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This paper develops a clear procedure for solving the nonlinear Vlasov–Maxwell equations for a one-component intense charged particle beam or finite-length charge bunch propagating through a cylindrical conducting pipe (radius r=rw=const), and confined by an applied focusing force Ffoc. In particular, the nonlinear Vlasov–Maxwell equations are Lorentz transformed to the beam frame (“primed” variables) moving with axial velocity Vb=βbc=const relative to the laboratory. In the beam frame, the particle motions are nonrelativistic for the applications of practical interest, already a major simplification. Then, in the beam frame, the electrostatic approximation (Es′=−∇′φ′, ET′≃0≃Bs′) is made, which fully incorporates beam space-charge effects, but neglects any fast electromagnetic processes with transverse polarization (e.g., light waves). The resulting Vlasov–Maxwell equations are then Lorentz transformed back to the laboratory frame, and properties of the self-generated fields and resulting nonlinear Vlasov...

Implications of the electrostatic approximation in the beam frame on the nonlinear Vlasov–Maxwell equations for intense beam propagation