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Painleve Transcendents
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Book Synopsis Painlevé Transcendents by : Athanassios S. Fokas
Download or read book Painlevé Transcendents written by Athanassios S. Fokas and published by American Mathematical Society. This book was released on 2023-11-20 with total page 570 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Painlevé Transcendents by : Decio Levi
Download or read book Painlevé Transcendents written by Decio Levi and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 454 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Painlevé Differential Equations in the Complex Plane by : Valerii I. Gromak
Download or read book Painlevé Differential Equations in the Complex Plane written by Valerii I. Gromak and published by Walter de Gruyter. This book was released on 2008-08-22 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Discrete Painlevé Equations by : Nalini Joshi
Download or read book Discrete Painlevé Equations written by Nalini Joshi and published by American Mathematical Soc.. This book was released on 2019-05-30 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Painlevé Handbook by : Robert Conte
Download or read book The Painlevé Handbook written by Robert Conte and published by Springer Nature. This book was released on 2020-11-07 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Orthogonal Polynomials and Special Functions by : Francisco Marcellàn
Download or read book Orthogonal Polynomials and Special Functions written by Francisco Marcellàn and published by Springer. This book was released on 2006-10-18 with total page 422 pages. Available in PDF, EPUB and Kindle. Book excerpt: Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? The present set of lecture notes contains seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions.
Book Synopsis Painleve Equations in the Differential Geometry of Surfaces by : Alexander I. Bobenko TU Berlin
Download or read book Painleve Equations in the Differential Geometry of Surfaces written by Alexander I. Bobenko TU Berlin and published by Springer. This book was released on 2003-07-01 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book brings together two different branches of mathematics: the theory of Painlev 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 Painlev 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 Painlev equations: the theory of isomonodromic deformation and the Painlev 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 Painlev equations. Researchers working in one of these areas can become familiar with another relevant branch of mathematics.
Book Synopsis The Kowalevski Property by : Vadim B. Kuznetsov
Download or read book The Kowalevski Property written by Vadim B. Kuznetsov and published by American Mathematical Soc.. This book was released on with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of survey articles on several topics related to the general notion of integrability. It stems from a workshop on ''Mathematical Methods of Regular Dynamics'' dedicated to Sophie Kowalevski. Leading experts introduce corresponding areas in depth. The book provides a broad overview of research, from the pioneering work of the nineteenth century to the developments of the 1970s through the present. The book begins with two historical papers by R. L. Cooke onKowalevski's life and work. Following are 15 research surveys on integrability issues in differential and algebraic geometry, classical complex analysis, discrete mathematics, spinning tops, Painleve equations, global analysis on manifolds, special functions, etc. It concludes with Kowalevski's famouspaper published in Acta Mathematica in 1889, ''Sur le probleme de la rotation d'un corps solide autour d'un point fixe''. The book is suitable for graduate students in pure and applied mathematics, the general mathematical audience studying integrability, and research mathematicians interested in differential and algebraic geometry, analysis, and special functions.
Book Synopsis Geometric Methods in Physics XXXVII by : Piotr Kielanowski
Download or read book Geometric Methods in Physics XXXVII written by Piotr Kielanowski and published by Springer Nature. This book was released on 2019-11-26 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book consists of articles based on the XXXVII Białowieża Workshop on Geometric Methods in Physics, 2018. The series of Białowieża workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. This edition of the workshop featured a special session dedicated to Professor Daniel Sternheimer on the occasion of his 80th birthday. The previously unpublished papers present cutting-edge current research, typically grounded in geometry and analysis, with applications to classical and quantum physics. For the past seven years, the Białowieża Workshops have been complemented by a School on Geometry and Physics comprising a series of advanced lectures for graduate students and early-career researchers. The book also includes abstracts of the five lecture series that were given at the seventh school.
Book Synopsis The Painlevé Property by : Robert Conte
Download or read book The Painlevé Property written by Robert Conte and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 828 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 :AMS Special Session on Special Functions and Orthogonal Polynomials Publisher :American Mathematical Soc. ISBN 13 :0821846507 Total Pages :226 pages Book Rating :4.8/5 (218 download)
Book Synopsis Special Functions and Orthogonal Polynomials by : AMS Special Session on Special Functions and Orthogonal Polynomials
Download or read book Special Functions and Orthogonal Polynomials written by AMS Special Session on Special Functions and Orthogonal Polynomials and published by American Mathematical Soc.. This book was released on 2008 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This volume contains fourteen articles that represent the AMS Special Session on Special Functions and Orthogonal Polynomials, held in Tucson, Arizona in April of 2007. It gives an overview of the modern field of special functions with all major subfields represented, including: applications to algebraic geometry, asymptotic analysis, conformal mapping, differential equations, elliptic functions, fractional calculus, hypergeometric and q-hypergeometric series, nonlinear waves, number theory, symbolic and numerical evaluation of integrals, and theta functions. A few articles are expository, with extensive bibliographies, but all contain original research." "This book is intended for pure and applied mathematicians who are interested in recent developments in the theory of special functions. It covers a wide range of active areas of research and demonstrates the vitality of the field."--BOOK JACKET.
