[PDF] The Painleve Property eBook

Author : Robert Conte
Publisher : Springer Science & Business Media
Page : 828 pages
File Size : 39,50 MB
Release : 2012-12-06
Category : Science
ISBN : 1461215323

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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 : A. S. Fokas
Publisher : American Mathematical Soc.
Page : 570 pages
File Size : 32,52 MB
Release : 2006
Category : Mathematics
ISBN : 082183651X

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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 : N Euler
Publisher : World Scientific
Page : 345 pages
File Size : 37,35 MB
Release : 1988-10-01
Category : Mathematics
ISBN : 9814520233

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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.

Author : Robert M. Conte
Publisher : Springer Science & Business Media
Page : 271 pages
File Size : 27,63 MB
Release : 2008-11-23
Category : Science
ISBN : 1402084919

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Nonlinear differential or difference equations are encountered not only in mathematics, but also in many areas of physics (evolution equations, propagation of a signal in an optical fiber), chemistry (reaction-diffusion systems), and biology (competition of species). This book introduces the reader to methods allowing one to build explicit solutions to these equations. A prerequisite task is to investigate whether the chances of success are high or low, and this can be achieved without any a priori knowledge of the solutions, with a powerful algorithm presented in detail called the Painlevé test. If the equation under study passes the Painlevé test, the equation is presumed integrable. If on the contrary the test fails, the system is nonintegrable or even chaotic, but it may still be possible to find solutions. The examples chosen to illustrate these methods are mostly taken from physics. These include on the integrable side the nonlinear Schrödinger equation (continuous and discrete), the Korteweg-de Vries equation, the Hénon-Heiles Hamiltonians, on the nonintegrable side the complex Ginzburg-Landau equation (encountered in optical fibers, turbulence, etc), the Kuramoto-Sivashinsky equation (phase turbulence), the Kolmogorov-Petrovski-Piskunov equation (KPP, a reaction-diffusion model), the Lorenz model of atmospheric circulation and the Bianchi IX cosmological model. Written at a graduate level, the book contains tutorial text as well as detailed examples and the state of the art on some current research.

Author : Robert Conte
Publisher : Springer Nature
Page : 389 pages
File Size : 31,68 MB
Release : 2020-11-07
Category : Science
ISBN : 3030533409

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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 : A. A. Coley
Publisher : American Mathematical Soc.
Page : 460 pages
File Size : 23,56 MB
Release : 2001-01-01
Category : Mathematics
ISBN : 9780821870259

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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 : Anton Dzhamay
Publisher : American Mathematical Soc.
Page : 210 pages
File Size : 20,26 MB
Release : 2015-10-28
Category : Mathematics
ISBN : 1470416549

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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 : Valerii I. Gromak
Publisher : Walter de Gruyter
Page : 313 pages
File Size : 47,83 MB
Release : 2008-08-22
Category : Mathematics
ISBN : 3110198096

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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 : Decio Levi
Publisher : Springer Science & Business Media
Page : 454 pages
File Size : 27,83 MB
Release : 2013-11-11
Category : Science
ISBN : 1489911588

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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.