Title: Discrete Painlevé equations /
Authors: Joshi, Nalini; Conference Board of the Mathematical Sciences; National Science Foundation (U.S.)
Year: 2019
Subjects: Painlevé equations; Differential equations, Nonlinear; Differential equations, Nonlinear; Painlevé equations; Special functions (33-XX deals with the properties of functions as functions) -- Other special functions -- PainlevÃ©-type functions.; Difference and functional equations -- Difference equations -- Equations in the complex domain.; Algebraic geometry -- Birational geometry -- Birational automorphisms, Cremona group and generalizations.; Algebraic geometry -- Birational geometry -- Global theory and resolution of singularities.; Several complex variables and analytic spaces -- Singularities -- Modifications; resolution of singularities.
Abstract: "Discrete Painlevé equations are nonlinear difference equations, which arise from translations on crystallographic lattices. The deceptive simplicity of this statement hides immensely rich mathematical properties, connecting dynamical systems, algebraic geometry, Coxeter groups, topology, special functions theory, and mathematical physics. This book necessarily starts with introductory material to give the reader an accessible entry point to this vast subject matter. It is based on lectures that the author presented as principal lecturer at a Conference Board of Mathematical Sciences and National Science Foundation conference in Texas in 2016. Instead of technical theorems or complete proofs, the book relies on providing essential points of many arguments through explicit examples, with the hope that they will be useful for applied mathematicians and physicists."--Back cover.
Contents: 1. Introduction -- 2. A dynamical systems approach -- 3. Initial value spaces -- 4. Foliated initial value spaces -- 5. Cremona mappings -- 6. Asymptotic analysis -- 7. Lax pairs -- 8. Riemann-Hilbert problems -- Appendix A. Foliations and vector bundles -- Appendix B. Projective spaces -- Appendix C. Reflections groups -- Appendix D. Lists of discrete Painlevé equations -- Appendix E. Asymptotics of discrete equations.