The nonlinear stationary states of an electron beam moving in a homogeneous positive background are calculated for the full range of amplitudes Em of a longitudinal self-induced electric field in the collisionless limit. The parameter that controls the system is κ=Em/(4πn0mev02)1/2, where n0,v0 are the number density and velocity of the beam when electrons are submitted to maximum force. If κ≪1 the beam variables vary harmonically in space. As κ increases within 0<κ⩽1, the beam variables become gradually anharmonic but their wavelength remains constant and independent of κ. If κ>1 it is shown that no wave breaking occurs. Instead, the electric field becomes discontinuous at certain points and the electrons delay there forming periodic electrostatic (Langmuir) structures centered around negatively charged planes. The size and charge of the above structures as well as their wavelength, which now depends on κ, are derived.
The evolution (damping or growing) of electrostatic plasma waves (EPW) is governed by the self-consistent Vlasov-Poisson system [1] which can be solved in the linear limit by using the Landau method [2] or the Van Kampen technique [3]. The nonlinear evolution [4] of EPW is still under investigation [5]. On the other hand, the interaction
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