Enhancement of the non-linear frequency shift due to electron-electron collisions

During the evolution of a weakly-collisional non-linear electron plasma, an enhancement of the non-linear frequency shift is observed in kinetic simulations. Theory and simulations of damping due to electron-electron collisions modeled using a Fokker-Planck operator show that the wave damps weakly for a range of physically relevant parameters. By performing perturbative expansions on the kinetic dispersion relation in frequency, time, and velocity, we show that the growth rate of the frequency is directly proportional to the damping rate of the electric field amplitude. A relation for the complex non-linear frequency of quasi-steady-state, weakly-collisional, non-linear electron plasma waves is provided.