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Lectures on Integrable Systems

Author : Jens Hoppe
Publisher : Springer
Release : 2014-08-23
Category : Science
ISBN : 3662138824

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Book Lectures on Integrable Systems Description/Summary:

Mainly 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.

Lectures on Integrable Systems

Author : Jens Hoppe
Publisher : Springer Science & Business Media
Release : 2008-09-15
Category : Science
ISBN : 9783540472742

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Book Lectures on Integrable Systems Description/Summary:

Mainly 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.

Geometry and Dynamics of Integrable Systems

Author : Alexey Bolsinov,Juan J. Morales-Ruiz,Nguyen Tien Zung
Publisher : Birkhäuser
Release : 2016-10-27
Category : Mathematics
ISBN : 9783319335032

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Book Geometry and Dynamics of Integrable Systems Description/Summary:

Based on lectures given at an advanced course on integrable systems at the Centre de Recerca Matemàtica in Barcelona, these lecture notes address three major aspects of integrable systems: obstructions to integrability from differential Galois theory; the description of singularities of integrable systems on the basis of their relation to bi-Hamiltonian systems; and the generalization of integrable systems to the non-Hamiltonian settings. All three sections were written by top experts in their respective fields. Native to actual problem-solving challenges in mechanics, the topic of integrable systems is currently at the crossroads of several disciplines in pure and applied mathematics, and also has important interactions with physics. The study of integrable systems also actively employs methods from differential geometry. Moreover, it is extremely important in symplectic geometry and Hamiltonian dynamics, and has strong correlations with mathematical physics, Lie theory and algebraic geometry (including mirror symmetry). As such, the book will appeal to experts with a wide range of backgrounds.

Integrability of Nonlinear Systems

Author : Yvette Kosmann-Schwarzbach,Basil Grammaticos,K.M. Tamizhmani
Publisher : Springer Science & Business Media
Release : 2004-02-17
Category : Science
ISBN : 3540206302

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Book Integrability of Nonlinear Systems Description/Summary:

The lectures that comprise this volume constitute a comprehensive survey of the many and various aspects of integrable dynamical systems. The present edition is a streamlined, revised and updated version of a 1997 set of notes that was published as Lecture Notes in Physics, Volume 495. This volume will be complemented by a companion book dedicated to discrete integrable systems. Both volumes address primarily graduate students and nonspecialist researchers but will also benefit lecturers looking for suitable material for advanced courses and researchers interested in specific topics.

Quantum Group And Quantum Integrable Systems - Nankai Lectures On Mathematical Physics

Author : Ge Mo-lin
Publisher : World Scientific
Release : 1992-05-30
Category : Uncategorized
ISBN : 9789814555838

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Book Quantum Group And Quantum Integrable Systems - Nankai Lectures On Mathematical Physics Description/Summary:

This volume provides a self-contained survey of the mechanisms presiding information processing and communication. The main thesis is that chaos and complexity are the basic ingredients allowing systems composed of interesting subunits to generate and process information and communicate in a meaningful way. Emphasis is placed on communication in the form of games and on the related issue of decision making under conditions of uncertainty. Biological, cognitive, physical, engineering and societal systems are approached from a unifying point of view, both analytically and by numerical simulation, using the methods of nonlinear dynamics and probability theory. Epistemological issues in connection with incompleteness and self-reference are also addressed.

Integrable Systems

Author : X C Song
Publisher : World Scientific
Release : 1989-11-01
Category : Uncategorized
ISBN : 9789814644099

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Book Integrable Systems Description/Summary:

This workshop is part of a series of annual workshops organised by the Nankai Institute of Mathematics. Prominent scientists from abroad are invited to deliver the main lectures. Contents:Speeches:On the Relation between Mathematics and Physics (L D Faddeev)Journey through Statistical Mechanics (C N Yang)Invited Lectures:Quantum Inverse Scattering Method (L D Faddeev)Geometrical Integrability and Equations of Motion in Physics: A Unifying View (Ling-Lie Chau)and 10 contributed papers Readership: Mathematical and theoretical physicists.

Integrable Systems

Author : N.J. Hitchin,G. B. Segal,R.S. Ward
Publisher : Oxford University Press
Release : 2013-03-14
Category : Mathematics
ISBN : 9780199676774

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Book Integrable Systems Description/Summary:

Designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors, this book has its origins in a lecture series given by the internationally renowned authors. Written in an accessible, informal style, it fills a gap in the existing literature.

