Nonlinear Regime of a Single-mode CSR Instability

Close to the shielding threshold of synchrotron radiation in a toroidal waveguide the coherent synchrotron radiation (CSR) instability may be driven by a single synchronous mode. The linear analysis of the problem is presented elsewhere. In this paper we study a nonlinear regime of the single-mode CSR instability. We show that in the nonlinear regime the instability is described by a system of equations which, after a proper scaling, coincides with the 1D FEL instability theory. We present numeric and analytic analysis of the non-linear regime for the time interval large compared with the damping time taking into account radiation damping and diffusion due to the quantum fluctuations