Covariant kinetic theory for nonlinear plasma waves interaction

A rigorous and most general covariant kinetic formalism is developed to study the nonlinear-wave interaction in relativistic Vlasov plasmas. The typical nonlinear plasma reaction is a nonlinear current measured by the nonlinear plasma conductivity, and these quantities are derived here on the basis of relativistic Vlasov–Maxwell equations. Knowing the nonlinear plasma conductivity allows us to determine all plasma modes nonlinearly excited in plasma. The general covariant form of nonlinear conductivity is provided first for any value of plasma temperature and for the whole complex frequency plane by a correct analytical continuation. The further analysis is restricted to a correct relativistic particle distribution, which is vanishing for particle speeds greater than the speed of light. In the limit of non-relativistic plasma temperatures, the covariant nonlinear conductivity is significantly different from the standard non-covariant non-relativistic results, which are reached only in the formal limit of an infinitely large speed of light $c \to \infty$.