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 rw. The average axial electric field is expressed as $⟨{E}zˢ⟩={-}{e}b{g}₀{∂}{{λ}}b/{∂}z{-}{e}b{g}₂{r}w²{{∂}}³{{λ}}b/{∂}{z}³$, where ${g}₀$ and ${g}₂$ are constant geometric factors, and ${{λ}}b(z,t)={∫}d{p}z{F}b(z,{p}z,t)$ 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.