Lie Algebras, Geometry, and Toda-Type Systems

Alexander V. Razumov; Mikhail V. Saveliev

doi:10.1017/CBO9780511599927

This book introduces the use of Lie algebra and differential geometry methods to study nonlinear integrable systems of Toda type. Many challenging problems in theoretical physics are related to the solution of nonlinear systems of partial differential equations. One of the most fruitful approaches in recent years has resulted from a merging of group algebraic and geometric techniques. The book gives a comprehensive introduction to this exciting branch of science. Chapters 1 and 2 review basic notions of Lie algebras and differential geometry with an emphasis on further applications to integrable nonlinear systems. Chapter 3 contains a derivation of Toda type systems and their general solutions based on Lie algebra and differential geometry methods. The last chapter examines explicit solutions of the corresponding equations. The book is written in an accessible 'lecture note' style with many examples and exercises to illustrate key points and to reinforce understanding.

‘In all this book is to be recommended to anyone wishing to learn the mathematical machinery that underlies so much of the modern theory of nonlinear integrable systems.’

F. Burstall Source: Contemporary Physics

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Lie Algebras, Geometry, and Toda-Type Systems

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Contents

Frontmatter
pp i-viii

Contents
pp ix-xii

Preface
pp xiii-xviii

Acknowledgements
pp xix-xx

1 - Introductory data on Lie algebras
pp 1-54

2 - Basic notions of differential geometry
pp 55-128

3 - Differential geometry of Toda-type systems
pp 129-207

4 - Toda-type systems and their explicit solutions
pp 208-236

References
pp 237-241

Index
pp 242-247

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