Painlevé Differential Equations In The Complex Plane

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Painlevé Differential Equations in the Complex Plane

Valerii I. Gromak,Ilpo Laine,Shun Shimomura
Painlev Differential Equations in the Complex Plane Book PDF

Author : Valerii I. Gromak,Ilpo Laine,Shun Shimomura
Publisher : Walter de Gruyter
Page : 313 pages
File Size : 47,6 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.

Complex Variables

Mark J. Ablowitz,A. S. Fokas
Complex Variables Book PDF

Author : Mark J. Ablowitz,A. S. Fokas
Publisher : Cambridge University Press
Page : 656 pages
File Size : 47,5 Mb
Release : 2003
Category : Functions of complex variables
ISBN : 0521534291

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Complex Variables by Mark J. Ablowitz,A. S. Fokas Pdf

Complex variables provide powerful methods for attacking many difficult problems, and it is the aim of this book to provide a thorough grounding in these methods and their application. This new edition has been improved throughout and is ideal for use in undergraduate and introductory graduate courses in complex variables.

Painlevé Transcendents

Athanassios S. Fokas,Alexander R. Its,Andrei A. Kapaev,Victor Yu. Novokshenov
Painlev Transcendents Book PDF

Author : Athanassios S. Fokas,Alexander R. Its,Andrei A. Kapaev,Victor Yu. Novokshenov
Publisher : American Mathematical Society
Page : 570 pages
File Size : 52,8 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.

Painleve Transcendents

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

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 : 49,5 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.

Invertible Point Transformations and Nonlinear Differential Equations

Willi-Hans Steeb
Invertible Point Transformations and Nonlinear Differential Equations Book PDF

Author : Willi-Hans Steeb
Publisher : World Scientific
Page : 188 pages
File Size : 44,7 Mb
Release : 1993-06-04
Category : Science
ISBN : 9789814504362

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Invertible Point Transformations and Nonlinear Differential Equations by Willi-Hans Steeb Pdf

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

Handbook of Differential Equations: Ordinary Differential Equations

Flaviano Battelli,Michal Feckan
Handbook of Differential Equations Ordinary Differential Equations Book PDF

Author : Flaviano Battelli,Michal Feckan
Publisher : Elsevier
Page : 719 pages
File Size : 42,7 Mb
Release : 2008-08-19
Category : Mathematics
ISBN : 9780080559469

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Handbook of Differential Equations: Ordinary Differential Equations by Flaviano Battelli,Michal Feckan Pdf

This handbook is the fourth volume in a series of volumes devoted to self-contained and up-to-date surveys in the theory of ordinary differential equations, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience. Covers a variety of problems in ordinary differential equations Pure mathematical and real-world applications Written for mathematicians and scientists of many related fields

Divergent Series, Summability and Resurgence III

Eric Delabaere
Divergent Series Summability and Resurgence III Book PDF

Author : Eric Delabaere
Publisher : Springer
Page : 230 pages
File Size : 46,7 Mb
Release : 2016-06-28
Category : Mathematics
ISBN : 9783319290003

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Divergent Series, Summability and Resurgence III by Eric Delabaere Pdf

The aim of this volume is two-fold. First, to show how the resurgent methods introduced in volume 1 can be applied efficiently in a non-linear setting; to this end further properties of the resurgence theory must be developed. Second, to analyze the fundamental example of the First Painlevé equation. The resurgent analysis of singularities is pushed all the way up to the so-called “bridge equation”, which concentrates all information about the non-linear Stokes phenomenon at infinity of the First Painlevé equation. The third in a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists who are interested in divergent power series and related problems, such as the Stokes phenomenon. The prerequisites are a working knowledge of complex analysis at the first-year graduate level and of the theory of resurgence, as presented in volume 1.

The Painlevé Handbook

Robert Conte,Micheline Musette
The Painlev Handbook Book PDF

Author : Robert Conte,Micheline Musette
Publisher : Springer Nature
Page : 389 pages
File Size : 55,6 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.

Differential Algebra, Complex Analysis and Orthogonal Polynomials

Primitivo B. Acosta Humanez,Francisco Marcellán
Differential Algebra Complex Analysis and Orthogonal Polynomials Book PDF

Author : Primitivo B. Acosta Humanez,Francisco Marcellán
Publisher : American Mathematical Soc.
Page : 241 pages
File Size : 44,8 Mb
Release : 2010
Category : Differentiable dynamical systems
ISBN : 9780821848869

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Differential Algebra, Complex Analysis and Orthogonal Polynomials by Primitivo B. Acosta Humanez,Francisco Marcellán Pdf

Presents the 2007-2008 Jairo Charris Seminar in Algebra and Analysis on Differential Algebra, Complex Analysis and Orthogonal Polynomials, which was held at the Universidad Sergio Arboleda in Bogota, Colombia.

