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Divergent Series, Summability and Resurgence II

Author: Michèle Loday-Richaud

Publisher: Springer

Published: 2016-06-28

Total Pages: 286

ISBN-13: 3319290754

Addressing the question how to “sum” a power series in one variable when it diverges, that is, how to attach to it analytic functions, the volume gives answers by presenting and comparing the various theories of k-summability and multisummability. These theories apply in particular to all solutions of ordinary differential equations. The volume includes applications, examples and revisits, from a cohomological point of view, the group of tangent-to-identity germs of diffeomorphisms of C studied in volume 1. With a view to applying the theories to solutions of differential equations, a detailed survey of linear ordinary differential equations is provided, which includes Gevrey asymptotic expansions, Newton polygons, index theorems and Sibuya’s proof of the meromorphic classification theorem that characterizes the Stokes phenomenon for linear differential equations. This volume is the second in a series of three, entitled Divergent Series, Summability and Resurgence. It is aimed at graduate students and researchers in mathematics and theoretical physics who are interested in divergent series, Although closely related to the other two volumes, it can be read independently.

Divergent Series, Summability and Resurgence III

Author: Eric Delabaere

Publisher: Springer

Published: 2016-06-28

Total Pages: 252

ISBN-13: 3319290002

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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.

Divergent Series, Summability and Resurgence

Author:

Publisher:

Published: 2016

Total Pages:

ISBN-13:

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Divergent Series, Summability and Resurgence I

Author: Claude Mitschi

Publisher: Springer

Published: 2016-08-27

Total Pages: 314

ISBN-13: 3319287362

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Providing an elementary introduction to analytic continuation and monodromy, the first part of this volume applies these notions to the local and global study of complex linear differential equations, their formal solutions at singular points, their monodromy and their differential Galois groups. The Riemann-Hilbert problem is discussed from Bolibrukh’s point of view. The second part expounds 1-summability and Ecalle’s theory of resurgence under fairly general conditions. It contains numerous examples and presents an analysis of the singularities in the Borel plane via “alien calculus”, which provides a full description of the Stokes phenomenon for linear or non-linear differential or difference equations. The first of a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists interested in geometric, algebraic or local analytic properties of dynamical systems. It includes useful exercises with solutions. The prerequisites are a working knowledge of elementary complex analysis and differential algebra.

Divergent Series, Summability and Resurgence I-III

Author: Eric Delabaere

Publisher: Springer

Published: 2017-09

Total Pages:

ISBN-13: 9783319595276

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This three-volume work treats divergent series in one variable, especially those arising as solutions to complex ordinary differential or difference equations, and methods for extracting their analytic information. It provides a systematic construction, illustrated with examples, of the various theories of summability and the theory of resurgence developed since the 1980s. The Stokes phenomenon, for both linear and non-linear equations, plays an underlying and unifying role throughout the volumes. Applications presented include resurgent analyses of the First Painlevé equation and of the tangent-to-identity germs of diffeomorphisms of C, and links to differential Galois theory and the Riemann-Hilbert problem for linear differential equations. The volumes are aimed at graduate students, mathematicians in general, and theoretical physicists who are interested in the theories of monodromy, summability, and resurgence, as well as the current problems in the field.Although the three volumes are closely related, they have been organized to be read independently. The prerequisites are advanced calculus, especially holomorphic functions in one complex variable, and differential algebra. Moreover; the various themes are presented thoroughly step-by-step so as to be accessible to first-year graduate students in mathematics. This three-volume treatise should become a reference on summability and resurgence.

Complex Differential and Difference Equations

Author: Galina Filipuk

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2019-11-18

Total Pages: 473

ISBN-13: 3110611422

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With a balanced combination of longer survey articles and shorter, peer-reviewed research-level presentations on the topic of differential and difference equations on the complex domain, this edited volume presents an up-to-date overview of areas such as WKB analysis, summability, resurgence, formal solutions, integrability, and several algebraic aspects of differential and difference equations.

The Painlevé Handbook

Author: Robert Conte

Publisher: Springer Nature

Published: 2020-11-07

Total Pages: 389

ISBN-13: 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.

Resurgence, Physics and Numbers

Author: Frédéric Fauvet

Publisher: Springer

Published: 2017-11-17

Total Pages: 384

ISBN-13: 8876426132

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This book is issued from a conference around resurgent functions in Physics and multiple zetavalues, which was held at the Centro di Ricerca Matematica Ennio de Giorgi in Pisa, on May 18-22, 2015. This meeting originally stemmed from the impressive upsurge of interest for Jean Ecalle's alien calculus in Physics, in the last years – a trend that has considerably developed since then. The volume contains both original research papers and surveys, by leading experts in the field, reflecting the themes that were tackled at this event: Stokes phenomenon and resurgence, in various mathematical and physical contexts but also related constructions in algebraic combinatorics and results concerning numbers, specifically multiple zetavalues.

Integrability, Quantization, and Geometry: I. Integrable Systems

Author: Sergey Novikov

Publisher: American Mathematical Soc.

Published: 2021-04-12

Total Pages: 516

ISBN-13: 1470455919

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This book is a collection of articles written in memory of Boris Dubrovin (1950–2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions to this collection of papers are split into two parts: “Integrable Systems” and “Quantum Theories and Algebraic Geometry”, reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.

Formal And Analytic Solutions Of Differential Equations

Author: Galina Filipuk

Publisher: World Scientific

Published: 2022-03-03

Total Pages: 400

ISBN-13: 1800611374

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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.