Spectral Transform and Solitons
Published on 14. May 2014
533 pages
978-0-08-087534-7 (ISBN)
Description
Spectral Transform and Solitons
More details
Series
Language
English
Place of publication
Burlington
Netherlands
Product notice
Windows
ISBN-13
978-0-08-087534-7 (9780080875347)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
F. Calogero | A. Degasperis
Spectral Transform and Solitons: Volume 13
Book
04/2000
North-Holland
€107.70
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Content
- Front Cover
- Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations
- Copyright Page
- Contents
- Contents of Volume II
- Preface
- Foreword
- Chapter 0. Introduction
- O.N. Notes to chapter 0
- Chapter 1. The Main Idea and Results: An Overview
- 1.1. Solution of linear evolution equations by Fourier transform
- 1.2. A class of solvable nonlinear evolution equations
- 1.3. The spectral transform
- 1.4. Solution of nonlinear evolution equations via the spectral transform
- 1.5. Relation to the Fourier transform technique to solve linear evolution equations
- 1.6. Qualitative behaviour of the solutions: solitons and background
- 1.7. Additional properties of the solutions
- 1.8. A list of solvable equations
- l.N. Notes to chapter 1
- Chapter 2. The Schroedinger Spectral Problem on the Line
- 2.1. Direct spectral problem
- 2.2. Inverse spectral problem
- 2.3. The spectral transform
- 2.4. Formulae relating two functions to the corresponding spectral transforms
- 2.N. Notes to chapter 2
- Chapter 3. Nonlinear Evolution Equations Solvable by the (Schroedinger) Spectral Transform
- 3.1. KdV and higher KdV's
- 3.2. Analysis of the solutions
- 3.N. Notes to chapter 3
- Chapter 4. Bäcklund Transformations and Related Results
- 4.1. Bäcklund transformations
- 4.2. Commutativity of Bäcklund transformations and nonlinear superposition principle
- 4.3. Resolvent formula
- 4.4. Nonlinear operator identities
- 4.5. Generalized Bäcklund transformations and resolvent formula
- 4.N. Notes to chapter 4
- Chapter 5. Conservation Laws
- 5.N. Notes to chapter 5
- Chapter 6. Extensions
- 6.1. More variables
- 6.2. Coefficients depending linearly on x
- 6.3. Solutions of the KdV equation that are asymptotically linear in x
- 6.4. Solutions of the KdV equation with one real double pole
- 6.5. Evolution equations associated with the spectral problem based on the ODE-?xx(x)+u(x)?(x)=k2p2(x)?(x)
- 6.N. Notes to chapter 6
- Appendices
- A.1. On the number of discrete eigenvalues of the Schroedinger spectral problem on the whole line
- A.2. Orthogonality and completeness relations for the Schroedinger spectral problem on the whole line
- A.3. Asymptotic behaviour (in k) of the transmission imd reflection coefficients
- A.4. Dispersion relations for the transmission coefficient
- A.5. The inverse spectral Schroedinger problem on the whole line
- A.6. Wronskian integral relations: proofs
- A.7. Spectral integral relations: proofs
- A.8. A formula for the variation of the coefficients of the asymptotic expansion of the phase of the transmission coefficient
- A.9. Properties of the operators ?,?,L, L, and other formulae
- A.10. The two-soliton solution of the KdV and higher KdV equations
- A.11. Miura and Gardner transformations and related results
- A.12. Bäcklund transformations, Darboux transformations and Bargmann strip
- A.13. Asymptotic expansion of C(k) = 2ik[1 - 1/T(k)]
- A.14. Conserved quantities for generalized KdV equations
- A.15. Reflection and transmission coefficients at k= 0
- A.16. The spectral transform outside of the class of bona fide potentials
- A.17. Applications of the wronskian and spectral integral relations to the Schroedinger scattering problem on the whole line
- A.18. On the class of equations ?(L)ut=a(L)ux
- A.19. Examples of functions with explicitly known spectral transform
- A.20. A general approach based on the algebra of differential operators
- connections with, and amongst the spectral transform method, the Lax approach and the AKNS technique
- A.21. Local conservation laws: proofs
- A.22. "Variable phase approach" to the Schroedinger scattering problem on the whole line
- A.23. KdV and higher KdV equations as hamiltonian flows: an outline
- References
- Subject Index
- List of Symbols
