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

摘要

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 ${r}_{w}$. The average axial electric field is expressed as $?{E}_{z}?=ensuremath{-}(ensuremath{partial}/ensuremath{partial}z)?ensuremath{phi}?=ensuremath{-}{e}_{b}{g}_{0}ensuremath{partial}{ensuremath{lambda}}_{b}/ensuremath{partial}zensuremath{-}{e}_{b}{g}_{2}{r}_{w}^{2}{ensuremath{partial}}^{3}{ensuremath{lambda}}_{b}/ensuremath{partial}{z}^{3}$, where ${g}_{0}$ and ${g}_{2}$ are constant geometric factors, ${ensuremath{lambda}}_{b}(z,t)=ensuremath{int}d{p}_{z}{F}_{b}(z,{p}_{z},t)$ is the line density of beam particles, and ${F}_{b}(z,{p}_{z},t)$ 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 ${F}_{b}=ext{const}$ in a bounded region of ${p}_{z}$-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 $?{E}_{z}?$.