Abstract
We study a nonlinear integral equation that is a necessary condition on the equilibrium phase space distribution function of stored, colliding electron beams. It is analogous to the Haissinski equation, being derived from Vlasov-Fokker-Planck theory, but is quite different in form. The equation is analyzed for the case of the Chao-Ruth model of the beam-beam interaction in one degree of freedom, a so-called strong-strong model with nonlinear beam-beam force. We prove existence of a unique solution, for sufficiently small beam current, by an application of the implicit function theorem. We have not yet proved that this solution is positive, as would be required to establish existence of an equilibrium. There is, however, numerical evidence of a positive solution. We expect that our analysis can be extended to more realistic models.
Highlights
In the theory of stability of stored beams a primary step should be the study of equilibrium states, expected to exist at low current
An equilibrium state should become unstable at some threshold in current, but in order to compute the threshold we must linearize the kinetic equation [Vlasov or Vlasov-Fokker-Planck (VFP)] about the equilibrium phase-space distribution
There is one case in which there is a widely known theory of equilibrium, which makes some contact with experiment, namely, the case of longitudinal motion of a single bunched electron beam in a storage ring, subject to a wake field [1,2]
Summary
In the theory of stability of stored beams a primary step should be the study of equilibrium states, expected to exist at low current. If the wake field satisfies a mild restriction, it is not difficult to prove that the equation has a unique solution in a large function space S, at sufficiently small current [2]. The analysis gives results on existence and uniqueness of equilibria, using factorization and an integral equation on d-dimensional configuration space, quite in analogy to Haıssinski theory. Between beam-beam collisions the phase-space distribution function for beam i, denoted by fi q; p; , propagates according to the Fokker-Planck equation,. The implicit function theorem takes into account the nonlinear term, and assures us that for sufficiently small there will be a unique exact solution of (21) close to the approximation (25). Our existence proof of a unique solution works for any a > 2, but the requirement on smallness of the beam current parameter may become more strict as a is increased.
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