The Problem of Integrable Discretization

The Problem of Integrable Discretization PDF

Author: Yuri B. Suris

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 1078

ISBN-13: 3034880162

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An 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.

Discrete Integrable Geometry and Physics

Discrete Integrable Geometry and Physics PDF

Author: Alexander I. Bobenko

Publisher: Clarendon Press

Published: 1999

Total Pages: 466

ISBN-13: 9780198501602

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Recent interactions between the fields of geometry, classical and quantum dynamical systems, and visualization of geometric objects such as curves and surfaces have led to the observation that most concepts of surface theory and of the theory of integrable systems have natural discreteanalogues. These are characterized by the property that the corresponding difference equations are integrable, and has led in turn to some important applications in areas of condensed matter physics and quantum field theory, amongst others. The book combines the efforts of a distinguished team ofauthors from various fields in mathematics and physics in an effort to provide an overview of the subject. The mathematical concepts of discrete geometry and discrete integrable systems are firstly presented as fundamental and valuable theories in themselves. In the following part these concepts areput into the context of classical and quantum dynamics.

Discrete Systems and Integrability

Discrete Systems and Integrability PDF

Author: J. Hietarinta

Publisher: Cambridge University Press

Published: 2016-09

Total Pages: 461

ISBN-13: 1107042720

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A first introduction to the theory of discrete integrable systems at a level suitable for students and non-experts.

Integrable Hamiltonian Hierarchies

Integrable Hamiltonian Hierarchies PDF

Author: Vladimir Gerdjikov

Publisher: Springer

Published: 2008-12-02

Total Pages: 645

ISBN-13: 3540770542

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This book presents a detailed derivation of the spectral properties of the Recursion Operators allowing one to derive all the fundamental properties of the soliton equations and to study their hierarchies.

Continuous Symmetries and Integrability of Discrete Equations

Continuous Symmetries and Integrability of Discrete Equations PDF

Author: Decio Levi

Publisher: American Mathematical Society, Centre de Recherches Mathématiques

Published: 2023-01-23

Total Pages: 520

ISBN-13: 0821843540

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This book on integrable systems and symmetries presents new results on applications of symmetries and integrability techniques to the case of equations defined on the lattice. This relatively new field has many applications, for example, in describing the evolution of crystals and molecular systems defined on lattices, and in finding numerical approximations for differential equations preserving their symmetries. The book contains three chapters and five appendices. The first chapter is an introduction to the general ideas about symmetries, lattices, differential difference and partial difference equations and Lie point symmetries defined on them. Chapter 2 deals with integrable and linearizable systems in two dimensions. The authors start from the prototype of integrable and linearizable partial differential equations, the Korteweg de Vries and the Burgers equations. Then they consider the best known integrable differential difference and partial difference equations. Chapter 3 considers generalized symmetries and conserved densities as integrability criteria. The appendices provide details which may help the readers' understanding of the subjects presented in Chapters 2 and 3. This book is written for PhD students and early researchers, both in theoretical physics and in applied mathematics, who are interested in the study of symmetries and integrability of difference equations.

Asymptotic, Algebraic and Geometric Aspects of Integrable Systems

Asymptotic, Algebraic and Geometric Aspects of Integrable Systems PDF

Author: Frank Nijhoff

Publisher: Springer Nature

Published: 2020-10-23

Total Pages: 240

ISBN-13: 3030570002

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This proceedings volume gathers together selected works from the 2018 “Asymptotic, Algebraic and Geometric Aspects of Integrable Systems” workshop that was held at TSIMF Yau Mathematical Sciences Center in Sanya, China, honoring Nalini Joshi on her 60th birthday. The papers cover recent advances in asymptotic, algebraic and geometric methods in the study of discrete integrable systems. The workshop brought together experts from fields such as asymptotic analysis, representation theory and geometry, creating a platform to exchange current methods, results and novel ideas. This volume's articles reflect these exchanges and can be of special interest to a diverse group of researchers and graduate students interested in learning about current results, new approaches and trends in mathematical physics, in particular those relevant to discrete integrable systems.

Discrete Mechanics, Geometric Integration and Lie–Butcher Series

Discrete Mechanics, Geometric Integration and Lie–Butcher Series PDF

Author: Kurusch Ebrahimi-Fard

Publisher: Springer

Published: 2018-11-05

Total Pages: 361

ISBN-13: 3030013979

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This volume resulted from presentations given at the international “Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie–Butcher Series”, that took place at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid, Spain. It combines overview and research articles on recent and ongoing developments, as well as new research directions. Why geometric numerical integration? In their article of the same title Arieh Iserles and Reinout Quispel, two renowned experts in numerical analysis of differential equations, provide a compelling answer to this question. After this introductory chapter a collection of high-quality research articles aim at exploring recent and ongoing developments, as well as new research directions in the areas of geometric integration methods for differential equations, nonlinear systems interconnections, and discrete mechanics. One of the highlights is the unfolding of modern algebraic and combinatorial structures common to those topics, which give rise to fruitful interactions between theoretical as well as applied and computational perspectives. The volume is aimed at researchers and graduate students interested in theoretical and computational problems in geometric integration theory, nonlinear control theory, and discrete mechanics.

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