“…According to the method of integration over initial data [2,3], the general solution to the Vlasov equation for the distribution functions f α can be written as (3) Here, z α ( t , z 0 , p 0 ) and p α ( t , z 0 , p 0 ) are the solutions to the set of characteristic relativistic Vlasov equations (4) with the initial conditions z α | t = 0 = z 0 ∈ (-∞ , + ∞ ), p α | t = 0 = p 0 ∈ (-∞ , + ∞ ) , and the velocities v α of the beam and plasma electrons are described by the expressions (5) We assume that, in the system under consideration, the initial perturbation occurs on a characteristic longitudinal scale l and also that the eigenfunctions ϕ s ( r ⊥ ) and eigenvalues of the waveguide cross section are known. Under these assumptions, the polarization potential ψ can be represented as a double series, (6) where k z = 2π/l is the fundamental longitudinal wavenumber.…”