Mathematical Structures in Continuous Dynamical Systems

This work addresses several aspects of continuous dynamical systems, all of which can be viewed as generalizations of methods from classical mechnics. Equations such as the Korteweg-de Vries, non-linear Schrodinger, Sine-Gordon and Boussinesq equations are treated in detail.

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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.