An analytical description of the saturation stage of the self-modulation instability in proton-driven plasma wakefield accelerators

The goal of this work is to provide an analytical description of the saturation or nonlinear stage of the self-modulation instability (SMI), a critical phenomenon in oscillating unstable systems. In many such systems, the nonlinear stage is marked by a shift in the resonant frequency, which begins to depend on the amplitude of oscillation. This amplitude dependence fundamentally alters the system's behavior, leading to complex dynamics. To investigate this, we propose introducing a dependence of the resonant frequency, kp, on the oscillation amplitude, specifically using the plasma density perturbation, δn as a key parameter. To approach this problem, we utilize the Green's function method in cases where the operator of the differential equation remains linear. With this we accomplish to model a realistic behaviour of the plasma density perturbation when the SMI goes through the saturation stage. The oscillations of the perturbation of the plasma density takes longer to disappear as the density spread of the bunch increases.