| Paper | Title | Page |
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| WEPLT092 | Equilibrium Longitudinal Distribution for Localized Regularized Inductive Wake | 2062 |
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In a recent paper [*] we have shown that a localized wake assumption and the Gaussian approximation for the longitudinal beam distribution function can be used to understand the nature of the stationary solutions for the inductive wake, by comparison between the resulting map and the Haissinski equation, which rules the (less realistic) case of a uniformly distributed wake. In particular we showed the non-existence of solutions of Haissinski's equation when the inductive wake strength exceeds a certain threshold [**] to correspond to the onset of chaos in the map evolving the moments of the beam distribution from turn to turn. In this paper we use the same formalism to confirm that as noted in [**] for Haissinski's equation, a steady state solution for the longitudinal phase space distribution function always exists if a physically regularized inductive wake, which satisfies an obvious causality condition, is used.
* S. Petracca and Th. Demma, Proc. of the 2003 PAC, IEEE Press, New York, 2003, ISBN 0-7803-7739-9, p.2996.** Y. Shobuda and K. Hirata, Part. Accel. vol. 62, 165 (1999). |