Abstract : This thesis is devoted to the study of
the stability of small stationary solutions of a nonlinear time
dependent equation coming from relativistic quantum mechanics: the
nonlinear Dirac equation.
In this study, non linear equations are viewed as small nonlinear
perturbations of linear systems. A part of this thesis is hence
devoted to the study of linear problems. We prove that for a Dirac
operator, with no resonance at thresholds nor eigenvalue at
thresholds, the propagator satisfies propagation and dispersive
estimates. We also deduce smoothness estimates in the sense of Kato
and Strichartz estimates.
With some ad hoc assumptions on the discrete spectrum of a
Dirac operator, we build small manifolds of stationary states. Then
with small variations on these assumptions, we can highlight some
stabilization process and orbital instability phenomena for some