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    <subfield code="a">Octupole magnets are a central mitigation method against the transverse collective instabilities expected for the high-intensity operation of the SIS18 and SIS100 synchrotrons in the FAIR project. For these beam parameters, the self-field space-charge effect dominates the betatron footprint, and strongly modifies the instability drive and the Landau damping properties. The space-charge tune shifts are related to all three incoherent amplitudes, and is an intrinsic interaction. We consider all these effects and study Landau damping of head-tail modes due to the combination of octupoles and space-charge. Using the data from experimental instability observations, and particle tracking simulations, we provide estimations for the expected high-intensity operation of the SIS synchrotrons.</subfield>
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