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In recent decades, the study of nonlinear integrable systems has grown into a full-fledged research topic. The ideas associated with Lie algebra and groups can form a particularly elegant approach to the properties of nonlinear systems. In this volume, the author presents the techniques for using Lie algebraic concepts to explore nonlinear integrable systems. His emphasis is not on developing a rigorous mathematical basis, but on using Lie algebraic methods as a tool. The text summarizes the use of these methods in various fields of theoretical research and serves mathematicians and theoretical physicists working in nonlinear integrable systems, dynamical systems, and chaos.
List of contents
INTRODUCTION,Lax Equation and IST,Conserved Densities and Hamiltonian Structure,Symmetry Aspects,Observations,LIE ALGEBRA,Introduction,Structure Constants and Basis of Lie Algebra,Lie Groups and Lie Algebra,Representation of a Lie Algebra,Cartan-Killing Form,Roots Space Decomposition,Lie Groups: Finite and Infinite Dimensional,Loop Groups,Virasoro Group,Quantum Tori Algebra,Kac-Moody Algebra,Serre's Approach to Kac-Moody Algebra,Gradation,Other Infinite Dimensional Lie Algebras,PROLONGATION THEORY,Introduction,Sectioning of Forms,The KdV Problem,The Method of the Hall Structure,Prolongation in (2+1) Dimension,Method of Pseudopotentials,Prolongation Structure and the Backlund Transformation,Constant Coefficient Ideal,Connections,Morphisms and Prolongation,Principal Prolongation Structure,Prolongations and Isovectors,Vessiot's Approach,Observations,CO-ADJOINT ORBITS,Introduction,The Kac-Moody Algebra,Integrability Theorem: Adler, Kostant, Symes,Superintegrable Systems,Nonlinear Partial Differential Equation,Extended AKS Theorem,Space-Dependent Integrable Equation,The Moment Map,Moment Map in Relation to Integrable Nonlinear Equation,Co-Adjoint Orbit of the Volterra Group,SYMMETRIES OF INTEGRABLE SYSTEMS,Introduction,Lie Point and Lie Backlund Symmetry,Lie Backlund Transformation,Some New Ideas in Symmetry Analysis,Non-Local Symmetries,Observations,HAMILTONIAN STRUCTURE,Introduction,Drinfeld Sokolob Approach,The Lie Algebraic Approach,Example of Hamiltonian Structure and Reduction,Hamiltonian Reduction in (2+1) Dimension,Hamiltonian Reduction of Drinfeld and Sokolov,Kupershmidt's Approach,Gelfand Dikii Formula,Trace Identity and Hamiltonian Structure,Symmetry and Hamiltonian Structure,CLASSICAL r-MATRIX,Introduction,Double Lie Algebra,Classical r-Matrix,The Use of r-Matrix,The r-Matrix and KP Equation
About the author
Amit K. Roy-Chowdhury (University of California, Riverside, USA) (Author)
Summary
Presents the basic techniques of using Lie algebraic concepts to explore the domain of nonlinear integrable systems. This book begins by establishing a practical basis in Lie algebra, including discussions of structure Lie, loop, and Virasor groups, quantum tori and Kac-Moody algebras, and gradation.