Book Synopsis Handbook of Nonlinear Partial Differential Equations, Second Edition by : Andrei D. Polyanin
Download or read book Handbook of Nonlinear Partial Differential Equations, Second Edition written by Andrei D. Polyanin and published by CRC Press. This book was released on 2016-04-19 with total page 1878 pages. Available in PDF, EPUB and Kindle. Book excerpt: New to the Second Edition More than 1,000 pages with over 1,500 new first-, second-, third-, fourth-, and higher-order nonlinear equations with solutions Parabolic, hyperbolic, elliptic, and other systems of equations with solutions Some exact methods and transformations Symbolic and numerical methods for solving nonlinear PDEs with MapleTM, Mathematica®, and MATLAB® Many new illustrative examples and tables A large list of references consisting of over 1,300 sources To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology. They outline the methods in a schematic, simplified manner and arrange the material in increasing order of complexity.
Book Synopsis Painleve Transcendents by : A. S. Fokas
Download or read book Painleve Transcendents written by A. S. Fokas and published by American Mathematical Soc.. This book was released on 2006 with total page 570 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Algebraic Analysis of Differential Equations by : T. Aoki
Download or read book Algebraic Analysis of Differential Equations written by T. Aoki and published by Springer Science & Business Media. This book was released on 2009-03-15 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains 23 articles on algebraic analysis of differential equations and related topics, most of which were presented as papers at the conference "Algebraic Analysis of Differential Equations – from Microlocal Analysis to Exponential Asymptotics" at Kyoto University in 2005. This volume is dedicated to Professor Takahiro Kawai, who is one of the creators of microlocal analysis and who introduced the technique of microlocal analysis into exponential asymptotics.
Book Synopsis The Oxford Handbook of Random Matrix Theory by : Gernot Akemann
Download or read book The Oxford Handbook of Random Matrix Theory written by Gernot Akemann and published by Oxford Handbooks. This book was released on 2015-08-09 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: With a foreword by Freeman Dyson, the handbook brings together leading mathematicians and physicists to offer a comprehensive overview of random matrix theory, including a guide to new developments and the diverse range of applications of this approach.In part one, all modern and classical techniques of solving random matrix models are explored, including orthogonal polynomials, exact replicas or supersymmetry.
Book Synopsis Invertible Point Transformations and Nonlinear Differential Equations by : Willi-Hans Steeb
Download or read book Invertible Point Transformations and Nonlinear Differential Equations written by Willi-Hans Steeb and published by World Scientific. This book was released on 1993-06-04 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: The invertible point transformation is a powerful tool in the study of nonlinear differential and difference equations. This book gives a comprehensive introduction to this technique. Ordinary and partial differential equations are studied with this approach. The book also covers nonlinear difference equations. The connections with Lie symmetries, the Painlevé property, first integrals and the Cartan equivalence method are discussed in detail. Most of the evaluations are checked with the computer language REDUCE; the book includes 30 REDUCE programs. A short introduction to the jet bundle formalism is given. Contents:First-Order Ordinary Differential EquationSecond-Order Ordinary Differential EquationsThird-Order Differential EquationsLie Point SymmetriesFirst Integrals and Differential EquationCartan Equivalence MethodPainlevé Test and LinearizationPainlevé Test and Partial Differential EquationsPartial Differential EquationsDifference EquationsREDUCE ProgramsJet Bundle Formalism Readership: Mathematicians, physicists and engineers. keywords:Nonlinear Differential Equations;Invertible Point Transformation;Lie Point Symmetries;Painleve Test;Jet Bundle Formalism “The text is well written, and fairly elementary from a mathematical standpoint. The concepts are clearly illustrated; there are numerous examples of interest to applied mathematicians and physicists.” SIAM Review
Book Synopsis Nonlinear Evolution Equations and Painlevé Test by : W-H Steeb
Download or read book Nonlinear Evolution Equations and Painlevé Test written by W-H Steeb and published by World Scientific. This book was released on 1988-10-01 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
