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Painleve Analysis And Its Applications


Painleve Analysis And Its Applications
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Painleve Analysis And Its Applications


Painleve Analysis And Its Applications
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Author : Amit K. Roy-Chowdhury
language : en
Publisher: CRC Press
Release Date : 1999-12-27

Painleve Analysis And Its Applications written by Amit K. Roy-Chowdhury and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-12-27 with Mathematics categories.


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.



Painleve Analysis And Its Applications


Painleve Analysis And Its Applications
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Author : A.R. Chowdhury
language : en
Publisher:
Release Date : 1999-05-01

Painleve Analysis And Its Applications written by A.R. Chowdhury and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-05-01 with categories.




Painlev Transcendents


Painlev Transcendents
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Author : Decio Levi
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-11

Painlev Transcendents written by Decio Levi and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-11 with Science categories.


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.



The Painlev Handbook


The Painlev Handbook
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Author : Robert Conte
language : en
Publisher: Springer Nature
Release Date : 2020-11-07

The Painlev Handbook written by Robert Conte and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-11-07 with Science categories.


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.



Advances In Nonlinear Partial Differential Equations And Related Areas


Advances In Nonlinear Partial Differential Equations And Related Areas
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Author : Gui-Qiang Chen
language : en
Publisher: World Scientific
Release Date : 1998

Advances In Nonlinear Partial Differential Equations And Related Areas written by Gui-Qiang Chen and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with Mathematics categories.


This volume is a collection of research papers on nonlinear partial differential equations and related areas, representing many aspects of the most recent developments in these important areas. In particular, the following are included: nonlinear conservation laws, semilinear elliptic equations, nonlinear hyperbolic equations, nonlinear parabolic equations, singular limit problems, and analysis of exact and numerical solutions. Important areas such as numerical analysis, relaxation theory, multiphase theory, kinetic theory, combustion theory, dynamical systems, and quantum field theory are also covered.



Painleve Transcendents


Painleve Transcendents
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Author : A. S. Fokas
language : en
Publisher: American Mathematical Soc.
Release Date : 2006

Painleve Transcendents written by A. S. Fokas and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.


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.



Painlev Transcendents


Painlev Transcendents
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Author : Athanassios S. Fokas
language : en
Publisher: American Mathematical Society
Release Date : 2023-11-20

Painlev Transcendents written by Athanassios S. Fokas and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-11-20 with Mathematics categories.


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.



Computer Algebra And Geometric Algebra With Applications


Computer Algebra And Geometric Algebra With Applications
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Author : Hongbo Li
language : en
Publisher: Springer
Release Date : 2005-06-20

Computer Algebra And Geometric Algebra With Applications written by Hongbo Li and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-06-20 with Computers categories.


MathematicsMechanization consistsoftheory,softwareandapplicationofc- puterized mathematical activities such as computing, reasoning and discovering. ItsuniquefeaturecanbesuccinctlydescribedasAAA(Algebraization,Algori- mization, Application). The name “Mathematics Mechanization” has its origin in the work of Hao Wang (1960s), one of the pioneers in using computers to do research in mathematics, particularly in automated theorem proving. Since the 1970s, this research direction has been actively pursued and extensively dev- oped by Prof. Wen-tsun Wu and his followers. It di?ers from the closely related disciplines like Computer Mathematics, Symbolic Computation and Automated Reasoning in that its goal is to make algorithmic studies and applications of mathematics the major trend of mathematics development in the information age. The International Workshop on Mathematics Mechanization (IWMM) was initiated by Prof. Wu in 1992, and has ever since been held by the Key L- oratory of Mathematics Mechanization (KLMM) of the Chinese Academy of Sciences. There have been seven workshops of the series up to now. At each workshop, several experts are invited to deliver plenary lectures on cutting-edge methods and algorithms of the selected theme. The workshop is also a forum for people working on related subjects to meet, collaborate and exchange ideas.



Applications Of Analytic And Geometric Methods To Nonlinear Differential Equations


Applications Of Analytic And Geometric Methods To Nonlinear Differential Equations
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Author : P.A. Clarkson
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Applications Of Analytic And Geometric Methods To Nonlinear Differential Equations written by P.A. Clarkson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Science categories.


In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years. (1) The inverse scattering transform (IST), using complex function theory, which has been employed to solve many physically significant equations, the `soliton' equations. (2) Twistor theory, using differential geometry, which has been used to solve the self-dual Yang--Mills (SDYM) equations, a four-dimensional system having important applications in mathematical physics. Both soliton and the SDYM equations have rich algebraic structures which have been extensively studied. Recently, it has been conjectured that, in some sense, all soliton equations arise as special cases of the SDYM equations; subsequently many have been discovered as either exact or asymptotic reductions of the SDYM equations. Consequently what seems to be emerging is that a natural, physically significant system such as the SDYM equations provides the basis for a unifying framework underlying this class of integrable systems, i.e. `soliton' systems. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. The majority of nonlinear evolution equations are nonintegrable, and so asymptotic, numerical perturbation and reduction techniques are often used to study such equations. This book also contains articles on perturbed soliton equations. Painlevé analysis of partial differential equations, studies of the Painlevé equations and symmetry reductions of nonlinear partial differential equations. (ABSTRACT) In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years; the inverse scattering transform (IST), for `soliton' equations and twistor theory, for the self-dual Yang--Mills (SDYM) equations. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. Additionally, it contains articles on perturbed soliton equations, Painlevé analysis of partial differential equations, studies of the Painlevé equations and symmetry reductions of nonlinear partial differential equations.



Painleve Analysis Lie Symmetries And Integrability Of Nonlinear Ordinary Differential Equations


Painleve Analysis Lie Symmetries And Integrability Of Nonlinear Ordinary Differential Equations
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Author :
language : en
Publisher:
Release Date : 2005

Painleve Analysis Lie Symmetries And Integrability Of Nonlinear Ordinary Differential Equations written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with categories.


The Painleve analysis plays an important role in investigating local structure of the solutions of differential equations, while Lie symmetries provide powerful tools in global solvability of equations. In this research, the method of Painleve analysis is applied to discrete nonlinear Schrodinger equations and to a family of second order nonlinear ordinary differential equations. Lie symmetries are studied together with the Painleve property for second order nonlinear ordinary differential equations. In the study of the local singularity of discrete nonlinear Schrodinger equations, the Painleve method shows the existence of solution blow up at finite time. It also determines the rate of blow-up. For second order nonlinear ordinary differential equations, the Painleve test is introduced and demonstrated in detail using several examples. These examples are used throughout the research. The Painleve property is shown to be significant for the integrability of a differential equation. After introducing one-parameter groups, a family of differential equations is determined for discussing solvability and for drawing more meaningful conclusions. This is the most general family of differential equations invariant under a given one-parameter group. The first part of this research is the classification of the integrals in the general solutions of differential equations obtained by quadratures. The second part is the application of Riemann surfaces and algebraic curves in the projective complex space to the integrands. The theories of Riemann surfaces and algebraic curves lead us to an effective way to understand the nature of the integral defined on a curve. Our theoretical work then concentrates on the blowing-up of algebraic curves at singular points. The calculation of the genus, which essentially determines the shape of a curve, becomes possible after a sequence of blowing-ups. The research shows that when combining both the Painleve property and Lie symmetries possessed by the differential equations studied in the thesis, the general solutions can be represented by either elementary functions or elliptic integrals.