Algebraic Integrability, Painlevé Geometry and Lie Algebras

ISBN-13: 9783642061288 / Angielski / Miękka / 2010 / 484 str.

Mark Adler; Pierre Van Moerbeke; Pol Vanhaecke

Opis

This Ergebnisse volume is aimed at a wide readership of mathematicians and physicists, graduate students and professionals. The main thrust of the book is to show how algebraic geometry, Lie theory and Painleve analysis can be used to explicitly solve integrable differential equations and construct the algebraic tori on which they linearize; at the same time, it is, for the student, a playing ground to applying algebraic geometry and Lie theory. The book is meant to be reasonably self-contained and presents numerous examples. The latter appear throughout the text to illustrate the ideas, and make up the core of the last part of the book. The first part of the book contains the basic tools from Lie groups, algebraic and differential geometry to understand the main topic. "

Recenzje

From the reviews of the first edition:

"The aim of this book is to explain 'how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations'. ... One of the main advantages of this book is that the authors ... succeeded to present the material in a self-contained manner with numerous examples. As a result it can be also used as a reference book for many subjects in mathematics. In summary ... a very good book which covers many interesting subjects in modern mathematical physics." (Vladimir Mangazeev, The Australian Mathematical Society Gazette, Vol. 33 (4), 2006)

"This is an extensive volume devoted to the integrability of nonlinear Hamiltonian differential equations. The book is designed as a teaching textbook and aims at a wide readership of mathematicians and physicists, graduate students and professionals. ... The book provides many useful tools and techniques in the field of completely integrable systems. It is a valuable source for graduate students and researchers who like to enter the integrability theory or to learn fascinating aspects of integrable geometry of nonlinear differential equations." (Ma Wen-Xiu, Zentralblatt MATH, Vol. 1083, 2006)

Spis treści

1 Introduction.- 2 Lie Algebras.- 3 Poisson Manifolds.- 4 Integrable Systems on Poisson Manifolds.- 5 The Geometry of Abelian Varieties.- 6 A.c.i. Systems.- 7 Weight Homogeneous A.c.i. Systems.- 8 Integrable Geodesic Flow on SO(4).- 9 Periodic Toda Lattices Associated to Cartan Matrices.- 10 Integrable Spinning Tops.- References.