Solitary Waves in Particle Beams

Since space charge waves on a particle beam exhibit both dispersive and nonlinear character, solitary waves or solitons are possible. Dispersive, nonlinear wave propagation in high current beams is found to be similar to ion-acoustic waves in plasmas with an analogy between Debye screening and beam pipe shielding. Exact longitudinal solitary wave propagation is found for potentials associated with certain transverse distributions which fill the beam pipe. For weak dispersion, the waves satisfy the Korteweg-deVries (KdV) equation, but for strong dispersion they exhibit breaking. More physically realizable distributions which do not fill the beam pipe are investigated and shown to also satisfy a KdV equation for weak dispersion if averaging over rapid transverse motion is physically justified. Scaling laws are presented to explore likely parameter regimes where these phenomena may be observed experimentally. 1 SOLITARY WAVES AND SOLITONS Nonlinearity in wave propagation typically leads to steepening phenomena. For example, consider the simple wave equation ut + (1 + u)ux = 0 (1) which has the implicit solution [1] u(x, t) = f (x − (1 + u)t) (2) where f is an arbitrary differentiable function. Note that the velocity, (1 + u ), depends on the amplitude, and in particular, higher amplitudes propagate faster. If f describes a localized distribution, the peak value will tend to overtake lower values, and steepening and breaking of the pulse and formation of a shock will result. On the other hand, if the velocity depends strongly on wavelength (dispersion), a localized distribution tends to spread. A solitary wave results when the nonlinear steepening is canceled by the dispersive spreading, yielding a localized disturbance which propagates without distortion. Since solitary waves of different heights will generally travel with different velocities collisions can occur. The term soliton describes solitary waves which maintain their identity and shape after collision. 2 SPACE CHARGE WAVES Space charge forces can produce longitudinal density waves in low momentum spread, charged particle beams. For a uniform, non-relativistic beam of radius a in a uniform beam pipe of radius b, the propagation is nondispersive in the linear, long wavelength approximation. The wave velocity vp is vp = ω k = e λog 4πεom (3) where ω is the mode frequency for wave number k, e is the electron charge, λo is the unperturbed linear charge density, g is a geometric factor depending on particle distribution and pipe radius, εo is the permittivity of free space, and m is the mass of the beam particles. The associated force is given by F = − ge 2 4πεo ∂λ