Derivation of nonlinear Schrödinger equation for electrostatic and electromagnetic waves in fully relativistic two-fluid plasmas by the reductive perturbation method

Abstract

The reductive perturbation method is used to derive a generic form of nonlinear Schrödinger equation (NLSE) that describes the nonlinear evolution of electrostatic (ES)/electromagnetic (EM) waves in fully relativistic two-fluid plasmas. The matrix eigenvector analysis shows that there are two mutually exclusive modes of waves, each mode involving only either one of two electric potentials, A and ϕ. The general result is applied to the electromagnetic mode in electron-ion plasmas with relativistically high electron temperature (Te≫mec2). In the limit of high frequency (ck≫ωe), the NLSE predicts bump type electromagnetic soliton structures having width scaling as ∼kTe5/2. It is shown that, in electron-positron pair plasmas with high temperature, dip type electromagnetic solitons can exist. The NLSE is also applied to electrostatic (Langmuir) wave and it is shown that dip type solitons can exist if kλD≪1, where λD is the electron’s Debye length. For the kλD≫1, however, the solution is of bump type soliton with width scaling as ∼1/(k5Te). It is also shown that dip type solitons can exist in cold plasmas having relativistically high streaming speed.