Nonlinear Dirac Equation: Spectral Stability of Solitary Waves
Nov 2019 · Mathematical Surveys and Monographs Book 244 · American Mathematical Soc.
Ebook
297
Pages
About this ebook
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.
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 quantizati...
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About the author
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
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
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