The Painlevé Handbook

2. Auflage 2020
ISBN: 978-3-030-53342-7
Verlag: Springer International Publishing

Buch, Englisch, 389 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 639 g

Reihe: Mathematical Physics Studies

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.

Zielgruppe

Research

Autoren/Hrsg.

Musette, Micheline

Conte, Robert

Fachgebiete

• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Naturwissenschaften Chemie Chemie Allgemein Chemometrik, Chemoinformatik
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Geometrie Dynamische Systeme

Inhaltsverzeichnis

1. Introduction.- 2. Singularity analysis: Painlevé test.- 3. Integrating ordinary differential equations.- 4. Partial Differential Equations: Painlevé test.- 5. From the test to explicit solutions of PDEs.- 6. Integration of Hamiltonian Systems.- 7. Discrete nonlinear equations.- 8. FAQ (Frequently asked questions).- 9. Selected Problems Integrated by Painlevé functions. A. The classical results of Painlevé and followers. B. More on the Painlevé transcendents. C. Brief presentation of the elliptic functions. D. Basic introduction to the Nevanlinna theory. E. The bilinear formalism. F. Algorithm for computing the Laurent series. Index.

Über Autor(innen)

Robert Conte is associate director of research at the Centre de mathématiques et de leurs applications, École normale supérieure de Cachan, CNRS, Université Paris-Saclay. He is also an honorary professor in the Department of Mathematics at the University of Hong Kong, and an associate external member of the Centre de recherches mathématiques, Université de Montréal, Canada. He received his PhD from Université Paris VI and held positions at IBM France, UC Berkeley, and the Commissariat à l’énergie atomique, Saclay, before taking on his current role. He has co-authored and edited six books and published nearly 100 articles in refereed journals. Trained in both mathematics and physics, the main theme of his research is the mathematical solution of theoretical problems arising from physics.

Micheline Musette is professor emerita at the Vrije Universiteit, Dienst Theoretische Natuurkunde (TENA) Brussels, Belgium. Prior to joining the Vrije Universiteit, she completed a PhD at Université Libre, Brussels, and held positions at the Inter University Institute for Nuclear Sciences and the National Fund for Scientific Research, Belgium. She has published around 60 papers in refereed journals, and is a member of the American Physical Society.