalised variational principle for an action involving a certain differential form whose coefficients are interpreted as Lagrangians for a hierarchy. Since its proposal, this idea has flourished in various directions and I will review the main developments for classical field theories in 1+1 dimensions.
Two key ingredients are the multiform Euler-Lagrange equations and the so-called closure relation, both of which derive from the generalised variational principle. In this talk, I will present the connection between Lagrangian multiform theory and the well-established theory of the classical r-matrix which had a purely Hamiltonian interpretation so far. I will explain how the classical Yang-Baxter equation underpins the fundamental properties of a certain Lagrangian multiform and the corresponding zero curvature equations. A large variety of known hierarchies are contained as special cases, such as the Ablowitz-Kaup-Newell-Segur hierarchy, the sine-Gordon (sG) hierarchy and various hierarchies related to Zakharov-Mikhailov type models which contain the Faddeev-Reshetikhin (FR) model and recently introduced deformed sigma/Gross-Neveu models as particular cases.
Time permitting, I will also illustrate the versatility of our method by showing how to construct new examples of integrable field theories and their hierarchies by coupling integrable hierarchies together. We provide two examples: the coupling of the nonlinear Schrödinger system to the FR model and the coupling of sG with the anisotropic FR model.
This most recent results are based on the joint work arXiv:2201.08286 with M. Stoppato and B. Vicedo.
About the Speaker
Dr. Vincent Caudrelier is currently a lecturer in Mathematical Physics at the School of Mathematics, University of Leeds, UK. He has an MSc from the French 'Grande Ecole' Supaero and an MSc from the University of Cambridge, both obtained in 2002. He obtained a PhD in Theoretical Physics in 2005 from the Laboratoire d'Annecy-le-vieux de Physique Théorique, Université de Savoie. He went on to do a post-doc at the Department of Mathematics, University of York, as an EPSRC research fellow. This led to his hiring at City University London in 2007. In 2016, he joined the School of Mathematics at the University of Leeds.
His areas of expertise include classical and quantum integrable systems; classical and quantum inverse scattering method; boundaries and defects; integrable PDEs on graphs; Yang-Baxter and reflection equations.
The seminar will be broadcasted online by Zoom.
Interested people are free to join, without registration in advance.
The Zoom info is
URL: https://us02web.zoom.us/j/3854420225?pwd=SXY4eWJKOTBFZWJDaE16aXpTamY1QT09
Meeting ID: 385 442 0225
Passcode: yauc
Slides--Vincent Caudrelier--Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies-20220510.pdf