Korteweg–deVries equation for longitudinal disturbances in coasting charged-particle beams

This paper employs 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 r_w. The average axial electric field is expressed as ⟨E_z^s⟩=-e_bg₀∂λ_b/∂z-e_bg₂r_w²∂³λ_b/∂z³, where g₀ and g₂ are constant geometric factors, and λ_b(z,t)=∫dp_zF_b(z,p_z,t) is the line density. Assuming a waterbag distribution for the longitudinal distribution function F_b(z,p_z,t), it is shown that weakly nonlinear disturbances moving near the sound speed evolve according to the Korteweg–deVries equation.