  We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} \psi =
- \Delta_d \psi$ where $\psi$ is the dual 4-form of a $G_2$-structure $\phi$
and $\Delta_d$ is the Hodge Laplacian on forms. This flow preserves the
condition of the $G_2$-structure being coclosed ($d\psi =0$). We study this
flow for two explicit examples of coclosed $G_2$-structures with symmetry.
These are given by warped products of an interval or a circle with a compact
6-manifold $N$ which is taken to be either a nearly K\"ahler manifold or a
Calabi-Yau manifold. In both cases, we derive the flow equations and also the
equations for soliton solutions. In the Calabi-Yau case, we find all the
soliton solutions explicitly. In the nearly K\"ahler case, we find several
special soliton solutions, and reduce the general problem to a single
\emph{third order} highly nonlinear ordinary differential equation.
