Modified Laplacian coflow of $G_{2}$-structures on manifolds with symmetry

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.