Nonlinear Dirac Equation: Spectral Stability of Solitary Waves (Mathematical Surveys and Monographs) - Hardcover

Nabile Boussaid (author) & Andrew Comech (author)

Hardcover

ISBN 10: 1470443953 ISBN 13: 9781470443955

Publisher: American Mathematical Society, 2020

Synopsis

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 Schrodinger 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.

About this title

About the Author

Nabile Boussaid, Universite de Franche-Comte, Besancon, France.

Andrew Comech, Texas A&M University, College Station, TX.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2020
• Language: English
• ISBN 10: 1470443953
• ISBN 13: 9781470443955
• Binding: Hardcover
• Number of pages: 297