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Author: J. Hietarinta
Publisher: Cambridge University Press
Published: 2016-09
Total Pages: 461
ISBN-13: 1107042720
DOWNLOAD EBOOKA first introduction to the theory of discrete integrable systems at a level suitable for students and non-experts.
Author: Alexander I. Bobenko
Publisher: American Mathematical Society
Published: 2023-09-14
Total Pages: 432
ISBN-13: 1470474565
DOWNLOAD EBOOKAn emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best? This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question “How do we discretize differential geometry?” arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful.
Author: Yuri B. Suris
Publisher: Birkhäuser
Published: 2012-12-06
Total Pages: 1078
ISBN-13: 3034880162
DOWNLOAD EBOOKAn exploration of the theory of discrete integrable systems, with an emphasis on the following general problem: how to discretize one or several of independent variables in a given integrable system of differential equations, maintaining the integrability property? This question (related in spirit to such a modern branch of numerical analysis as geometric integration) is treated in the book as an immanent part of the theory of integrable systems, also commonly termed as the theory of solitons. Most of the results are only available from recent journal publications, many of them are new. Thus, the book is a kind of encyclopedia on discrete integrable systems. It unifies the features of a research monograph and a handbook. It is supplied with an extensive bibliography and detailed bibliographic remarks at the end of each chapter. Largely self-contained, it will be accessible to graduate and post-graduate students as well as to researchers in the area of integrable dynamical systems.
Author: Alain Goriely
Publisher: World Scientific
Published: 2001
Total Pages: 435
ISBN-13: 981023533X
DOWNLOAD EBOOKThis invaluable book examines qualitative and quantitative methods for nonlinear differential equations, as well as integrability and nonintegrability theory. Starting from the idea of a constant of motion for simple systems of differential equations, it investigates the essence of integrability, its geometrical relevance and dynamical consequences. Integrability theory is approached from different perspectives, first in terms of differential algebra, then in terms of complex time singularities and finally from the viewpoint of phase geometry (for both Hamiltonian and non-Hamiltonian systems). As generic systems of differential equations cannot be exactly solved, the book reviews the different notions of nonintegrability and shows how to prove the nonexistence of exact solutions and/or a constant of motion. Finally, nonintegrability theory is linked to dynamical systems theory by showing how the property of complete integrability, partial integrability or nonintegrability can be related to regular and irregular dynamics in phase space.
Author: Basil Grammaticos
Publisher:
Published: 2014-01-15
Total Pages: 460
ISBN-13: 9783662144602
DOWNLOAD EBOOKAuthor: Jens Hoppe
Publisher: Springer Science & Business Media
Published: 2008-09-15
Total Pages: 109
ISBN-13: 3540472746
DOWNLOAD EBOOKMainly drawing on explicit examples, the author introduces the reader to themost recent techniques to study finite and infinite dynamical systems. Without any knowledge of differential geometry or lie groups theory the student can follow in a series of case studies the most recent developments. r-matrices for Calogero-Moser systems and Toda lattices are derived. Lax pairs for nontrivial infinite dimensionalsystems are constructed as limits of classical matrix algebras. The reader will find explanations of the approach to integrable field theories, to spectral transform methods and to solitons. New methods are proposed, thus helping students not only to understand established techniques but also to interest them in modern research on dynamical systems.
Author: Kenji Iohara
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 633
ISBN-13: 1447148630
DOWNLOAD EBOOKThis volume is the result of two international workshops; Infinite Analysis 11 – Frontier of Integrability – held at University of Tokyo, Japan in July 25th to 29th, 2011, and Symmetries, Integrable Systems and Representations held at Université Claude Bernard Lyon 1, France in December 13th to 16th, 2011. Included are research articles based on the talks presented at the workshops, latest results obtained thereafter, and some review articles. The subjects discussed range across diverse areas such as algebraic geometry, combinatorics, differential equations, integrable systems, representation theory, solvable lattice models and special functions. Through these topics, the reader will find some recent developments in the field of mathematical physics and their interactions with several other domains.
Author: Nalini Joshi
Publisher: American Mathematical Soc.
Published: 2019-05-30
Total Pages: 154
ISBN-13: 1470450380
DOWNLOAD EBOOKDiscrete 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.
Author: Boris A Kuperschmidt
Publisher: World Scientific
Published: 1990-10-25
Total Pages: 399
ISBN-13: 9814506737
DOWNLOAD EBOOKSome of the most active practitioners in the field of integrable systems have been asked to describe what they think of as the problems and results which seem to be most interesting and important now and are likely to influence future directions. The papers in this collection, representing their authors' responses, offer a broad panorama of the subject as it enters the 1990's.
Author: Peter A. Clarkson
Publisher: Cambridge University Press
Published: 1999-02-04
Total Pages: 444
ISBN-13: 9780521596992
DOWNLOAD EBOOKThis volume comprises state-of-the-art articles in discrete integrable systems.
