Effect of space charge tune shift on transverse instabilities

Transverse instabilities in synchrotrons with large space charge tune shift are considered. In particular the variation of the incoherent tune with longitudinal position in the bunch is included. Expansion in an appropriate basis set results in an eigenvalue problem which reduces to previously derived expressions when the space charge tune spread is ignored. The regime where the synchrotron tune is negligible compared to the space charge tune shift is also considered. Under appropriate conditions this results in a significantly smaller growth rate than the ones obtained using the weak coupling formalism. Finally, the effect of octupole induced betatron tune spread is included, resulting in a technique for estimating the maximum stable current.