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

Abstract

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 $⟨E_z⟩=-(\partial /\partial z)⟨\varphi ⟩=-e_b{g}_0\partial {\lambda }_b/\partial z-e_b{g}_2{r}_w^2{\partial }^3{\lambda }_b/\partial z^3$, where $g_0$ and $g_2$ are constant geometric factors, ${\lambda }_b(z,t)=\int d{p}_z{F}_b(z,p_z,t)$ 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 (solitons) 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: (a) the nonlinear waterbag distribution, where $F_b=const.$ in a bounded region of $p_z$-space; and (b) nonlinear Bernstein-Green-Kruskal (BGK)-like solutions, allowing for both trapped and untrapped particle distributions to interact with the self-generated electric field $⟨E_z⟩$. .

arXiv comment42 pages, 17 figures