Download or read book Analytic Bilinear Approach to Integrable Hierarchies written by L.V. Bogdanov and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of this book is the hierarchies of integrable equations connected with the one-component and multi component loop groups. There are many publications on this subject, and it is rather well defined. Thus, the author would like t.o explain why he has taken the risk of revisiting the subject. The Sato Grassmannian approach, and other approaches standard in this context, reveal deep mathematical structures in the base of the integrable hi erarchies. These approaches concentrate mostly on the algebraic picture, and they use a language suitable for applications to quantum field theory. Another well-known approach, the a-dressing method, developed by S. V. Manakov and V.E. Zakharov, is oriented mostly to particular systems and ex act classes of their solutions. There is more emphasis on analytic properties, and the technique is connected with standard complex analysis. The language of the a-dressing method is suitable for applications to integrable nonlinear PDEs, integrable nonlinear discrete equations, and, as recently discovered, for t.he applications of integrable systems to continuous and discret.e geometry. The primary motivation of the author was to formalize the approach to int.e grable hierarchies that was developed in the context of the a-dressing method, preserving the analytic struetures characteristic for this method, but omitting the peculiarit.ies of the construetive scheme. And it was desirable to find a start.
Download or read book Integrable Hierarchies and Modern Physical Theories written by Henrik Aratyn and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proceedings of the NATO Advanced Research Workshop, Chicago, USA, July 22-26, 2000
Download or read book Generalized Lie Theory in Mathematics Physics and Beyond written by Sergei D. Silvestrov and published by Springer Science & Business Media. This book was released on 2008-11-18 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the cutting edge of the fundamental role of generalizations of Lie theory and related non-commutative and non-associative structures in mathematics and physics.
Download or read book Discrete Systems and Integrability written by J. Hietarinta and published by Cambridge University Press. This book was released on 2016-08-19 with total page 461 pages. Available in PDF, EPUB and Kindle. Book excerpt: This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. While treating the material at an elementary level, the book also highlights many recent developments. Topics include: Darboux and Bäcklund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Padé approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painlevé equations; and the novel theory of Lagrangian multiforms. The book builds the material in an organic way, emphasizing interconnections between the various approaches, while the exposition is mostly done through explicit computations on key examples. Written by respected experts in the field, the numerous exercises and the thorough list of references will benefit upper-level undergraduate, and beginning graduate students as well as researchers from other disciplines.
Download or read book SIDE III written by Decio Levi and published by American Mathematical Soc.. This book was released on 2000-06-15 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the third meeting on ``Symmetries and Integrability of Difference Equations'' (SIDE III). The collection includes original results not published elsewhere and articles that give a rigorous but concise overview of their subject, and provides a complete description of the state of the art. Research in the field of difference equations--often referred to more generally as discrete systems--has undergone impressive development in recent years. In this collection the reader finds the most important new developments in a number of areas, including: Lie-type symmetries of differential-difference and difference-difference equations, integrability of fully discrete systems such as cellular automata, the connection between integrability and discrete geometry, the isomonodromy approach to discrete spectral problems and related discrete Painleve equations, difference and q-difference equations and orthogonal polynomials, difference equations and quantum groups, and integrability and chaos in discrete-time dynamical systems. The proceedings will be valuable to mathematicians and theoretical physicists interested in the mathematical aspects and/or in the physical applications of discrete nonlinear dynamics, with special emphasis on the systems that can be integrated by analytic methods or at least admit special explicit solutions. The research in this volume will also be of interest to engineers working in discrete dynamics as well as to theoretical biologists and economists.
Download or read book The Problem of Integrable Discretization written by Yuri B. Suris and published by Birkhäuser. This book was released on 2012-12-06 with total page 1078 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Geometry and Integrability written by Lionel Mason and published by Cambridge University Press. This book was released on 2003-11-20 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: Articles from leading researchers to introduce the reader to cutting-edge topics in integrable systems theory.
Download or read book Quantum Theory Deformation and Integrability written by R. Carroll and published by Elsevier. This book was released on 2000-11-09 with total page 421 pages. Available in PDF, EPUB and Kindle. Book excerpt: About four years ago a prominent string theorist was quoted as saying that it might be possible to understand quantum mechanics by the year 2000. Sometimes new mathematical developments make such understanding appear possible and even close, but on the other hand, increasing lack of experimental verification make it seem to be further distant. In any event one seems to arrive at new revolutions in physics and mathematics every year. This book hopes to convey some of the excitment of this period, but will adopt a relatively pedestrian approach designed to illuminate the relations between quantum and classical. There will be some discussion of philosophical matters such as measurement, uncertainty, decoherence, etc. but philosophy will not be emphasized; generally we want to enjoy the fruits of computation based on the operator formulation of QM and quantum field theory. In Chapter 1 connections of QM to deterministic behavior are exhibited in the trajectory representations of Faraggi-Matone. Chapter 1 also includes a review of KP theory and some preliminary remarks on coherent states, density matrices, etc. and more on deterministic theory. We develop in Chapter 4 relations between quantization and integrability based on Moyal brackets, discretizations, KP, strings and Hirota formulas, and in Chapter 2 we study the QM of embedded curves and surfaces illustrating some QM effects of geometry. Chapter 3 is on quantum integrable systems, quantum groups, and modern deformation quantization. Chapter 5 involves the Whitham equations in various roles mediating between QM and classical behavior. In particular, connections to Seiberg-Witten theory (arising in N = 2 supersymmetric (susy) Yang-Mills (YM) theory) are discussed and we would still like to understand more deeply what is going on. Thus in Chapter 5 we will try to give some conceptual background for susy, gauge theories, renormalization, etc. from both a physical and mathematical point of view. In Chapter 6 we continue the deformation quantization then by exhibiting material based on and related to noncommutative geometry and gauge theory.
Download or read book Journal of Physics A written by and published by . This book was released on 1999 with total page 772 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focuses on fundamental mathematical and computational methods underpinning physics. Relevant to statistical physics, chaotic and complex systems, classical and quantum mechanics, classical and quantum integrable systems and classical and quantum field theory.
Download or read book Mathematical Reviews written by and published by . This book was released on 2003 with total page 900 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Applications of Analytic and Geometric Methods to Nonlinear Differential Equations written by P.A. Clarkson and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Journal of the Physical Society of Japan written by and published by . This book was released on 2017 with total page 940 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Proceedings written by and published by . This book was released on 2009 with total page 1036 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Proceedings A publishes refereed research papers in the mathematical, physical, and engineering sciences. The emphasis is on new, emerging areas of interdisciplinary and multidisciplinary research." Continues: Proceedings. Mathematical and physical sciences.
Download or read book American Book Publishing Record written by and published by . This book was released on 2000 with total page 1886 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Integrability written by Alexander Mikhailov and published by Springer. This book was released on 2008-11-05 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book The British National Bibliography written by Arthur James Wells and published by . This book was released on 2000 with total page 1270 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Physics Briefs written by and published by . This book was released on 1994 with total page 1146 pages. Available in PDF, EPUB and Kindle. Book excerpt:
