  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 $\langle E_{z}\rangle=-(\partial/\partial
z)\langle\phi\rangle=-e_{b}g_{0}\partial\lambda_{b}/\partial
z-e_{b}g_{2}r_{w}^{2}\partial^{3}\lambda_{b}/\partial z^{3}$, where $g_{0}$ and
$g_{2}$ are constant geometric factors, $\lambda_{b}(z,t)=\int
dp_{z}F_{b}(z,p_{z},t)$ 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 $\langle E_{z}\rangle$.
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