Robert Conte - The Painlevé Handbook

The series publishes original research monographs on contemporary mathematical physics. The focus is on important recent developments at the interface of Mathematics, and Mathematical and Theoretical Physics. These will include, but are not restricted to: application of algebraic geometry, D-modules and symplectic geometry, category theory, number theory, low-dimensional topology, mirror symmetry, string theory, quantum field theory, noncommutative geometry, operator algebras, functional analysis, spectral theory, and probability theory.

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As compared to the rather compact first edition, several important topics are now added, updated, or extended.

The two main additions are (1) the description of the -method of Painlev, by far the most powerful method of the Painlev test and (2) a new chapter presenting important problems from quite various domains (geometry, random matrices, superintegrability, etc.) whose solution is achieved by Painlev functions.

The essential update is an overhaul of Chap. , which presents recently introduced constructive methods able to build all meromorphic solutions of a large class of partially integrable ordinary differential equations.

Finally, the growing interest in Painlev functions has convinced us to extend the previous appendix into a presentation of both classical and recently found properties, to serve as a reference for both graduate students and researchers.

Nonlinear differential or difference equations are encountered not only in mathematics but also in many areas of physics (evolution equations and propagation of a signal in an optical fiber), chemistry (reactiondiffusion systems), biology (competition of species), etc.

and similar elements in the discrete situation. If, on the contrary, the test fails, the system is nonintegrable or even chaotic, but it may still be possible to find solutions. Indeed, the methods developed for the integrable case still apply and may in principle produce all the available pieces of integrability, such as the solitary waves of evolution equations, or solutions describing the collision of solitary waves, or the first integrals of dynamical systems, etc. The examples chosen to illustrate these methods are mostly taken from physics. These include on the integrable side the nonlinear Schrdinger equation (continuous and discrete), the Kortewegde Vries equation, the Boussinesq equation, the HnonHeiles Hamiltonians and on the nonintegrable side the complex GinzburgLandau equation (encountered in optical fibers, turbulence, etc.), the KuramotoSivashinsky equation (phase turbulence), the reactiondiffusion model of KolmogorovPetrovskiPiskunov (KPP), the Lorenz model of atmospheric circulation and the Bianchi IX cosmological model which are both chaotic, etc.

Written at a graduate level, the book contains tutorial text as well as detailed examples and the state of the art on some current research.

In Chap. , we insist that a nonlinear equation should not be considered as the perturbation of a linear equation. We illustrate on two simple examples the importance of taking account of the singularity structure in the complex plane to determine the general solution of nonlinear equations. We then present the point of view of the Painlev school to define new functions from nonlinear ordinary differential equations (ODEs) possessing a general solution which can be made single-valued in its domain of definition (