Calogero-Moser Systems and Representation Theory

Author : Pavel Etingof,Pavel I. Etingof
Publisher : European Mathematical Society
Release : 2007
Category : Mathematics
ISBN : 3037190345

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Book Calogero-Moser Systems and Representation Theory Description/Summary:

Calogero-Moser systems, which were originally discovered by specialists in integrable systems, are currently at the crossroads of many areas of mathematics and within the scope of interests of many mathematicians. More specifically, these systems and their generalizations turned out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), representation theory (double affine Hecke algebras, Lie groups, quantum groups), deformation theory (symplectic reflection algebras), homological algebra (Koszul algebras), Poisson geometry, etc. The goal of the present lecture notes is to give an introduction to the theory of Calogero-Moser systems, highlighting their interplay with these fields. Since these lectures are designed for non-experts, the author gives short introductions to each of the subjects involved and provides a number of exercises.

Integrability

Author : Alexander Mikhailov
Publisher : Springer Science & Business Media
Release : 2008-11-25
Category : Science
ISBN : 9783540881100

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Book Integrability Description/Summary:

The principal aim of the book is to give a comprehensive account of the variety of approaches to such an important and complex concept as Integrability. Dev- oping mathematical models, physicists often raise the following questions: whether the model obtained is integrable or close in some sense to an integrable one and whether it can be studied in depth analytically. In this book we have tried to c- ate a mathematical framework to address these issues, and we give descriptions of methods and review results. In the Introduction we give a historical account of the birth and development of the theory of integrable equations, focusing on the main issue of the book – the concept of integrability itself. A universal de nition of Integrability is proving to be elusive despite more than 40 years of its development. Often such notions as “- act solvability” or “regular behaviour” of solutions are associated with integrable systems. Unfortunately these notions do not lead to any rigorous mathematical d- inition. A constructive approach could be based upon the study of hidden and rich algebraic or analytic structures associated with integrable equations. The requi- ment of existence of elements of these structures could, in principle, be taken as a de nition for integrability. It is astonishing that the nal result is not sensitive to the choice of the structure taken; eventually we arrive at the same pattern of eq- tions.

Integrability

Author : Patrick Dorey,Gregory Korschemsky,Nikita Nekrasov,Didina Serban,Volker Schomerus,Leticia Cugliandolo
Publisher : Lecture Notes of the Les Houch
Release : 2019-07-19
Category : Mathematics
ISBN : 9780198828150

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Book Integrability Description/Summary:

This volume, 106 of the Les Houches Summer School series, brings together applications of integrability to supersymmetric gauge and string theory. The book focuses on the application of integrability and problems in quantum field theory. Particular emphasis is given to the exact solution of planar N=4 super-Yang-Mills theory and its relation with string theory on the one hand, and the exact determination of the low-energy physics of N=2 super-Yang-Mills theories on the other; links with other domains are also explored. The purpose of the Les Houches Summer School was to bring together young researchers and specialists from statistical physics, condensed matter physics, gauge and string theory, and mathematics, to stimulate discussion across these different research areas.

Integrable Systems and Quantum Groups

Author : Ron Donagi,Boris Dubrovin,Edward Frenkel,Emma Previato
Publisher : Springer
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540477068

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Book Integrable Systems and Quantum Groups Description/Summary:

The aim of this CIME Session was to review the state of the art in the recent development of the theory of integrable systems and their relations with quantum groups. The purpose was to gather geometers and mathematical physicists to allow a broader and more complete view of these attractive and rapidly developing fields. The papers contained in this volume have at the same time the character of survey articles and of research papers, since they contain both a survey of current problems and a number of original contributions to the subject.

Complex Analysis, Riemann Surfaces and Integrable Systems

Author : Sergey M. Natanzon
Publisher : Springer Nature
Release : 2020-01-03
Category : Mathematics
ISBN : 9783030346409

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Book Complex Analysis, Riemann Surfaces and Integrable Systems Description/Summary:

This book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided. We begin with a brief but fairly complete course on the theory of holomorphic, meromorphic, and harmonic functions. We then present a uniformization theory, and discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as a factor space of a contracted space by a discrete group. Next, we consider compact Riemann surfaces and prove the classical theorems of Riemann-Roch, Abel, Weierstrass, etc. We also construct theta functions that are very important for a range of applications. After that, we turn to modern applications of this theory. First, we build the (important for mathematics and mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at important solutions to these differential equations. We subsequently use the theory of harmonic functions and the theory of differential hierarchies to explicitly construct a conformal mapping that translates an arbitrary contractible domain into a standard disk – a classical problem that has important applications in hydrodynamics, gas dynamics, etc. The book is based on numerous lecture courses given by the author at the Independent University of Moscow and at the Mathematics Department of the Higher School of Economics.

