Statistical properties of shocks in Burgers turbulence, II: Tail probabilities for velocities, shock-strengths and rarefaction intervals

This paper studies the structure of the random "sawtooth" profile corresponding to the solution of the inviscid Burgers equation with white-noise initial data. This function consists of a countable sequence of rarefaction waves separated by shocks. We are concerned here with calculating the probabilities of rare events associated with the occurrence of very large values of the normalized velocity, shock-strength and rarefaction intervals. We find that these quantities have tail probabilities of the form exp{-Cx³}, x≫1. This "cubic exponential" decay of probabilities was conjectured in the companion paper [1]. The calculations are done using a representation of the shock-strength and length of rarefaction intervals in terms of the statistics of certain conditional diffusion processes.