Discrete Painlevé Equations

Author : Nalini Joshi
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 44,7 Mb
Release : 2019-05-30
Category : Differential equations, Nonlinear
ISBN : 9781470450380

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Discrete Painlevé Equations by Nalini Joshi Pdf

Discrete 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 : Nalini Joshi
Publisher : Unknown
Page : 154 pages
File Size : 43,7 Mb
Release : 2019
Category : Differential equations, Nonlinear
ISBN : 1470452359

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Discrete Painlevé Equations by Nalini Joshi Pdf

Discrete 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 Nation.

Author : Valerii I. Gromak,Ilpo Laine,Shun Shimomura
Publisher : Walter de Gruyter
Page : 313 pages
File Size : 45,5 Mb
Release : 2008-08-22
Category : Mathematics
ISBN : 9783110198096

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Painlevé Differential Equations in the Complex Plane by Valerii I. Gromak,Ilpo Laine,Shun Shimomura Pdf

This book is the first comprehensive treatment of Painlevé differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution of Painlevé transcendents. The subsequent main part of the book is devoted to topics of classical background such as representations and expansions of solutions, solutions of special type like rational and special transcendental solutions, Bäcklund transformations and higher order analogues, treated separately for each of these six equations. The final chapter offers a short overview of applications of Painlevé equations, including an introduction to their discrete counterparts. Due to the present important role of Painlevé equations in physical applications, this monograph should be of interest to researchers in both mathematics and physics and to graduate students interested in mathematical physics and the theory of differential equations.

Author : Robert Conte
Publisher : Springer Science & Business Media
Page : 828 pages
File Size : 53,9 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781461215325

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The Painlevé Property by Robert Conte Pdf

The subject this volume is explicit integration, that is, the analytical as opposed to the numerical solution, of all kinds of nonlinear differential equations (ordinary differential, partial differential, finite difference). Such equations describe many physical phenomena, their analytic solutions (particular solutions, first integral, and so forth) are in many cases preferable to numerical computation, which may be long, costly and, worst, subject to numerical errors. In addition, the analytic approach can provide a global knowledge of the solution, while the numerical approach is always local. Explicit integration is based on the powerful methods based on an in-depth study of singularities, that were first used by Poincar and subsequently developed by Painlev in his famous Leons de Stockholm of 1895. The recent interest in the subject and in the equations investigated by Painlev dates back about thirty years ago, arising from three, apparently disjoint, fields: the Ising model of statistical physics and field theory, propagation of solitons, and dynamical systems. The chapters in this volume, based on courses given at Cargse 1998, alternate mathematics and physics; they are intended to bring researchers entering the field to the level of present research.

Author : Basil Grammaticos,Yvette Kosmann-Schwarzbach,Thamizharasi Tamizhmani
Publisher : Unknown
Page : 460 pages
File Size : 40,7 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662144603

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Discrete Integrable Systems by Basil Grammaticos,Yvette Kosmann-Schwarzbach,Thamizharasi Tamizhmani Pdf

Author : Walter Van Assche
Publisher : Cambridge University Press
Page : 192 pages
File Size : 48,5 Mb
Release : 2018
Category : Mathematics
ISBN : 9781108441940

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Orthogonal Polynomials and Painlevé Equations by Walter Van Assche Pdf

There are a number of intriguing connections between Painlev equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these. Researchers in integrable systems and non-linear equations will find the many explicit examples where Painlev equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painlev transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painlev equations are worked out in detail, with a survey of recent results and an outline of their close relationship with orthogonal polynomials. Exercises throughout the book help the reader to get to grips with the material. The author is a leading authority on orthogonal polynomials, giving this work a unique perspective on Painlev equations.

Author : Hidehito Nagao,Yasuhiko Yamada
Publisher : Springer Nature
Page : 94 pages
File Size : 55,6 Mb
Release : 2021-09-01
Category : Science
ISBN : 9789811629983

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Padé Methods for Painlevé Equations by Hidehito Nagao,Yasuhiko Yamada Pdf

The isomonodromic deformation equations such as the Painlevé and Garnier systems are an important class of nonlinear differential equations in mathematics and mathematical physics. For discrete analogs of these equations in particular, much progress has been made in recent decades. Various approaches to such isomonodromic equations are known: the Painlevé test/Painlevé property, reduction of integrable hierarchy, the Lax formulation, algebro-geometric methods, and others. Among them, the Padé method explained in this book provides a simple approach to those equations in both continuous and discrete cases. For a given function f(x), the Padé approximation/interpolation supplies the rational functions P(x), Q(x) as approximants such as f(x)~P(x)/Q(x). The basic idea of the Padé method is to consider the linear differential (or difference) equations satisfied by P(x) and f(x)Q(x). In choosing the suitable approximation problem, the linear differential equations give the Lax pair for some isomonodromic equations. Although this relation between the isomonodromic equations and Padé approximations has been known classically, a systematic study including discrete cases has been conducted only recently. By this simple and easy procedure, one can simultaneously obtain various results such as the nonlinear evolution equation, its Lax pair, and their special solutions. In this way, the method is a convenient means of approaching the isomonodromic deformation equations.

