Nonlinear electromagnetic theory of low-frequency instability of a relativistic electron beam in plasma with strongly nonpotential plasma and beam waves

A substantially nonpotential nonlinear theory of instability of a straight high-current relativistic electron beam propagating in a plasma waveguide under the conditions of the collective stimulated Cherenkov effect is elaborated. In the most general statement of problem, relativistic nonlinear equations for the temporal dynamics of this instability are obtained. The case of a low-density plasma is considered when both the plasma and beam waves are essentially nonpotential. Under the assumption that weak resonance plasma-beam interaction and linearity of plasma electrons are present, the general equations are transformed (by expanding the electron trajectories and momenta) to cubically nonlinear relativistic equations, which describe nonlinear shifts in frequencies of the plasma and beam waves. Analytic solutions to these equations are obtained with allowance for the dependence of the electron mass on the electron velocity and for the nonpotentiality of the interacting waves. The results are in good agreement with the numerical solutions of the general nonlinear equations in a wide range of values for the parameters.