| Summary: | 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_{b}g_{0}∂λ_{b}/∂z-e_{b}g_{2}r_{w}^{2}∂^{3}λ_{b}/∂z^{3}, where g_{0} and g_{2} are constant geometric factors, and λ_{b}(z,t)=∫dp_{z}F_{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.
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