The Painlevé Property

Author : Robert Conte
Publisher : Springer Science & Business Media
Page : 828 pages
File Size : 50,6 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 : Robert Conte,Micheline Musette
Publisher : Springer Nature
Page : 389 pages
File Size : 53,7 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 : Athanassios S. Fokas,Alexander R. Its,Andrei A. Kapaev,Victor Yu. Novokshenov
Publisher : American Mathematical Society
Page : 570 pages
File Size : 44,7 Mb
Release : 2023-11-20
Category : Mathematics
ISBN : 9781470475567

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Painlevé Transcendents by Athanassios S. Fokas,Alexander R. Its,Andrei A. Kapaev,Victor Yu. Novokshenov Pdf

At the turn of the twentieth century, the French mathematician Paul Painlevé and his students classified second order nonlinear ordinary differential equations with the property that the location of possible branch points and essential singularities of their solutions does not depend on initial conditions. It turned out that there are only six such equations (up to natural equivalence), which later became known as Painlevé I–VI. Although these equations were initially obtained answering a strictly mathematical question, they appeared later in an astonishing (and growing) range of applications, including, e.g., statistical physics, fluid mechanics, random matrices, and orthogonal polynomials. Actually, it is now becoming clear that the Painlevé transcendents (i.e., the solutions of the Painlevé equations) play the same role in nonlinear mathematical physics that the classical special functions, such as Airy and Bessel functions, play in linear physics. The explicit formulas relating the asymptotic behaviour of the classical special functions at different critical points play a crucial role in the applications of these functions. It is shown in this book that even though the six Painlevé equations are nonlinear, it is still possible, using a new technique called the Riemann-Hilbert formalism, to obtain analogous explicit formulas for the Painlevé transcendents. This striking fact, apparently unknown to Painlevé and his contemporaries, is the key ingredient for the remarkable applicability of these “nonlinear special functions”. The book describes in detail the Riemann-Hilbert method and emphasizes its close connection to classical monodromy theory of linear equations as well as to modern theory of integrable systems. In addition, the book contains an ample collection of material concerning the asymptotics of the Painlevé functions and their various applications, which makes it a good reference source for everyone working in the theory and applications of Painlevé equations and related areas.

Author : Katsunori Iwasaki,Hironobu Kimura,Shun Shimemura,Masaaki Yoshida
Publisher : Springer Science & Business Media
Page : 355 pages
File Size : 47,5 Mb
Release : 2013-04-17
Category : Technology & Engineering
ISBN : 9783322901637

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From Gauss to Painlevé by Katsunori Iwasaki,Hironobu Kimura,Shun Shimemura,Masaaki Yoshida Pdf

This book gives an introduction to the modern theory of special functions. It focuses on the nonlinear Painlevé differential equation and its solutions, the so-called Painlevé functions. It contains modern treatments of the Gauss hypergeometric differential equation, monodromy of second order Fuchsian equations and nonlinear differential equations near singular points.The book starts from an elementary level requiring only basic notions of differential equations, function theory and group theory. Graduate students should be able to work with the text."The authors do an excellent job of presenting both the historical and mathematical details of the subject in a form accessible to any mathematician or physicist." (MPR in "The American Mathematical Monthly" März 1992.

Author : Nalini Joshi
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 51,9 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 : Decio Levi,Pavel Winternitz
Publisher : Springer Science & Business Media
Page : 454 pages
File Size : 54,9 Mb
Release : 2013-11-11
Category : Science
ISBN : 9781489911582

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Painlevé Transcendents by Decio Levi,Pavel Winternitz Pdf

The NATO Advanced Research Workshop "Painleve Transcendents, their Asymp totics and Physical Applications", held at the Alpine Inn in Sainte-Adele, near Montreal, September 2 -7, 1990, brought together a group of experts to discuss the topic and produce this volume. There were 41 participants from 14 countries and 27 lectures were presented, all included in this volume. The speakers presented reviews of topics to which they themselves have made important contributions and also re sults of new original research. The result is a volume which, though multiauthored, has the character of a monograph on a single topic. This is the theory of nonlinear ordinary differential equations, the solutions of which have no movable singularities, other than poles, and the extension of this theory to partial differential equations. For short we shall call such systems "equations with the Painleve property". The search for such equations was a very topical mathematical problem in the 19th century. Early work concentrated on first order differential equations. One of Painleve's important contributions in this field was to develop simple methods applicable to higher order equations. In particular these methods made possible a complete analysis of the equation ;; = f(y',y,x), where f is a rational function of y' and y, with coefficients that are analytic in x. The fundamental result due to Painleve (Acta Math.

Author : Alexei Cheviakov
Publisher : Springer Nature
Page : 322 pages
File Size : 41,8 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 9783031530746

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Analytical Properties of Nonlinear Partial Differential Equations by Alexei Cheviakov Pdf

Author : Amit K. Roy-Chowdhury
Publisher : CRC Press
Page : 312 pages
File Size : 53,5 Mb
Release : 1999-12-27
Category : Mathematics
ISBN : 0849306388

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Painleve Analysis and Its Applications by Amit K. Roy-Chowdhury Pdf

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.

Author : Anton Dzhamay,Kenichi Maruno,Christopher M. Ormerod
Publisher : American Mathematical Soc.
Page : 194 pages
File Size : 50,8 Mb
Release : 2015-10-28
Category : Algebra
ISBN : 9781470416546

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Algebraic and Analytic Aspects of Integrable Systems and Painleve Equations by Anton Dzhamay,Kenichi Maruno,Christopher M. Ormerod Pdf

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.

