A new non-linear instability for scalar fields
  In this letter we introduce the non-linear partial differential equation
(PDE) $\partial^2_{\tau} \pi \propto (\vec\nabla \pi)^2$ showing a new type of
instability. Such equations appear in the effective field theory (EFT) of dark
energy for the $k$-essence model as well as in many other theories based on the
EFT formalism. We demonstrate the occurrence of instability in the cosmological
context using a relativistic $N$-body code, and we study it mathematically in
3+1 dimensions within spherical symmetry. We show that this term dominates for
the low speed of sound limit where some important linear terms are suppressed.
