The Painlevé handbook

• Main Authors: Conte, Robert, Muset, Micheline
• Format: Book
• Language: English
• Published: Springer 2017
• Subjects: Physics; Astronomy ; Painlevé equations ; Mathematical physics.; 518.6
• Online Access: http://repository.vnu.edu.vn/handle/VNU_123/25112
• Institution: Vietnam National University, Hanoi

Chap. 1, we insist that a nonlinear equation shouldnotbe considered as the pertur-bation of a linear equation. We illustrate using two simple examples the importanceof taking account of the singularity structure in the complex plane to determine thegeneral solution of nonlinear equations. We then present the point of view of thePainlev ́e school todefine new functionsfrom nonlinear ordinary differential equa-tions (ODEs) possessing a general solution which can be made single valued in itsdomain of definition (Painlev ́e property, PP).In Chap. 2, we present a local analysis, called thePainlev ́etest, in order to in-vestigate the nature of the movable singularities (i.e. whose location depends onthe initial conditions) of the general solution of a nonlinear differential equation.The simplest of the methods involved in this test was historically introduced bySophie Kowalevski [257] and later turned into an algorithm by Bertrand Gambier[163]. For equations possessing the Painlev ́e property, the test is by constructionsatisfied, therefore we concentrate on equations which generically fail the test, inorder to extract some constructive information on cases of partial integrability. Wefirst choose four examples describing physical phenomena, for which the test selectscases which may admit closed form particular solutions1or first integrals