Vlasov description of the effects of nonlinear chromaticity on transverse coherent beam instabilities

This paper extends the existing Vlasov theory for transverse coherent beam instabilities to account for nonlinear chromaticity. It confirms the hypothesis that nonlinear chromaticity has two effects on the beam dynamics: (i) it changes the effective impedance which results in a modification of the coherent frequencies of the head-tail modes, and (ii) it can introduce betatron detuning with longitudinal action and provide Landau damping of transverse modes. Based on the new formalism, these two mechanisms are identified and studied analytically with both an airbag and a Gaussian beam. To complete the study, the theory is successfully benchmarked against the circulant matrix solver BimBim as well as the PyHEADTAIL macroparticle tracking code.