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Lie-Backlund Transformations in Applications

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By Robert L. Anderson and Nail H. Ibragimov


1979 / x + 124 pages / Hardcover / ISBN: 978-0-898711-51-6 / List Price $60.50 / SIAM Member Price $42.35 / Order Code AM01

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.

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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