  A recent paper by Gross and Erler (hep-th/0406199) showed that by making a
certain well-defined, unitary transformation on the mode basis for the open
bosonic string--one that identifies the lightcone component of position with
the string midpoint--it is possible to render the action for cubic string field
theory local in lightcone time. In this basis, then, cubic string field theory
possesses a well-defined initial value formulation and a conserved Hamiltonian.
With this new understanding it seems natural to study time dependent solutions
representing the the decay of an unstable D-branes. In this paper we study such
solutions using level truncation of mode oscillators in the lightcone basis,
finding both homogenous solutions by perturbatively expanding the string field
in modes $e^{nt}$, and inhomogenous solutions by integrating the equations of
motion on a lattice. Truncating the theory to level $(\tilde{2},\tilde{4})$ in
$\alpha^+$ oscillators, we find time dependent solutions whose behavior seems
to converge to that of earlier solutions constructed in the center of mass
basis, where the cubic action contains an infinite number of time derivatives.
We further construct time-dependent inhomogeneous solutions including all
fields up to level $(\tilde{2},\tilde{4})$. These solutions at the outset
display rather erratic behavior due to an unphysical instability introduced by
truncating the theory at the linear level. However upon truncating away the
field responsible for the instability, we find more reasonable solutions which
may possibly represent an approximation to tachyon matter. We conclude with
some discussion of future directions.
