An error was encountered while trying to add the item to the cart. Please try again.
Copy To Clipboard
Successfully Copied!
Nonlinear Dirac Equation: Spectral Stability of Solitary Waves

Nabile Boussaïd Université de Franche-Comté, Besançon, France
Andrew Comech Texas A&M University, College Station, TX and Institute for Information Transmission Problems, Moscow, Russia
Available Formats:
Hardcover ISBN: 978-1-4704-4395-5
Product Code: SURV/244
List Price: $129.00 MAA Member Price:$116.10
AMS Member Price: $103.20 Electronic ISBN: 978-1-4704-5422-7 Product Code: SURV/244.E List Price:$129.00
MAA Member Price: $116.10 AMS Member Price:$103.20
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $193.50 MAA Member Price:$174.15
AMS Member Price: $154.80 Click above image for expanded view Nonlinear Dirac Equation: Spectral Stability of Solitary Waves Nabile Boussaïd Université de Franche-Comté, Besançon, France Andrew Comech Texas A&M University, College Station, TX and Institute for Information Transmission Problems, Moscow, Russia Available Formats:  Hardcover ISBN: 978-1-4704-4395-5 Product Code: SURV/244  List Price:$129.00 MAA Member Price: $116.10 AMS Member Price:$103.20
 Electronic ISBN: 978-1-4704-5422-7 Product Code: SURV/244.E
 List Price: $129.00 MAA Member Price:$116.10 AMS Member Price: $103.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$193.50 MAA Member Price: $174.15 AMS Member Price:$154.80
• Book Details

Mathematical Surveys and Monographs
Volume: 2442019; 297 pp
MSC: Primary 35; 37; 47; 81;

This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves.

The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrödinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.

Graduate students and researchers interested in the analysis of non-linear PDEs.

• Chapters
• Introduction
• Distributions and function spaces
• Spectral theory of nonselfadjoint operators
• Linear stability of NLS solitary waves
• Solitary waves of nonlinear Schrödinger equation
• Limiting absorption principle
• Carleman–Berthier–Georgescu estimates
• The Dirac matrices
• The Soler model
• Bi-frequency solitary waves
• Bifurcations of eigenvalues from the essential spectrum
• Nonrelativistic asymptotics of solitary waves
• Spectral stability in the nonrelativistic limit

• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2442019; 297 pp
MSC: Primary 35; 37; 47; 81;

This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves.

The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrödinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.

Graduate students and researchers interested in the analysis of non-linear PDEs.

• Chapters
• Introduction
• Distributions and function spaces
• Spectral theory of nonselfadjoint operators
• Linear stability of NLS solitary waves
• Solitary waves of nonlinear Schrödinger equation
• Limiting absorption principle
• Carleman–Berthier–Georgescu estimates
• The Dirac matrices
• The Soler model
• Bi-frequency solitary waves
• Bifurcations of eigenvalues from the essential spectrum
• Nonrelativistic asymptotics of solitary waves
• Spectral stability in the nonrelativistic limit
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
You may be interested in...
Please select which format for which you are requesting permissions.