The Painlevé Handbook

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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.

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.

"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)

“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)