Exact results for some linear and nonlinear beam-beam effects

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Abstract
The nonlinear equation of a motion for a particle’s displacement from its ideal orbit in a storage ring, when ’’kicked’’ by a second (periodically) colliding beam, can be translated into a nonlinear difference equation. Near the center of the beam this equation is virtually linear and we obtain the (Mathieu‐like) regions of (in)stability, and all other properties there, analytically and in closed form. Adding in the slow (synchrotron) oscillations along the (now) ’bunched’ beam, as a square wave modulation, another analytic expression is found which simply shows that if the p e a k beam‐beam force,during a synchrotron oscillation, exceeds the previous (in)stability border, instability arises in this case. The order periodic solutions (resonances) are obtained for any, odd, non‐linear b‐b force, indicating that the planned tune for ’’ISABELLE’’ may be too close to a major (period 4) resonances. Approximating the actual (error function) b‐b force by a (3‐) piece‐wise‐linear force (itself the exact b‐b force for a rectangular beam) a region of nonlinear stability for all time is found (in ISABELLE as ’’Δν’’?0.03) which is about 50% wider than the width of the beam. This sizable region of nonlinear stability is bordered by an invariant curve, constructed with the aid of a technique employed earlier by McMillan and by Laslett.

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