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) $ofthese$G{2}$-structures,where$\varphi $and$\psi $arethefundamental$3$-formand$4 $-form which define the $G_2$-structure and ${\mathrm{\Delta }}_{\psi }$ 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.