An Introduction to Integrable Techniques for One-Dimensional Quantum Systems

Author : Fabio Franchini
Publisher : Springer
Release : 2017-05-25
Category : Science
ISBN : 9783319484877

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Book An Introduction to Integrable Techniques for One-Dimensional Quantum Systems Description/Summary:

This book introduces the reader to basic notions of integrable techniques for one-dimensional quantum systems. In a pedagogical way, a few examples of exactly solvable models are worked out to go from the coordinate approach to the Algebraic Bethe Ansatz, with some discussion on the finite temperature thermodynamics. The aim is to provide the instruments to approach more advanced books or to allow for a critical reading of research articles and the extraction of useful information from them. We describe the solution of the anisotropic XY spin chain; of the Lieb-Liniger model of bosons with contact interaction at zero and finite temperature; and of the XXZ spin chain, first in the coordinate and then in the algebraic approach. To establish the connection between the latter and the solution of two dimensional classical models, we also introduce and solve the 6-vertex model. Finally, the low energy physics of these integrable models is mapped into the corresponding conformal field theory. Through its style and the choice of topics, this book tries to touch all fundamental ideas behind integrability and is meant for students and researchers interested either in an introduction to later delve in the advance aspects of Bethe Ansatz or in an overview of the topic for broadening their culture.

Integrable Systems And Quantum Groups

Author : Carfora Mauro,Martellini Maurizio,Marzuoli Annalisa
Publisher : World Scientific
Release : 1992-04-30
Category : Uncategorized
ISBN : 9789814554763

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Book Integrable Systems And Quantum Groups Description/Summary:

Integrable Systems

Author : John P. Harnad,Gert Sabidussi,Pavel Winternitz
Publisher : American Mathematical Soc.
Release : 2022-06-29
Category : Mathematics
ISBN : 082187022X

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Book Integrable Systems Description/Summary:

Integrable Systems, Quantum Groups, and Quantum Field Theories

Author : Alberto Ibort,M.A. Rodríguez
Publisher : Springer Science & Business Media
Release : 2012-12-06
Category : Science
ISBN : 9789401119801

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Book Integrable Systems, Quantum Groups, and Quantum Field Theories Description/Summary:

In many ways the last decade has witnessed a surge of interest in the interplay between theoretical physics and some traditional areas of pure mathematics. This book contains the lectures delivered at the NATO-ASI Summer School on `Recent Problems in Mathematical Physics' held at Salamanca, Spain (1992), offering a pedagogical and updated approach to some of the problems that have been at the heart of these events. Among them, we should mention the new mathematical structures related to integrability and quantum field theories, such as quantum groups, conformal field theories, integrable statistical models, and topological quantum field theories, that are discussed at length by some of the leading experts on the areas in several of the lectures contained in the book. Apart from these, traditional and new problems in quantum gravity are reviewed. Other contributions to the School included in the book range from symmetries in partial differential equations to geometrical phases in quantum physics. The book is addressed to researchers in the fields covered, PhD students and any scientist interested in obtaining an updated view of the subjects.

Algebraic Integrability, Painlevé Geometry and Lie Algebras

Author : Mark Adler,Pierre van Moerbeke,Pol Vanhaecke
Publisher : Springer Science & Business Media
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662056509

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Book Algebraic Integrability, Painlevé Geometry and Lie Algebras Description/Summary:

This Ergebnisse volume is aimed at a wide readership of mathematicians and physicists, graduate students and professionals. The main thrust of the book is to show how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations and construct the algebraic tori on which they linearize; at the same time, it is, for the student, a playing ground to applying algebraic geometry and Lie theory. The book is meant to be reasonably self-contained and presents numerous examples. The latter appear throughout the text to illustrate the ideas, and make up the core of the last part of the book. The first part of the book contains the basic tools from Lie groups, algebraic and differential geometry to understand the main topic.

Integrable Systems on Lie Algebras and Symmetric Spaces

Author : A. T. Fomenko,V. V. Trofimov
Publisher : CRC Press
Release : 1988
Category : Mathematics
ISBN : 2881241700

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Book Integrable Systems on Lie Algebras and Symmetric Spaces Description/Summary:

Second volume in the series, translated from the Russian, sets out new regular methods for realizing Hamilton's canonical equations in Lie algebras and symmetric spaces. Begins by constructing the algebraic embeddings in Lie algebras of Hamiltonian systems, going on to present effective methods for constructing complete sets of functions in involution on orbits of coadjoint representations of Lie groups. Ends with the proof of the full integrability of a wide range of many- parameter families of Hamiltonian systems that allow algebraicization. Annotation copyrighted by Book News, Inc., Portland, OR

Lectures on Symplectic Geometry

Author : Ana Cannas da Silva
Publisher : Springer
Release : 2004-10-27
Category : Mathematics
ISBN : 9783540453307

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Book Lectures on Symplectic Geometry Description/Summary:

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.