Nonlinear reactions between electrons and high frequency waves make change of DC-component of the electrons. ∂f₀/∂t = (e/m)Σ-iqφ_-q(∂f_q/∂v), where the φ is potential and the -iqφ means electric field. While the DC-component of electrons, f₀(v,t), sustains the neutrality condition n_e=n_i, the change will give rise to corresponding ion waves. We analize here the interactions between two beam electrons and an ion with renormalization theory [1]. As an envelope solitary wave (soliton) is third order nonlinear quantity, the retarded Green function, G(k,v,ω;k',v',ω'), includes a second order self self-energy term, Σk,v,ω;k',v',ω'), whose real part represents collision frequency and the imaginary part means frequency broadening or frequency sift. Then the soliton is conposed of the Green function times electron distribution, f(k',v',ω'), and a factor, -ike/(ε₀k²), where the Green function is written by G(k,v,ω;k',v',ω') = -i/{ω - k·v + Σ(k,v,ω;k',v',ω')} and the self energy part is given by :
Σ(k,v,ω;k',v',ω') = (e/m)²Σ_k1∫(dω/2&pi)E(k₁ω₁)·∂{G(k-k₁,v,ω-ω₁;;k',v',ω')E(-k₁,-ω)·∂/∂v}/∂v. The complex function Σ(k,v,ω;k',v',ω') is collision frequency and it depends on the intensity of the electric field |E(k₁,ω₁)|² [2]. If the wave, (k₁,ω₁), corresponds to an ion wave, then electron interaction, that one beam electron emits a phonon and another electron absorbs it, is possible. By this interaction, two electrons will not work as Fermion but work as Bosson, so that the soliton, which caused by the collision between the beam electron and negative phase of the ion wave, approaches each other to the negative phase of the ion wave. Thus the electron beam emits the soliton when it collides to the the negative phase of the ion wave by Bremsstrahlung. Our results fit well to the theory of coupled nonlinear electron-plasma and ion wave obtained by the fluid model [3].

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