Author: Anton DzhamayPublisher:
ISBN:
Category : Algebra
Languages : en
Pages :
Book Description
Algebraic and Analytic Aspects of Integrable Systems and Painleve Equations
Author: Anton DzhamayAlgebraic and Analytic Aspects of Integrable Systems and Painlevé Equations
Author: Anton DzhamayPublisher:
ISBN: 9781470427795
Category : Algebra
Languages : en
Pages : 194
Book Description
This volume contains the proceedings of the AMS Special Session on Algebraic and Analytic Aspects of Integrable Systems and Painlevé Equations, held on January 18, 2014, at the Joint Mathematics Meetings in Baltimore, MD. The theory of integrable systems has been at the forefront of some of the most important developments in mathematical physics in the last 50 years. The techniques to study such systems have solid foundations in algebraic geometry, differential geometry, and group representation theory. Many important special solutions of continuous and discrete integrable systems can be written in terms of special functions such as hypergeometric and basic hypergeometric functions. The analytic tools developed to study integrable systems have numerous applications in random matrix theory, statistical mechanics and quantum gravity. One of the most exciting recent developments has been the emergence of good and interesting discrete and quantum analogues of classical integrable differential equations, such as the Painlevé equations and soliton equations. Many algebraic and analytic ideas developed in the continuous case generalize in a beautifully natural manner to discrete integrable systems. The editors have sought to bring together a collection of expository and research articles that represent a good cross section of ideas and methods in these active areas of research within integrable systems and their applications
Algebraic and Analytic Aspects of Integrable Systems and Painleve Equations
Author: Anton DzhamayPublisher: American Mathematical Soc.
ISBN: 1470416549
Category : Algebra
Languages : en
Pages : 194
Book Description
This volume contains the proceedings of the AMS Special Session on Algebraic and Analytic Aspects of Integrable Systems and Painlevé Equations, held on January 18, 2014, at the Joint Mathematics Meetings in Baltimore, MD. The theory of integrable systems has been at the forefront of some of the most important developments in mathematical physics in the last 50 years. The techniques to study such systems have solid foundations in algebraic geometry, differential geometry, and group representation theory. Many important special solutions of continuous and discrete integrable systems can be written in terms of special functions such as hypergeometric and basic hypergeometric functions. The analytic tools developed to study integrable systems have numerous applications in random matrix theory, statistical mechanics and quantum gravity. One of the most exciting recent developments has been the emergence of good and interesting discrete and quantum analogues of classical integrable differential equations, such as the Painlevé equations and soliton equations. Many algebraic and analytic ideas developed in the continuous case generalize in a beautifully natural manner to discrete integrable systems. The editors have sought to bring together a collection of expository and research articles that represent a good cross section of ideas and methods in these active areas of research within integrable systems and their applications.
Algebraic and Geometric Aspects of Integrable Systems and Random Matrices
Author: Anton DzhamayPublisher: American Mathematical Soc.
ISBN: 0821887475
Category : Mathematics
Languages : en
Pages : 363
Book Description
This volume contains the proceedings of the AMS Special Session on Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, held from January 6-7, 2012, in Boston, MA. The very wide range of topics represented in this volume illustrates
Algebraic Aspects of Integrable Systems
Author: Irene DorfmanPublisher: Birkhauser
ISBN:
Category : Differentiable dynamical systems
Languages : en
Pages : 368
Book Description
Algebraic Integrability, Painlevé Geometry and Lie Algebras
Author: Mark AdlerPublisher: Springer Science & Business Media
ISBN: 366205650X
Category : Mathematics
Languages : en
Pages : 487
Book Description
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.
