Lie-Backlund Transformations in Applications

by Anderson, R. L.; Ibragimov, N. H.

• ISBN13: 9780898711516
• ISBN10: 0898711517
• Format: Hardcover
• Copyright: 1979-06-01
• Publisher: Society for Industrial & Applied

Summary

This title presents an introduction to the classical treatment of Backlund and general surface transformations; and includes detailed and accessible techniques for constructing both groups of tranformations which will be of great value to the scientist and engineer in the analysis of mathematical models of physical phenomena. Classical and recent examples of Backlund transformations as applied to geometry, nonlinear optics, turbulence models, nonlinear waves and quantum mechanics are given. The authors discuss applications of Lie-Backlund transformations in mechanics, quantum mechanics, gas dynamics, hydrodynamics, and relativity.

Table of Contents

Introduction

Classical foundations

Surface-transformations: Lie's first questions, finite-order Generalization, Infinite- order structure

Tranformation of families of surfaces: Lie's second question, Bianchi-Lie tranformation, Backlund transformations

Examples of Backlund transformations: invariance transformations

Transformations relating different differential equations

Tangent transformation groups: Finite-order tangent tranformations, tangent transformation groups of Sophus Lie, higher-order tangent transformation groups, infinite-order tangent transformations

Lie-Backlund tangent transformation Groups, Lie-Backlund equations

Application to differential equations: defining equations

Group theoretical nature of conservation laws, Lie via Lie-Backlund for ordinary differential equations, group theoretical equivalence of quantum-mechanical systems

Some applications of Backlund transformations: Nonlinear optics, solitons and the KdV equation, constants of the motion and conservation laws, weakly dispersive shallow-water waves in two space dimensions

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