  Out of time ordered correlators (OTOCs) are useful tools for investigating
foundational questions such as thermalization in closed quantum systems because
they can potentially distinguish between integrable and nonintegrable dynamics.
Here we discuss the properties of wavefronts of OTOCs by focusing on the region
around the main wavefront at $x=v_{B}t$, where $v_{B}$ is the butterfly
velocity. Using a Heisenberg spin model as an example, we find that the leading
edge of a propagating Gaussian with the argument $-m(x)\left( x-v_B t \right)^2
+b(x)t$ gives an excellent fit to the region around $x=v_{B}t$ for both the
free and chaotic cases. However, the scaling in these two regimes is very
different: in the free case the coefficients $m(x)$ and $b(x)$ have an inverse
power law dependence on $x$ whereas in the chaotic case they decay
exponentially. We conjecture that this result is universal by using catastrophe
theory to show that, on the one hand, the wavefront in the free case has to
take the form of an Airy function and its local expansion shows that the power
law scaling seen in the numerics holds rigorously, and on the other hand an
exponential scaling of the OTOC wavefront must be a signature of nonintegrable
dynamics. We find that the crossover between the two regimes is smooth and
characterized by an S-shaped curve giving the lifting of Airy nodes as a
function of a chaos parameter. This shows that the Airy form is qualitatively
stable against weak chaos and consistent with the concept of a quantum
Kolmogorov-Arnold-Moser theory.