Author : A. A. Coley
Publisher : American Mathematical Soc.
Page : 460 pages
File Size : 40,5 Mb
Release : 2001-01-01
Category : Mathematics
ISBN : 0821870254

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Bäcklund and Darboux Transformations by A. A. Coley Pdf

This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Backlund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory. The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Backlund transformations of Painleve equations, to the dressing methodand Backlund and Darboux transformations, and to the classical geometry of Backlund transformations and their applications to soliton theory. The second part contains original contributions that represent new developments in the theory and applications of these transformations. Both the introductorylectures and the original talks were presented at an International Workshop that took place in Halifax, Nova Scotia (Canada). This volume covers virtually all recent developments in the theory and applications of Backlund and Darboux transformations.

Author : W-H Steeb,N Euler
Publisher : World Scientific
Page : 344 pages
File Size : 46,6 Mb
Release : 1988-10-01
Category : Mathematics
ISBN : 9789814520232

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Nonlinear Evolution Equations and Painlevé Test by W-H Steeb,N Euler Pdf

This book is an edited version of lectures given by the authors at a seminar at the Rand Afrikaans University. It gives a survey on the Painlevé test, Painlevé property and integrability. Both ordinary differential equations and partial differential equations are considered. Contents:IntroductionPainlevé Test and Ordinary Differential EquationsApplicationsZiglin's Theorems and NonintegrabilityGroup Theoretical Reduction of Partial Differential Equations and Painlevé TestPainlevé Property and Painlevé Test for Partial Differential EquationPainlevé Property and IntegrabilityHirota Technique and Painlevé TestDeformation of Painlevé Series under Symmetry ReductionIntegrable Field EquationsNonintegrable Field EquationsPainlevé Transcendents in Statistical Mechanics Readership: Mathematicians and physicists. Keywords:Nonlinear Differential Equations;Integrability;Painleve Test;Backlund Transformation;Soliton Equations;Symmetry SolutionsReview: “This excellent book is more than a survey on the Painlevé test, Painlevé property and integrability of both ordinary and partial differential equations; it also presents the recent progress in a rapidly growing field.” Mathematics Abstracts

Author : Alexander D. Bruno,Alexander B. Batkhin
Publisher : Walter de Gruyter
Page : 288 pages
File Size : 49,6 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 : A. S. Fokas,Alexander R. Its,Andrei A. Kapaev,Victor Yu Novokshenov,Andrei I. Kapaev,V. IU. Novokshenov
Publisher : American Mathematical Soc.
Page : 570 pages
File Size : 55,6 Mb
Release : 2006
Category : Differential equations, Nonlinear
ISBN : 9780821836514

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Painleve Transcendents by A. S. Fokas,Alexander R. Its,Andrei A. Kapaev,Victor Yu Novokshenov,Andrei I. Kapaev,V. IU. Novokshenov Pdf

At the turn of the twentieth century, the French mathematician Paul Painleve and his students classified second order nonlinear ordinary differential equations with the property that the location of possible branch points and essential singularities of their solutions does not depend on initial conditions. It turned out that there are only six such equations (up to natural equivalence), which later became known as Painleve I-VI. Although these equations were initially obtainedanswering a strictly mathematical question, they appeared later in an astonishing (and growing) range of applications, including, e.g., statistical physics, fluid mechanics, random matrices, and orthogonal polynomials. Actually, it is now becoming clear that the Painleve transcendents (i.e., the solutionsof the Painleve equations) play the same role in nonlinear mathematical physics that the classical special functions, such as Airy and Bessel functions, play in linear physics. The explicit formulas relating the asymptotic behaviour of the classical special functions at different critical points, play a crucial role in the applications of these functions. It is shown in this book, that even though the six Painleve equations are nonlinear, it is still possible, using a new technique called theRiemann-Hilbert formalism, to obtain analogous explicit formulas for the Painleve transcendents. This striking fact, apparently unknown to Painleve and his contemporaries, is the key ingredient for the remarkable applicability of these ``nonlinear special functions''. The book describes in detail theRiemann-Hilbert method and emphasizes its close connection to classical monodromy theory of linear equations as well as to modern theory of integrable systems. In addition, the book contains an ample collection of material concerning the asymptotics of the Painleve functions and their various applications, which makes it a good reference source for everyone working in the theory and applications of Painleve equations and related areas.

Author : Robert L Dewar,N Joshi
Publisher : World Scientific
Page : 130 pages
File Size : 55,5 Mb
Release : 1991-01-14
Category : Electronic
ISBN : 9789814569743

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Chaos And Order, Miniconference On - Proceedings Of The Centre For Mathematical Analysis, Australian National University by Robert L Dewar,N Joshi Pdf

This volume is dedicated to Dr Ding Lee for his untiring efforts in promoting the advancement of theoretical and computational acoustics.This proceedings volume provides a forum for active researchers to discuss the state-of-the-art developments and results in theoretical and computational acoustics, covering aero-, seismo- and ocean acoustics and related topics. It discusses multidimensional wave propagation modeling, methods of computational acoustics, wave propagation in rocks, fluid-solid interfaces, nonlinear acoustics, neural networks, real applications and experimental results.

Author : Sandra Carillo
Publisher : Unknown
Page : 30 pages
File Size : 42,8 Mb
Release : 1988
Category : Differential equations, Nonlinear
ISBN : UOM:39015019835274

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Painlevé Property by Sandra Carillo Pdf