VIRTUAL SEMINAR SERIES -- INTEGRABLE SYSTEMS: Covariant Poisson bracket and r-matrix structure for an integrable hierarchy: role of Lagrangian and Hamiltonian multiforms
Vincent Caudrelier (University of Leeds)
Wednesday 14th October, 2020 13:30-14:30 Zoom seminar hosted by ICMS
I will present new developments coming from the convergence of initially separate motivations. One was the question of formulating a given integrable classical field theory in 1+1 dimensions as a covariant Hamiltonian field theory and obtaining a classical r-matrix structure for the corresponding covariant Poisson bracket. This originated from earlier observations of a space-time duality in the r-matrix structure of a Lax pair which begged for a treatment where space and time variables are treated on equal footing, as opposed to the dominating standard Hamiltonian approach to integrable systems. Another motivation was the problem of encoding integrability in a purely variational fashion which led to the notion of Lagrangian multiforms introduced by Lobb and Nijhoff. It turns out that when one wants to extend the covariant Hamiltonian picture of an integrable classical field theory to the entire hierarchy it belongs to, Lagrangian multiforms play a crucial role in defining the required generalization of the covariant Poisson bracket which is called a multi-time Poisson bracket. The associated notion of Hamiltonian multiform, introduced by Caudrelier and Stoppato, is then needed to formulate the entire set of zero curvature equations as (multiform) Hamilton equations. All along, I will use the example of the AKNS hierarchy which contains the nonlinear Schrödinger equation and modified KdV equation as its most famous examples. This is joint work with M. Stoppato.
For more information, and how to access the seminar through Zoom, see the webpage of the events here.