Spectral Transform and Solitons

Name: Spectral Transform and Solitons
Brand: North-Holland
Price: 108.0 EUR
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Calogero F. Calogero A. Degasperis(Autor*in)

North-Holland (Verlag)

Erschienen am 14. Mai 2014

533 Seiten

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978-0-08-087534-7 (ISBN)

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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

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