Abstract

I will first give an overview of some aspects of integrable quad-graph systems . This includes the notion of multi-dimensional consistency, classification results of quad-equations, connections to complex analysis. Then, I will show how to implement initial-boundary value problems for quad-graph systems. It relies on the notion of boundary consistency that defines integrable boundary conditions for quad-graph systems. I will also discuss classification of integrable boundary conditions; their Lax formulations; and the so-called open boundary reduction technique as a systematic means to construct integrable mappings. This talk is mainly based on [Caudrelier, Crampe, CZ, Sigma, 10(014), 2014], [Caudrelier, van der Kamp, CZ, IMRN, rnac207, 2021] and [Sun, CZ, IMRN, rnab188, 2022].