Density waves in the shearing sheet V. Feedback cycle for swing amplification by non–linear effects

Non–linear effects in the dynamical evolution of a shearing sheet made of stars are studied. First the implications of hitherto neglected non–linearities of the Boltzmann equation for the dynamical evolution of the shearing sheet are investigated. Using a formalism developed previously on the basis of the linearized Boltzmann equation it is demonstrated that the inclusion of the non–linear term leads to a feedback cycle for swing amplified density waves in the unbounded shearing sheet. Such a feedback is unique to star disks and is not known for gas disks. In order to present concrete examples of the non–linear feedback cycle a SCF code was developed and numerical simulations of the dynamical evolution of the shearing sheet are performed. The numerical results seem to confirm the theoretical predictions. The evolution of the shearing sheet resembles closely and might actually explain the recurrent spiral instabilities found in large–scale numerical simulations of the dynamical evolution of galactic disks.