  We consider $G_{2}$-structures on $7$-manifolds that are warped products of
an interval and a six-manifold, which is either a Calabi-Yau manifold, or a
nearly K\"{a}hler manifold. We show that in these cases the $G_{2}$-structures
are determined by their torsion components up to a phase factor. We then study
the modified Laplacian coflow $\frac{d\psi }{dt}=\Delta _{\psi }\psi +2d\left(
\left( C-Tr T\right) \varphi \right) $ of these $G_{2}$-structures, where
$\varphi $ and $\psi $ are the fundamental $3$-form and $4 $-form which define
the $G_{2}$-structure and $\Delta _{\psi }$ is the Hodge Laplacian associated
with the $G_{2}$-structure. This flow is known to have short-time existence and
uniqueness. We analyse the soliton equations for this flow and obtain new
compact soliton solutions.