Group Theory and Numerical Analysis

Pavel Winternitz
Group Theory and Numerical Analysis Book PDF

Author : Pavel Winternitz
Publisher : American Mathematical Soc.
Page : 316 pages
File Size : 54,8 Mb
Release : 2024-06-29
Category : Mathematics
ISBN : 0821870343

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Group Theory and Numerical Analysis by Pavel Winternitz Pdf

The Workshop on Group Theory and Numerical Analysis brought together scientists working in several different but related areas. The unifying theme was the application of group theory and geometrical methods to the solution of differential and difference equations. The emphasis was on the combination of analytical and numerical methods and also the use of symbolic computation. This meeting was organized under the auspices of the Centre de Recherches Mathematiques, Universite de Montreal (Canada). This volume has the character of a monograph and should represent a useful reference book for scientists working in this highly topical field.

Formal And Analytic Solutions Of Differential Equations

Galina Filipuk,Alberto Lastra,Slawomir Michalik
Formal And Analytic Solutions Of Differential Equations Book PDF

Author : Galina Filipuk,Alberto Lastra,Slawomir Michalik
Publisher : World Scientific
Page : 400 pages
File Size : 48,8 Mb
Release : 2022-03-03
Category : Mathematics
ISBN : 9781800611375

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Formal And Analytic Solutions Of Differential Equations by Galina Filipuk,Alberto Lastra,Slawomir Michalik Pdf

The book provides the reader with an overview of the actual state of research in ordinary and partial differential equations in the complex domain. Topics include summability and asymptotic study of both ordinary and partial differential equations, and also q-difference and differential-difference equations. This book will be of interest to researchers and students who wish to expand their knowledge of these fields.With the latest results and research developments and contributions from experts in their field, Formal and Analytic Solutions of Differential Equations provides a valuable contribution to methods, techniques, different mathematical tools, and study calculations.

Handbook of Nonlinear Partial Differential Equations

Andrei D. Polyanin,Valentin F. Zaitsev
Handbook of Nonlinear Partial Differential Equations Book PDF

Author : Andrei D. Polyanin,Valentin F. Zaitsev
Publisher : CRC Press
Page : 835 pages
File Size : 40,5 Mb
Release : 2004-06-02
Category : Mathematics
ISBN : 9781135440817

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Handbook of Nonlinear Partial Differential Equations by Andrei D. Polyanin,Valentin F. Zaitsev Pdf

The Handbook of Nonlinear Partial Differential Equations is the latest in a series of acclaimed handbooks by these authors and presents exact solutions of more than 1600 nonlinear equations encountered in science and engineering--many more than any other book available. The equations include those of parabolic, hyperbolic, elliptic and other types, and the authors pay special attention to equations of general form that involve arbitrary functions. A supplement at the end of the book discusses the classical and new methods for constructing exact solutions to nonlinear equations. To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology, outline some of the methods in a schematic, simplified manner, and arrange the equations in increasing order of complexity. Highlights of the Handbook:

Handbook of Nonlinear Partial Differential Equations, Second Edition

Andrei D. Polyanin,Valentin F. Zaitsev
Handbook of Nonlinear Partial Differential Equations Second Edition Book PDF

Author : Andrei D. Polyanin,Valentin F. Zaitsev
Publisher : CRC Press
Page : 1878 pages
File Size : 40,7 Mb
Release : 2016-04-19
Category : Mathematics
ISBN : 9781420087246

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Handbook of Nonlinear Partial Differential Equations, Second Edition by Andrei D. Polyanin,Valentin F. Zaitsev Pdf

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.

Special Functions and Orthogonal Polynomials

AMS Special Session on Special Functions and Orthogonal Polynomials
Special Functions and Orthogonal Polynomials Book PDF

Author : AMS Special Session on Special Functions and Orthogonal Polynomials
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 50,8 Mb
Release : 2008
Category : Functions, Special
ISBN : 9780821846506

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Special Functions and Orthogonal Polynomials by AMS Special Session on Special Functions and Orthogonal Polynomials Pdf

"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.

Orthogonal Polynomials and Painlevé Equations

Walter Van Assche
Orthogonal Polynomials and Painlev Equations Book PDF

Author : Walter Van Assche
Publisher : Cambridge University Press
Page : 192 pages
File Size : 42,8 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.