A method for solving integrable nonlinear PEs, Sotiris Konstantinou-Rizos

Sotiris Konstantinou-Rizos from Yaroslavl State University. Title: A method for solving integrable nonlinear PΔEs Abstract: It has become understood over the past few decades that integrable systems of partial difference equations (PΔEs) are interesting in their own right. On one hand, they model many processes in nature, the industry and the IT sector, and, on the other hand, they have many interesting algebro-geometric properties, as well as they are related to many important equations of Mathematical Physics such as the Yang-Baxter equation and the Zamolodchikov tetrahedron equation. In this talk we present a discrete Darboux-Lax scheme for constructing solutions to quad-graph equations that do not necessarily possess the 3D-consistency property. As an illustrative example we use an Adler-Yamilov type of system that is the compatibility condition of two Darboux transformations for the nonlinear Schroedinger (NLS) equation. For this Adler-Yamilov system we construct 1- and 2- soliton solutions, starting from a simple seed one. Moreover, we present integrable discretisations of a noncommutative NLS equation. This talk is based on some recent results obtained in collaboration with P. Xenitidis and X. Fisenko.