  We consider scalar Born-Infeld type theories with arbitrary potentials V(T)
of a scalar field T. We find that for models with runaway potentials V(T) the
generic inhomogeneous solutions after a short transient stage can be very well
approximated by the solutions of a Hamilton-Jacobi equation that describes free
streaming wave front propagation. The analytic solution for this wave
propagation shows the formation of caustics with multi-valued regions beyond
them. We verified that these caustics appear in numerical solutions of the
original scalar BI non-linear equations. Our results include the scalar BI
model with an exponential potential, which was recently proposed as an
effective action for the string theory tachyon in the approximation where
high-order spacetime derivatives of T are truncated. Since the actual string
tachyon dynamics contain derivatives of all orders, the tachyon BI model with
an exponential potential becomes inadequate when the caustics develop because
high order spatial derivatives of T become divergent. BI type tachyon theory
with a potential decreasing at large T could have interesting cosmological
applications because the tachyon field rolling towards its ground state at
infinity acts as pressureless dark matter. We find that inhomogeneous
cosmological tachyon fluctuations rapidly grow and develop multiple caustics.
Any considerations of the role of the tachyon field in cosmology will have to
involve finding a way to predict the behavior of the field at and beyond these
caustics.
