This book covers methods of building explicit solutions to nonlinear differential equations. Finding out whether the chances of success are high or low can be achieved with a powerful algorithm presented in detail called the Painlevé test.

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.

Provides an accessible yet thorough introduction to Painlevé analysis Presents expanded, revised, and updated material with comprehensive coverage of both classical and modern applications Includes many diverse examples that are relevant to physical scientists

Autorentext
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.

Zusammenfassung
"The new edition includes an overview of new results achieved since the first edition of the book, appendices, and comprehensive references. 'The Painlevé handbook' will be useful for anyone interested in the theory of integrable nonlinear differential equations and systems, as well as for graduate students and researchers in both mathematics and physical sciences." (V. A. Pron'ko, zbMATH 1489.34001, 2022)

Inhalt

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.