Symmetries and Integrability of Difference Equations: Lecture Notes of the Abecederian School of SIDE 12, Montreal 2016 (CRM Series in Mathematical Physics)

This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers in the specific field of their expertise and, in turn, written for early career researchers. As a survey of the current state of the art, this book will serve as a valuable reference and is particularly well suited as an introduction to the field of symmetries and integrability of difference equations. Therefore, the book will be welcomed by advanced undergraduate and graduate students as well as by more advanced researchers. Front Matter....Pages i-x Continuous, Discrete and Ultradiscrete Painlevé Equations....Pages 1-41 Elliptic Hypergeometric Functions....Pages 43-74 Integrability of Difference Equations Through Algebraic Entropy and Generalized Symmetries....Pages 75-151 Introduction to Linear and Nonlinear Integrable Theories in Discrete Complex Analysis....Pages 153-193 Discrete Integrable Systems, Darboux Transformations, and Yang–Baxter Maps....Pages 195-260 Symmetry-Preserving Numerical Schemes....Pages 261-324 Introduction to Cluster Algebras....Pages 325-357 An Introduction to Difference Galois Theory....Pages 359-390 Lectures on Quantum Integrability: Lattices, Symmetries and Physics....Pages 391-430 Back Matter....Pages 431-435