Lie Algebraic Methods in Integrable Systems

Amit K. Roy-Chowdhury

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Print ISBN: 9781138401921
• eBook ISBN: 9781000153330
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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.

Chapter 1

Introduction

An exciting and extremely active area of research investigation during the past twenty-five years has been the study of solitons and the related issues of various properties of non-linear integrable partial differential equations. In short, it may be summarized by saying that the research in the field of non-linear integrable systems has revolutionized the concepts of applied mathematics and theoretical physics. The fact that non-linearities can be treated exactly (though in (1 + 1) and in (2 + 1) dimensions) was really surprising when the pioneering papers by Lax,¹ Gardner, Green, Kruskal, and Miura,² and Zakharov and Shabat³ appeared. Initially the main attention was concentrated on the various types of waves observed in hydrodynamics, and the governing equations were of the KdV type and its different variants, such as the intermediate long wave equation, the Benjamin-Ono equation, etc. As Miura⁴ points out, despite the early derivation of the KdV equation, it was not until 1960 that new applications of the equation were found. (Gardner and Morikawa⁵ rediscovered it in the study of collision-free hydromagnetic waves.) Subsequently, the KdV equation was seen to play prominent roles in the fields of stratified internal waves, ion-acoustic waves, lattice dynamics, plasma physics, non-linear networks, and many others.⁶ Details of such studies can be found in the excellent monographs of Jeffrey and Kakutani,⁷ Dodd, Kilbeck, Gibbon, and Morris,⁸ Lamb,⁹ and also in Novikov, Manakov, Pitaevskei, and Zakharov.¹⁰

The other non-linear equation which is of immense importance is the non-linear Schrodinger equation, first analyzed by Zakharov and Shabat.³ It is found to be of prime importance in the fields of nonlinear optics, self-focusing, plasma-physics and in many other places. It was later observed that there exist many other non-linear equations which find applications in other regions of physical interest. There are the Sine-Gordon equation (in elementary particles, Josephson transmission lines), the Modified KdV equation (plasma physics, hydrodynamics), the Toda lattice (lattice dynamics, solid state physics), the Kadomstev–Petv–Ashville equation (also in hydrodynamics a...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• 1 Introduction
• 2 Lie Algebra
• 3 Prolongation Theory
• 4 Co-adjoint Orbits
• 5 Symmetries of Integrable Systems
• 6 Hamiltonian Structure
• 7 Classical r-matrix
• Index