Nonlinear Dirac equation : spectral stability of solitary waves /

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...

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Bibliographic Details
Main Authors:	Boussaïd, Nabile, 1978- (Author), Komech, Andrew (Author)
Format:	Book
Language:	English
Published:	Providence, Rhode Island : American Mathematical Society, [2019]
Series:	Mathematical surveys and monographs ; no. 244.
Subjects:	Dirac equation. Wave equation. Differential equations, Partial. Particles (Nuclear physics) Partial differential equations > Qualitative properties of solutions > Bifurcation [See also 37Gxx, 37K50]. Partial differential equations > Qualitative properties of solutions > Stability. Partial differential equations > Qualitative properties of solutions > Asymptotic behavior of solutions. Partial differential equations > Representations of solutions > Soliton solutions. Partial differential equations > Spectral theory and eigenvalue problems [See also 47Axx, 47Bxx, 47F05] > General topics in linear spectral theory. Partial differential equations > Spectral theory and eigenvalue problems [See also 47Axx, 47Bxx, 47F05] > Nonlinear eigenvalue problems, nonlinear spectral theory. Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] > Infinite-dimensional Hamiltonian systems [See also 35Axx, 35Qxx] > Soliton theory, asymptotic. Operator theory > Equations and inequalities involving nonlinear operators [See also 46Txx] {For global and geometric aspects, see 58-XX} > Nonlinear spectral theory, nonlinear eigenvalue problems [.

Description
Summary:	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.
Physical Description:	vi, 297 pages : illustrations ; 27 cm.
Bibliography:	Includes bibliographical references (pages 279-292) and index.
ISBN:	9781470443955 1470443953
ISSN:	0076-5376 ;