Discrete Systems and Integrability

Description

This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. While treating the material at an elementary level, the book also highlights many recent developments. Topics include: Darboux and Bäcklund transformations, difference equations and special functions, multidimensional consistency of integrable lattice equations, associated linear problems (Lax pairs), connections with Padé approximants and convergence algorithms, singularities and geometry, Hirota’s bilinear formalism for lattices, intriguing properties of discrete Painlevé equations and the novel theory of Lagrangian multiforms. The book builds the material in an organic way, emphasizing interconnections between the various approaches, while the exposition is mostly done through explicit computations on key examples. Written by respected experts in the field, the book’s numerous exercises and thorough list of references will benefit both upperlevel undergraduate and beginning graduate students as well as researchers from other disciplines.JARMO HIETARINTA is Professor Emeritus of Theoretical Physics at the Universityof Turku, Finland. His work has focused on the search for integrable systems of variousforms, including Hamiltonian mechanics, Hirota bilinear form, Yang–Baxter and tetrahedronequations as well as lattice equations. He was instrumental in the setting up of thenlin.SI category in arxiv.org and created the web pages for the SIDE (Symmetries andIntegrability of Difference Equations) conference series: http://side-conferences.netNALINI JOSHI is Professor of Applied Mathematics at the University of Sydney. Sheis best known for her work on the Painlevé equations and works at the leading edge ofinternational efforts to analyze discrete and continuous integrable systems in the geometricsetting of their initial-value spaces, constructed by resolving singularities in complex projectivespace. She was elected as a Fellow of the Australian Academy of Science in 2008,holds a Georgina Sweet Australian Laureate Fellowship and was awarded the special HardyFellowship of the London Mathematical Society in 2015.FRANK NIJHOFF is Professor of Mathematical Physics in the School of Mathematicsof the University of Leeds. His research focuses on nonlinear difference and differentialequations, symmetries and integrability of discrete systems, variational calculus, quantumintegrable systems and linear and nonlinear special functions. He was the principalorganizer of the 2009 six-month programme on Discrete Integrable Systems at the IsaacNewton Institute, and a Royal Society Leverhulme Trust Senior Research Fellow in 2011.