One-dimensional kinetic description of nonlinear traveling-pulse and traveling-wave disturbances in long coasting charged particle beams

This paper makes use of a one-dimensional kinetic model to investigate the nonlinear longitudinal dynamics of a long coasting beam propagating through a perfectly conducting circular pipe with radius rw . The average axial electric field is expressed as ⟨Ez⟩=−(∂/∂z)⟨ϕ⟩=−ebg0∂λb/∂z−ebg2rw2∂3λb/∂z3 , where g0 and g2 are constant geometric factors, λb(z,t)=∫dpzFb(z,pz,t) is the line density of beam particles, and Fb(z,pz,t) satisfies the 1D Vlasov equation. Detailed nonlinear properties of traveling-wave and traveling-pulse (soliton) solutions with time-stationary waveform are examined for a wide range of system parameters extending from moderate-amplitudes to large-amplitude modulations of the beam charge density. Two classes of solutions for the beam distribution function are considered, corresponding to: (i) the nonlinear waterbag distribution, where Fb=const in a bounded region of pz -space; and (ii) nonlinear Bernstein-Green-Kruskal (BGK)-like solutions, allowing for both trapped and untrapped particle distributions to interact with the self-generated electric field ⟨Ez⟩ .