Author : Robert Conte,Micheline Musette
Publisher : Springer Nature
Page : 389 pages
File Size : 43,7 Mb
Release : 2020-11-07
Category : Science
ISBN : 9783030533403

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The Painlevé Handbook by Robert Conte,Micheline Musette Pdf

This book, now in its second edition, introduces the singularity analysis of differential and difference equations via the Painlevé test and shows how Painlevé analysis provides a powerful algorithmic approach to building explicit solutions to nonlinear ordinary and partial differential equations. It is illustrated with integrable equations such as the nonlinear Schrödinger equation, the Korteweg-de Vries equation, Hénon-Heiles type Hamiltonians, and numerous physically relevant examples such as the Kuramoto-Sivashinsky equation, the Kolmogorov-Petrovski-Piskunov equation, and mainly the cubic and quintic Ginzburg-Landau equations. Extensively revised, updated, and expanded, this new edition includes: recent insights from Nevanlinna theory and analysis on both the cubic and quintic Ginzburg-Landau equations; a close look at physical problems involving the sixth Painlevé function; and an overview of new results since the book’s original publication with special focus on finite difference equations. The book features tutorials, appendices, and comprehensive references, and will appeal to graduate students and researchers in both mathematics and the physical sciences.

Author : Alexander D. Bruno,Alexander B. Batkhin
Publisher : Walter de Gruyter
Page : 288 pages
File Size : 46,8 Mb
Release : 2012-08-31
Category : Mathematics
ISBN : 9783110275667

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Painlevé Equations and Related Topics by Alexander D. Bruno,Alexander B. Batkhin Pdf

This is a proceedings of the international conference "Painlevé Equations and Related Topics" which was taking place at the Euler International Mathematical Institute, a branch of the Saint Petersburg Department of the Steklov Institute of Mathematics of the Russian Academy of Sciences, in Saint Petersburg on June 17 to 23, 2011. The survey articles discuss the following topics: General ordinary differential equations Painlevé equations and their generalizations Painlevé property Discrete Painlevé equations Properties of solutions of all mentioned above equations: – Asymptotic forms and asymptotic expansions – Connections of asymptotic forms of a solution near different points – Convergency and asymptotic character of a formal solution – New types of asymptotic forms and asymptotic expansions – Riemann-Hilbert problems – Isomonodromic deformations of linear systems – Symmetries and transformations of solutions – Algebraic solutions Reductions of PDE to Painlevé equations and their generalizations Ordinary Differential Equations systems equivalent to Painlevé equations and their generalizations Applications of the equations and the solutions

Author : Alexander R. Its,Victor Y. Novokshenov
Publisher : Springer
Page : 318 pages
File Size : 47,9 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540398233

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The Isomonodromic Deformation Method in the Theory of Painleve Equations by Alexander R. Its,Victor Y. Novokshenov Pdf

Author : Decio Levi,Orlando Ragnisco
Publisher : American Mathematical Soc.
Page : 468 pages
File Size : 48,8 Mb
Release : 2000-06-15
Category : Mathematics
ISBN : 0821870211

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SIDE III by Decio Levi,Orlando Ragnisco Pdf

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.

Author : Masatoshi Noumi
Publisher : Springer Science & Business
Page : 172 pages
File Size : 55,5 Mb
Release : 2004
Category : Mathematics
ISBN : 0821832212

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Painlevé Equations Through Symmetry by Masatoshi Noumi Pdf

This book is devoted to the symmetry of Painleve equations (especially those of types II and IV). The author studies families of transformations for several types of Painleve equationsQthe so-called Backlund transformationsQwhich transform solutions of a given Painleve equation to solutions of the same equation with a different set of parameters. It turns out that these symmetries can be interpreted in terms of root systems associated to affine Weyl groups. The author describes the remarkable combinatorial structures of these symmetries and shows how they are related to the theory of $\tau$-functions associated to integrable systems.

Author : Alexander I. Bobenko,Ulrich Eitner
Publisher : Springer Science & Business Media
Page : 125 pages
File Size : 53,6 Mb
Release : 2000-12-12
Category : Mathematics
ISBN : 9783540414148

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Painleve Equations in the Differential Geometry of Surfaces by Alexander I. Bobenko,Ulrich Eitner Pdf

This book brings together two different branches of mathematics: the theory of Painlevé and the theory of surfaces. Self-contained introductions to both these fields are presented. It is shown how some classical problems in surface theory can be solved using the modern theory of Painlevé equations. In particular, an essential part of the book is devoted to Bonnet surfaces, i.e. to surfaces possessing families of isometries preserving the mean curvature function. A global classification of Bonnet surfaces is given using both ingredients of the theory of Painlevé equations: the theory of isomonodromic deformation and the Painlevé property. The book is illustrated by plots of surfaces. It is intended to be used by mathematicians and graduate students interested in differential geometry and Painlevé equations. Researchers working in one of these areas can become familiar with another relevant branch of mathematics.

Author : Basil Grammaticos,Yvette Kosmann-Schwarzbach,Thamizharasi Tamizhmani
Publisher : Springer
Page : 472 pages
File Size : 53,7 Mb
Release : 2004-06-22
Category : Science
ISBN : 3540214259

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Discrete Integrable Systems by Basil Grammaticos,Yvette Kosmann-Schwarzbach,Thamizharasi Tamizhmani Pdf

This volume consists of a set of ten lectures conceived as both introduction and up-to-date survey on discrete integrable systems. It constitutes a companion book to "Integrability of Nonlinear Systems" (Springer-Verlag, 2004, LNP 638, ISBN 3-540-20630-2). 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.

Author : Decio Levi,Raphaël Rebelo,Pavel Winternitz
Publisher : Springer
Page : 435 pages
File Size : 45,9 Mb
Release : 2017-06-30
Category : Science
ISBN : 9783319566665

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Symmetries and Integrability of Difference Equations by Decio Levi,Raphaël Rebelo,Pavel Winternitz Pdf

This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers in the specific field of their expertise and, in turn, written for early career researchers. As a survey of the current state of the art, this book will serve as a valuable reference and is particularly well suited as an introduction to the field of symmetries and integrability of difference equations. Therefore, the book will be welcomed by advanced undergraduate and graduate students as well as by more advanced researchers.