Relativistic corrections to the nonlinear plasma permittivity: I. A covariant kinetic theory for longitudinal plasma wave interactions

Abstract

The resonant interaction between three longitudinal plasma waves is investigated on the basis of a relativistic kinetic theory. The corresponding nonlinear plasma permittivity is derived assuming that the initial plasma at thermal equilibrium is described by a relativistic Maxwellian distribution. Introducing the covariant variables, explicit and reduced forms of relativistically correct nonlinear permittivities are obtained which hold for arbitrary values of the complex wave-frequencies and the plasma temperatures. In the limit of nonrelativistic temperatures the covariant form of the nonlinear permittivity is expressed in terms of the plasma dispersion function which facilitates the evaluation of wave-coupling constant. In the limit of a nonrelativistic theory the covariant form is reduced to the noncovariant version which is also expressed in terms of the plasma dispersion function. The relativistic correction found in the covariant approach for nonlinear plasma permittivity is estimated to be generally small, but it could become important when superluminal plasma waves are involved in the nonlinear coupling processes.