Continuous and lattice integrable systems with a boundary

Cheng Zhang, Leeds & Shanghai.

Classical/quantum Yang-Baxter/reflection equations are key structures in integrability, which guarantee the existence of sufficiently many commuting flows. In this talk, I will give two applications of these structures.

First, I will present a systematic method for solving classes of integrable PDEs on the half-line, called "dressing the boundary". This approach provide a natural algorithm for tackling integrable boundary-value problems, and is also consistent with the unified inverse scattering picture.

Second, in the context of integrable lattice equations, the reflection equation is formulated as boundary consistency and dual boundary consistency, which allow to define dynamics of lattice equations on quad-graphs with boundary. As a by-product, the open-boundary reduction will be presented as a mean for deriving integrable mappings.