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.
General
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