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 $\langle E_{z}\rangle=-(\partial/\partial z)\langle\phi\rangle=-e_{b}g_{0}\partial\lambda_{b}/\partial z-e_{b}g_{2}r_{w}^{2}\partial^{3}\lambda_{b}/\partial z^{3}$, 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 $\langle E_{z}\rangle$. .
Highlights
High-energy accelerators and transport systems [1,2,3,4,5,6] have a wide variety of applications ranging from basic research in high energy and nuclear physics, to applications such as spallation neutron sources, medical physics, and heavy ion fusion
IV supports a broad range of nonlinear pulselike solutions and periodic traveling-wave solutions that have time-stationary waveform in the frame moving with velocity M 1⁄4 const relative to the beam frame
This paper makes use of a one-dimensional kinetic model [14] that describes the nonlinear dynamics of the longitudinal distribution function Fbðz; pz; tÞ, the average self-generated axiaRl electric field hEziðz; tÞ, and the line density λbðz; tÞ 1⁄4 dpzFbðz; pz; tÞ, for an intense charged particle beam propagating in the z-direction through a circular, perfectly conducting pipe with radius rw
Summary
High-energy accelerators and transport systems [1,2,3,4,5,6] have a wide variety of applications ranging from basic research in high energy and nuclear physics, to applications such as spallation neutron sources, medical physics, and heavy ion fusion. We remove the restriction to the weakly nonlinear regime, and make use of the 1D kinetic model developed in Ref. Model (Sec. III A) to investigate detailed properties of nonlinear pulselike (soliton) or periodic traveling-wave disturbances propagating with constant normalized velocity. Z0 1⁄4 Z − MT and T0 1⁄4 T, the waveform of the disturbance is assumed to be time-stationary (∂=∂T0 1⁄4 0) in the frame moving with velocity M 1⁄4 const relative to the beam frame. (9) and (10) to primed variables, and look for solutions that are time stationary (∂=∂T0 1⁄4 0) in the frame moving with velocity M 1⁄4 const relative to the beam frame.
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