Ba?cklund Transformations and Their Applications
Published on 18. June 1982
333 pages
978-0-08-095667-1 (ISBN)
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Ba?cklund Transformations and Their Applications
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Colin Rogers | W. F. Shadwick
Backlund Transformations and Their Applications
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01/1982
Academic Press
€100.90
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Content
- Front Cover
- Bäcklund Transformations and Their Applications
- Copyright Page
- Contents
- Preface
- Acknowledgments
- General Introduction and Outline
- CHAPTER 1. Bäcklund Transformations and Their Application to Nonlinear Equations of Mathematical Physics
- 1.1 The Origin and Importance of Bäcklund Transformations
- 1.2 The 1 + 1 Sine-Gordon Equation. The Permutability Theorem. Bianchi Diagrams
- 1.3 Ultrashort Optical Pulse Propagation in a Resonant Medium. 2Np Light Pulses
- 1.4 Propagation of Magnetic Flux through a Long Josephson Junction. An Elliptic Analog of the 1 + 1 Sine-Gordon Equation. Generation of Solutions via a Bäcklund Transformation
- 1.5 The Cole-Hopf Reduction of Burgers' Equation to the Heat Equation. Solution of Initial Value Problems
- 1.6 Finite Amplitude Dispersive Waves and the Korteweg-deVries Equation
- 1.7 The Wahlquist-Estabrook Invariant Transformation of the Korteweg-deVries Equation. The Soliton Ladder
- 1.8 An Auto-Bäcklund Transformation of the Modified Korteweg-deVries Equation. The Permutability Theorem and Generation of Solutions
- 1.9 The Miura Transformation
- 1.10 A Nonlinear Schrödinger Equation
- 1.11 Bäcklund Transformations and the AKNS System. An Invariant Transformation of the Cubic Schrödinger Equation
- 1.12 The Hirota Bilinear Operator Formulation of Bäcklund Transformations. The Boussinesq Equation. Kaup's Higher Order Water Wave Equation
- 1.13 Bäcklund Transformations for Higher Order Korteweg-deVries Equations
- 1.14 Bäcklund Transformations in Higher Dimensions. The Sine-Gordon Equation in 3 + 1 Dimensions. The Kadomtsev-Petviashvili Equation. The Yang Equations
- 1.15 The Benjamin-Ono Equation for Internal Deep Water Waves. The Nakamura Transformation
- 1.16 A Bäcklund Transformation for Joseph's Equation for Internal Waves in a Stratified Fluid with Finite Depth
- 1.17 Bäcklund Transformations of Nonlinear Lattice Equations in Their Continuum Approximation. The Konno-Sanuki Transformation
- 1.18 Bäcklund Transformations of Nonlinear Differential-Difference Equations. The Toda and Associated Lattices
- 1.19 Bäcklund Transformations in General Relativity. The Ernst Equations. Neugebauer's Permutability Theorem
- CHAPTER 2. A Local Jet-Bundle Formulation of Bäcklund Transformations
- 2.1 Preliminaries
- 2.2 Contact Structures
- 2.3 Partial Differential Equations on Jet Bundles
- 2.4 Fibered Products of Jet Bundles
- 2.5 Bäcklund Maps
- 2.6 Bäcklund Transformations Determined by Bäcklund Maps
- 2.7 Symmetries of Differential Equations and One-Parameter Families of Bäcklund Maps
- 2.8 The Wahlquist-Estabrook Procedure
- CHAPTER 3. Bäcklund Transformations in Gasdynamics, Nonlinear Heat Conduction, and Magnetogasdynamics
- 3.1 The Reciprocal Relations and the Haar Transformation
- 3.2 Properties of the Reciprocal Relations. Invariance of the Equation of State
- 3.3 Reciprocal Relations in Subsonic Gasdynamics
- 3.4 Reciprocal-Type Transformations in Steady Magnetogasdynamics
- 3.5 A Bäcklund Transformation in Nonlinear Heat Conduction
- 3.6 Bäcklund Transformations of the Loewner Type
- 3.7 Reduction of the Hodograph Equations to Canonical Form in Subsonic, Transsonic, and Supersonic Gasdynamics
- 3.8 Bäcklund Transformations in Aligned Nondissipative Magne togasdynamics
- 3.9 Invariant Transformations in Nonsteady Gasdynamics
- 3.10 Bäcklund Transformations in Lagrangian Gasdynamics. Reflection of a Centered Wave in a Shock Tube
- CHAPTER 4. Bäcklund Transformations and Wave Propagation in Nonlinear Elastic and Nonlinear Dielectric Media
- 4.1 Propagation of Large Amplitude Waves in Nonlinear Elastic Media. Reduction to Canonical Form of the Riemann Representation
- 4.2 The Model Constitutive Laws
- 4.3 Reflection and Transmission of a Large Amplitude Shockless Pulse at a Boundary
- 4.4 Electromagnetic Wave Propagation in Nonlinear Dielectric Media. Reduction to Canonical Form via Bäcklund Transformations
- 4.5 Evolution of a Large Amplitude Centered Fan in a Nonlinear Dielectric Slab
- CHAPTER 5. Bäcklund Transformations and Stress Distribution Theory in Elastostatics
- 5.1 Weinstein's Correspondence Principle in the Context of Bäcklund Transformations of the Generalized Stokes-Beltrami System
- 5.2 Application of Weinstein's Correspondence Principle in Elastostatics. Associated Axially Symmetric Punch, Crack, and Torsion Boundary-Value Problems
- 5.3 Antiplane Crack and Contact Problems in Layered Elastic Media. Bäcklund Transformations and the Bergman Series Method
- 5.4 Stress Concentration for Shear-Strained Prismatic Bodies with a Nonlinear Stress-Strain Law
- APPENDIX I: Properties of the Hirota Bilinear Operators
- APPENDIX II: Differential Forms
- II.1 Tangent Spaces and Vector Fields on Rk
- II.2 Differential p-Forms on Rk
- II.3 The Exterior Derivative
- II.4 Pull-Back Maps
- II.5 The Interior Product of Vector Fields and Differential p-Forms
- II.6 The Lie Derivative of Differential Forms
- APPENDIX III: Differential Forms on Jet Bundles
- III.1 Preliminaries
- III.2 Exterior Differential Systems on Jet Bundles
- III.3 The System of Differential Equations Associated with an Exterior Differential System
- III.4 An Exterior Differential System of m-Forms on Jk(M, N) Associated with a Quasi-Linear Equation on Jk+1(M, N )
- APPENDIX IV: The Derivation of the Equations that Define W ) Bk+1,1(?)
- APPENDIX V: Symmetries of Differential Equations and Exterior Systems
- V.1 Point Transformations
- V.2 Symmetries
- APPENDIX VI: Composition of Symmetries and Bäcklund Maps
- References
- Author Index
- Subject Index
