Analytical Estimate of Phase Mixing Time of Longitudinal Akhiezer - Polovin Wave

Phase mixing and eventual breaking of longitudinal Akhiezer - Polovin wave subjected to a small amplitude longitudinal perturbation is studied analytically. It is well known that longitudinal Akhiezer - Polovin wave breaks via the process of phase mixing at an amplitude well below its breaking amplitude, when subjected to arbitrarily small longitudinal perturbation[Phys. Rev. Lett.108, 125005 (2012)]. Here we analytically show that the phase mixing time (breaking time) has a linear dependence with phase velocity of the wave, inverse square dependence with the maximum fluid velocity and it is inversely proportional to the amplitude of perturbation. This analytical dependence of phase mixing time on the above three parameters is further verified using numerical simulations based on Dawson Sheet Model.