• Current filamentation instability
• Multi-stream model
• Plasma physic
• Vlasov-Maxwell model
• Weibel instability

The origin of magnetic fields observed in laboratory and astrophysical plasmas is one of
the most challenging problems in plasma physics. In this respect, the Weibel type instabilities
are considered of key importance. These instabilities are caused by a temperature anisotropy (Weibel instability) and electron momentum (current filamentation instability).
The main objective of this thesis is the theoretical and numerical study of these instabilities in a collisionless plasma in the relativistic regime.
The first aspect of this work is to study the nonlinear regime of these instabilities and the role of kinetic and relativistic effects on the structure of self-consistent electromagnetic fields. In this context, a key problem for the theory and applications, is the identification and analysis of coherent structures developed spontaneously in the nonlinear regime of kinetic scales.
A second aspect of the work is the development of analytical and numerical techniques for the study of collisionless plasmas. A mathematical model of reference is the Vlasov-Maxwell model, where the Vlasov equation (mean field theory) is coupled to the Maxwell equations in a self-consistent way. A one-dimensional model, the multi-stream model, is also introduced. Based on a dimensional reduction technique, it is both an analytical model "simple" having the advantage of being able to solve a 1D Vlasov equation for each particle beam, and a numerical model less expensive than a complete model.