Lie Algebraic Methods in Integrable Systems

by Roy-Chowdhury; Amit K.

• ISBN13: 9781584880370
• ISBN10: 1584880376
• Edition: 1st
• Format: Nonspecific Binding
• Copyright: 1999-09-28
• Publisher: Chapman & Hall/

Summary

Over the last thirty years, the subject of nonlinear integrable systems has grown into a full-fledged research topic. In the last decade, Lie algebraic methods have grown in importance to various fields of theoretical research and worked to establish close relations between apparently unrelated systems.The various ideas associated with Lie algebra and Lie groups can be used to form a particularly elegant approach to the properties of nonlinear systems. In this volume, the author exposes the basic techniques of using Lie algebraic concepts to explore the domain of nonlinear integrable systems. His emphasis is not on developing a rigorous mathematical basis, but on using Lie algebraic methods as an effective tool.The 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. It then offers a detailed discussion of prolongation structure and its representation theory, the orbit approach-for both finite and infinite dimension Lie algebra. The author also presents the modern approach to symmetries of integrable systems, including important new ideas in symmetry analysis, such as gauge transformations, and the "soldering" approach. He then moves to Hamiltonian structure, where he presents the Drinfeld-Sokolov approach, the Lie algebraic approach, Kupershmidt's approach, Hamiltonian reductions and the Gelfand Dikii formula. He concludes his treatment of Lie algebraic methods with a discussion of the classical r-matrix, its use, and its relations to double Lie algebra and the KP equation.

Table of Contents

Preface
Introduction	1	(12)
Lax Equation and IST	3	(4)
Conserved Densities and Hamiltonian Structure	7	(3)
Symmetry Aspects	10	(1)
Observations	11	(2)
Lie Algebra	13	(62)
Introduction	13	(2)
Structure Constants and Basis of Lie Algebra	15	(3)
Special Linear Lie Algebra	16	(2)
Lie Groups and Lie Algebra	18	(2)
Representation of a Lie Algebra	20	(4)
Cartan-Killing Form	24	(2)
Root Space Decomposition	26	(4)
Lie Groups: Finite and Infinite Dimensional	30	(2)
Loop Groups	32	(5)
The Virasoro Group	37	(5)
Quantum Tori Algebra	42	(6)
Kac-Moody Algebra	48	(3)
Serre's Approach to Kac Moody Algebra	51	(3)
Gradation	54	(3)
Other Infinite Dimensional Lie Algebras	57	(18)
Prolongation Theory	75	(64)
Introduction	75	(2)
Sectioning of Forms	77	(3)
The KdV Problem	80	(4)
The Method of the Hall Structure	84	(5)
Prolongation in (2+1) Dimension	89	(8)
Method of Pseudopotentials	97	(7)
Prolongation Structure and the Backlund Transformation	104	(5)
Constant Coefficient Ideal	109	(2)
Connections	111	(3)
Morphisms and Prolongation	114	(5)
Principal Prolongation Structure	119	(8)
Prolongations and Isovectors	127	(5)
Vessiot's Approach	132	(4)
Observations	136	(3)
Co-adjoint Orbits	139	(54)
Introduction	139	(2)
The Kac-Moody Algebra	141	(3)
Integrability Theorem: Adler, Kostant, Symes	144	(5)
Superingegrable Systems	149	(4)
Non-Linear Partial Differential Equation	153	(8)
Extended AKS Theorem	161	(3)
Space-Dependent Integrable Equation	164	(4)
The Moment Map	168	(7)
Moment Map in Relation to Integrable Non-Linear Equation	175	(12)
Co-Adjoint Orbit of the Volterra Group	187	(6)
Symmetries of Integrable Systems	193	(62)
Introduction	193	(1)
Lie Point and Lie Backlund Symmetry	193	(4)
Lie Backlund Transformation	197	(4)
Some New Ideas in Symmetry Analysis	201	(45)
Gauge Transformation (Polyakov Approach)	202	(8)
Algebra of Lax Operator	210	(5)
Method of Fourier Analysis and Small Amplitude Expansion	215	(9)
Hidden Symmetries	224	(6)
Approximate Symmetries	230	(10)
Kac-Moody Symmetry	240	(6)
Non-Local Symmetries	246	(6)
Observations	252	(3)
Hamiltonian Structure	255	(56)
Introduction	255	(1)
Drinfeld Sokolov Approach	255	(6)
The Lie Algebraic Approach	261	(4)
Example of Hamiltonian Structure and Reduction	265	(3)
Hamiltonian Reduction in (2+1) Dimension	268	(7)
Hamiltonian Reduction of Drinfeld and Sokolov	275	(6)
Krupershmidt's Approach	281	(8)
Gelfand Dikii Formula	289	(5)
Trace Identity and Hamiltonian Structure	294	(12)
Symmetry and Hamiltonian Structure	306	(5)
Classical r-matrix	311	(36)
Introduction	311	(2)
Double Lie Algebra	313	(6)
Classical r-matrix	319	(5)
The Use of r-matrix	324	(10)
The r-matrix and KP Equation	334	(13)
Index	347