  We extend the Keldysh technique to enable the computation of out-of-time
order correlators. We show that the behavior of these correlators is described
by equations that display initially an exponential instability which is
followed by a linear propagation of the decoherence between two initially
identically copies of the quantum many body systems with interactions. At large
times the decoherence propagation (quantum butterfly effect) is described by a
diffusion equation with non-linear dissipation known in the theory of
combustion waves. The solution of this equation is a propagating non-linear
wave moving with constant velocity despite the diffusive character of the
underlying dynamics. Our general conclusions are illustrated by the detailed
computations for the specific models describing the electrons interacting with
bosonic degrees of freedom (phonons, two-level-systems etc.) or with each
other.
