Nonlinear regimeof a single-mode CSR instability

In our previous work we found the growth rate of the beam instability driven by the coherent synchrotron radiation (CSR) by using the so called "CSR impedance" that neglects the shielding effect of the walls and assumes a continuous spectrum of radiation. In this paper, we study the nonlinear regime of this instability. We assume that the bunch is much longer than the wavelength of the modulation and consider a coasting beam model