Hardcover ISBN:  9781470443955 
Product Code:  SURV/244 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
Electronic ISBN:  9781470454227 
Product Code:  SURV/244.E 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 

Book DetailsMathematical Surveys and MonographsVolume: 244; 2019; 297 ppMSC: 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 DiracMaxwell and similar equations are not yet available, we can consider the Dirac equation with scalar selfinteraction, 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 DiracPauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.ReadershipGraduate students and researchers interested in the analysis of nonlinear PDEs.

Table of Contents

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

Bifrequency solitary waves

Bifurcations of eigenvalues from the essential spectrum

Nonrelativistic asymptotics of solitary waves

Spectral stability in the nonrelativistic limit


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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 DiracMaxwell and similar equations are not yet available, we can consider the Dirac equation with scalar selfinteraction, 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 DiracPauli 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 nonlinear 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

Bifrequency solitary waves

Bifurcations of eigenvalues from the essential spectrum

Nonrelativistic asymptotics of solitary waves

Spectral stability in the nonrelativistic limit