Korteweg-de Vries flux avec conditions initiales générales, couverture rigide par Kotani,...
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- Achat immédiatKorteweg-de Vries flux avec conditions initiales générales, couverture rigide par Kotani,...
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Numéro de l'objet eBay :364863843320
Dernière mise à jour le 18 sept. 2024 14:27:59 CEST. Afficher toutes les modificationsAfficher toutes les modifications
Caractéristiques de l'objet
- État
- ISBN
- 9789819997374
- Book Title
- Korteweg-De Vries Flows with General Initial Conditions
- Book Series
- Mathematical Physics Studies
- Publisher
- Springer
- Item Length
- 9.3 in
- Publication Year
- 2024
- Format
- Hardcover
- Language
- English
- Illustrator
- Yes
- Genre
- Mathematics, Science
- Topic
- Functional Analysis, Probability & Statistics / General, Physics / Mathematical & Computational
- Item Width
- 6.1 in
- Number of Pages
- X, 162 Pages
À propos de ce produit
Product Identifiers
Publisher
Springer
ISBN-10
9819997372
ISBN-13
9789819997374
eBay Product ID (ePID)
28064422765
Product Key Features
Book Title
Korteweg-De Vries Flows with General Initial Conditions
Number of Pages
X, 162 Pages
Language
English
Publication Year
2024
Topic
Functional Analysis, Probability & Statistics / General, Physics / Mathematical & Computational
Illustrator
Yes
Genre
Mathematics, Science
Book Series
Mathematical Physics Studies
Format
Hardcover
Dimensions
Item Length
9.3 in
Item Width
6.1 in
Additional Product Features
Number of Volumes
1 vol.
Table Of Content
Introduction.- Sato's Theory.- KdV Flow I: Reflectionless Case.- KdV Flow II: Extension.- Applications.- Further Topics.- Appendix.
Synopsis
Large numbers of studies of the KdV equation have appeared since the pioneering paper by Gardner, Greene, Kruskal, and Miura in 1967. Most of those works have employed the inverse spectral method for 1D Schrödinger operators or an advanced Fourier analysis. Although algebraic approaches have been discovered by Hirota-Sato and Marchenko independently, those have not been fully investigated and analyzed. The present book offers a new approach to the study of the KdV equation, which treats decaying initial data and oscillating data in a unified manner. The author's method is to represent the tau functions introduced by Hirota-Sato and developed by Segal-Wilson later in terms of the Weyl-Titchmarsh functions (WT functions, in short) for the underlying Schrödinger operators. The main result is stated by a class of WT functions satisfying some of the asymptotic behavior along a curve approaching the spectrum of the Schrödinger operators at + in an order of -( n -1/2)for the n th KdV equation. This class contains many oscillating potentials (initial data) as well as decaying ones. Especially bounded smooth ergodic potentials are included, and under certain conditions on the potentials, the associated Schrödinger operators have dense point spectrum. This provides a mathematical foundation for the study of the soliton turbulence problem initiated by Zakharov, which was the author's motivation for extending the class of initial data in this book. A large class of almost periodic potentials is also included in these ergodic potentials. P. Deift has conjectured that any solutions to the KdV equation starting from nearly periodic initial data are almost periodic in time. Therefore, our result yields a foundation for this conjecture. For the reader's benefit, the author has included here (1) a basic knowledge of direct and inverse spectral problem for 1D Schrödinger operators, including the notion of the WT functions; (2)Sato's Grassmann manifold method revised by Segal-Wilson; and (3) basic results of ergodic Schrödinger operators., Large numbers of studies of the KdV equation have appeared since the pioneering paper by Gardner, Greene, Kruskal, and Miura in 1967. Most of those works have employed the inverse spectral method for 1D Schrödinger operators or an advanced Fourier analysis. Although algebraic approaches have been discovered by Hirota-Sato and Marchenko independently, those have not been fully investigated and analyzed. The present book offers a new approach to the study of the KdV equation, which treats decaying initial data and oscillating data in a unified manner. The author's method is to represent the tau functions introduced by Hirota-Sato and developed by Segal-Wilson later in terms of the Weyl-Titchmarsh functions (WT functions, in short) for the underlying Schrödinger operators. The main result is stated by a class of WT functions satisfying some of the asymptotic behavior along a curve approaching the spectrum of the Schrödinger operators at +∞ in an order of -( n -1/2)for the n th KdV equation. This class contains many oscillating potentials (initial data) as well as decaying ones. Especially bounded smooth ergodic potentials are included, and under certain conditions on the potentials, the associated Schrödinger operators have dense point spectrum. This provides a mathematical foundation for the study of the soliton turbulence problem initiated by Zakharov, which was the author's motivation for extending the class of initial data in this book. A large class of almost periodic potentials is also included in these ergodic potentials. P. Deift has conjectured that any solutions to the KdV equation starting from nearly periodic initial data are almost periodic in time. Therefore, our result yields a foundation for this conjecture. For the reader's benefit, the author has included here (1) a basic knowledge of direct and inverse spectral problem for 1D Schrödinger operators, including the notion of the WT functions; (2)Sato's Grassmann manifold method revised by Segal-Wilson; and (3) basic results of ergodic Schrödinger operators.
LC Classification Number
QC19.2-20.85
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