Applications of Analytic and Geometric Methods to Nonlinear Differential Equations
Author: P.A. ClarksonPublisher: Springer Science & Business Media
ISBN: 940112082X
Category : Science
Languages : en
Pages : 466
Book Description
In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years. (1) The inverse scattering transform (IST), using complex function theory, which has been employed to solve many physically significant equations, the `soliton' equations. (2) Twistor theory, using differential geometry, which has been used to solve the self-dual Yang--Mills (SDYM) equations, a four-dimensional system having important applications in mathematical physics. Both soliton and the SDYM equations have rich algebraic structures which have been extensively studied. Recently, it has been conjectured that, in some sense, all soliton equations arise as special cases of the SDYM equations; subsequently many have been discovered as either exact or asymptotic reductions of the SDYM equations. Consequently what seems to be emerging is that a natural, physically significant system such as the SDYM equations provides the basis for a unifying framework underlying this class of integrable systems, i.e. `soliton' systems. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. The majority of nonlinear evolution equations are nonintegrable, and so asymptotic, numerical perturbation and reduction techniques are often used to study such equations. This book also contains articles on perturbed soliton equations. Painlevé analysis of partial differential equations, studies of the Painlevé equations and symmetry reductions of nonlinear partial differential equations. (ABSTRACT) In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years; the inverse scattering transform (IST), for `soliton' equations and twistor theory, for the self-dual Yang--Mills (SDYM) equations. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. Additionally, it contains articles on perturbed soliton equations, Painlevé analysis of partial differential equations, studies of the Painlevé equations and symmetry reductions of nonlinear partial differential equations.
Asymptotic, Algebraic and Geometric Aspects of Integrable Systems
Author: Frank NijhoffPublisher: Springer Nature
ISBN: 3030570002
Category : Mathematics
Languages : en
Pages : 240
Book Description
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.
Integrable Systems in the Realm of Algebraic Geometry
Author: Pol VanhaeckePublisher: Springer Verlag
ISBN:
Category : Mathematics
Languages : en
Pages : 240
Book Description
2. Divisors and line bundles 97 2.1. Divisors . . 97 2.2. Line bundles 98 2.3. Sections of line bundles 99 2.4. The Riemann-Roch Theorem 101 2.5. Line bundles and embeddings in projective space 103 2.6. Hyperelliptic curves 104 3. Abelian varieties 106 3.1. Complex tori and Abelian varieties 106 3.2. Line bundles on Abelian varieties 107 3.3. Abelian surfaces 109 4. Jacobi varieties . . . 112 4.1. The algebraic Jacobian 112 4.2. The analytic/trancendental Jacobian 112 4.3. Abel's Theorem and Jacobi inversion 116 4.4. Jacobi and Kummer surfaces 118 4.5. Abelian surfaces of type (1.4) 120 V. Algebraic completely integrable Hamiltonian systems 123 1. Introduction . 123 2. A.c.i. systems 125 3. Painleve analysis for a.c.i. systems 131 4. Linearization of two-dimensional a.c.i. systems 134 5. Lax equations 136 VI. The master systems 139 1. Introduction . . . . .
Painleve Analysis and Its Applications
Author: Amit K. Roy-ChowdhuryPublisher: CRC Press
ISBN: 9780849306389
Category : Mathematics
Languages : en
Pages : 312
Book Description
With interest in the study of nonlinear systems at an all-time high, researchers are eager to explore the mysteries behind the nonlinear equations that govern various physical processes. Painléve analysis may be the only tool available that allows the analysis of both integrable and non-integrable systems. With a primary objective of introducing the uninitiated to the various techniques of the Painlevé approach, this monograph brings together the results of the extensive research performed in the field over the last few decades. For the first time in a single volume, this book offers treatment of both the theory of Painlevé analysis and its practical applications. In it, the author addresses the soliton and nonlinearity, Painlevé analysis and the integrability of ordinary and partial differential equations, Painlevé properties, different forms of expansion, and the relation of Painlevé expansion with conformal invariance. He also gives a detailed account of negative resonances, explains the connection with monodromy, and demonstrates applications to specific important equations. Painlevé Analysis and Its Applications offers a clear presentation and down-to-earth approach that includes many examples and requires only a basic understanding of complex function theory and differential equations.