Mathematical Structures in Continuous Dynamical Systems: Poisson Systems and Complete Integrability With Applications from Fluid Dynamics (STUDIES IN MATHEMATICAL PHYSICS) - Hardcover

Groesen, E. Van; Jager, E.

Hardcover

ISBN 10: 0444821511 ISBN 13: 9780444821515

Publisher: North-Holland

Synopsis

In mathematical physics various phenomena from nature are described at each instant with an infinite-dimensional state variable (a function of spatial variables, in general), and basic laws of physics describe the evolution. One important area of research, both for physical reasons and for the advancement of mathematical methods is fluid dynamics. Mathematically speaking, the state variable evolves according to a partial differential equation, an "evolution equation" describing the dynamical system. Dynamical systems for discrete (finite dimensional) systems, have been studied at length in classical mechanics, and new results and ideas such as chaos, are abundant. Although much more complicated than discrete systems, new developments for continuous systems (with spatial variations) are impressive. This book addresses several aspects, all of which can be viewed as generalizations of methods from classical mechanics. It explains in various ways how physical structures can be expected as a consequence of the underlying mathematical structure of the equation. Complete integrability is one such mathematical structure, but systems with a less restrictive Poisson (or Hamiltonian) structure can also exhibit the same properties. Famous equations like the Korteweg - de Vries, nonlinear Schrodinger, Sine-Gordon, Boussinesq equations are treated in detail. The book is divided into two parts. Part I deals with (general) Poisson systems, mainly for problems from fluid dynamics. Wave equations and the equations for vortical flows are the prime examples. Part II provides an introduction to the mathematical theory of solitons.

About this title

Review

... this book provides an excellent review of the role of infinite-dimensional structure in the range of PDE problems. These concepts are finding increasingly many further uses, for example in atmospheric dynamics, magnetohydrodynamics and plasma physics, and merit wider and deeper familiarity. -- R.S. MacKay

This is a nice book that will be helpful to anyone interested in the subject. -- Mathematical Reviews

Edition details

• Publisher: North-Holland
• Language: English
• ISBN 10: 0444821511
• ISBN 13: 9780444821515
• Binding: Hardcover