Nonlinear dynamics of microwave instability in accelerators

We develop a nonlinear theory of the weak single bunch instability in electron and positron circular accelerators and damping rings. A nonlinear equation is derived that governs the evolution of the amplitude of unstable oscillations with account of quantum diffusion effects due to the synchrotron radiation. Numerical solutions to this equation show a large variety of nonlinear regimes depending on the growth rate of the instability and the diffusion coefficient. Comparison with the observation in the Stanford Linear Collider Damping Ring at the Stanford Linear Accelerator Center shows qualitative agreement with the patterns observed in the experiment. {copyright} {ital 1997} {ital The American Physical Society}