VIRTUAL SEMINAR SERIES -- INTEGRABLE SYSTEMS: Integrable boundary conditions for equations on quad-graphs, open boundary reductions and integrable mappings

Vincent Caudrelier (University of Leeds)

Wednesday 13th October 13:30-14:30 Zoom seminar hosted by ICMS

Abstract

I will present the notion of integrable boundary conditions for partial difference equations formulated on the so-called quad-graphs. It relies on an appropriate adaptation of the established notion of bulk integrability based on the consistency around the cube (or multidimensional consistency). I will explain the origin of these ideas. This will take me briefly into the realm of the set-theoretical Yang-Baxter equation and its companion, the set-theoretical reflection equation. I will then introduce the method of open (boundary) reductions, as an alternative to the well-known method of periodic reductions, for constructing discrete integrable mappings and their invariants. The invariants are constructed using Sklyanin's double-row monodromy matrix and this requires the introduction of the notion of reflection matrix and (discrete) boundary zero curvature condition in this context. We focus on examples from the Adler-Bobenko-Suris classication and associated
integrable boundary equations, and on the simplest case of the two-dimensional lattice. This
presentation is based on joint work with Nicolas Crampé, Peter van der Kamp and Cheng Zhang.

 

For more information, and how to access the seminar through Zoom, see the webpage of the events here.

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