Evolution of ion bunch profile in the presence of longitudinal coherent electron cooling

In the presence of longitudinal coherent electron cooling, the evolution of the line-density profile of a circulating ion bunch can be described by the 1D Fokker-Planck equation. We show that, in the absence of diffusion, the 1D equation can be solved analytically for a particular dependence of the cooling force on the synchrotron amplitude. For more general cases with arbitrary diffusion, we solved the 1D Fokker-Planck equation numerically, and the numerical solutions have been compared with results from macroparticle tracking.