Nonlinear Dirac Equation: Spectral Stability of Solitary Waves
About this Title
Nabile Boussaïd, Université de Franche-Comté, Besançon, France and Andrew Comech, Institute for Information Transmission Problems, Moscow, Russia
Publication: Mathematical Surveys and Monographs
Publication Year: 2019; Volume 244
ISBNs: 978-1-4704-4395-5 (print); 978-1-4704-5422-7 (online)
MathSciNet review: 3971006
MSC: Primary 35-02; Secondary 35C08, 35Q55, 37K40, 47F05, 81Q05, 81Q10
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.
Table of Contents
- 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