Interaction of an Electromagnetic Wave with a Counter-Propagating Electron Beam Under the Condition of Cyclotron Resonance Absorption: Nonlinear Periodic Waves, Modulation Instability, and Generation of Solitons

Transmission of a monochromatic wave through a counter-propagating electron beam under the condition of cyclotron resonance absorption is studied by theoretical analysis and numerical simulation. The nonlinear dispersion relation is derived showing that with an increase in the incident wave power, the non-transmission band shifts to lower frequencies, so wave propagation without attenuation becomes possible. We demonstrate that this propagation is accompanied by the modulation instability (MI), which leads to the wave disintegration into envelope solitons. When the wave frequency increases, the MI changes its nature from absolute to convective, and the generation of soliton trains is replaced by the steady-state transmission. We derive explicit stationary solutions, which turn out to be periodic and are expressed in terms of elliptic Jacobi functions. In the limit of an infinite period, they turn into bright or dark soliton solutions. The theoretical predictions are verified by numerical simulations. The simulations confirm that the boundaries of the non-transmission band are determined by the parameters of the bright and dark solitons. In this way, we obtain the practically interesting analytical condition for the parameters of the incident monochromatic wave, under which the system generates a close-to-periodic train of microwave soliton pulses.