Analytical theory of self-consistent current structures in a collisionless plasma

The most-studied classes of exact solutions to Vlasov – Maxwell equations for stationary neutral current structures in a collisionless relativistic plasma, which allow the particle distribution functions (PDFs) to be chosen at will, are reviewed. A general classification is presented of the current sheets and filaments described by the method of invariants of motion of particles whose PDF is symmetric in a certain way in coordinate and momentum spaces. The possibility is discussed of using these explicit solutions to model the observed and/or expected features of current structures in cosmic and laboratory plasmas. Also addressed are how the magnetic field forms and the analytical description of the so-called Weibel instability in a plasma with an arbitrary PDF.