  The out-of-time-ordered correlator (OTOC) has emerged as an interesting
object in both classical and quantum systems for probing the spatial spread and
temporal growth of initially local perturbations in spatially extended chaotic
systems. Here, we study the (classical) OTOC and its ``light-cone'' in the
nonlinear Kuramoto-Sivashinsky (KS) equation, using extensive numerical
simulations. We also show that the linearized KS equation exhibits a
qualitatively similar OTOC and light-cone, which can be understood via a
saddle-point analysis of the linearly unstable modes. Given the deep connection
between the KS (deterministic) and the Kardar-Parisi-Zhang (KPZ which is
stochastic) equations, we also explore the OTOC in the KPZ equation. While our
numerical results in the KS case are expected to hold in the continuum limit,
for the KPZ case it is valid in a discretized version of the KPZ equation. More
broadly, our work unravels the intrinsic interplay between noise/instability,
nonlinearity and dissipation in partial differential equations (deterministic
or stochastic) through the lens of OTOC.
