Soliton solutions for the Laplacian co-flow of some G ₂-structures with symmetry

We consider the Laplacian "co-flow" of G ₂-structures: ∂∂tψ=-δdψ where ψ is the dual 4-form of a G ₂-structure φ and δ _d is the Hodge Laplacian on forms. Assuming short-time existence and uniqueness, this flow preserves the condition of the G ₂-structure being coclosed (dψ=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 Kähler 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 Kähler case, we find several special soliton solutions, and reduce the general problem to a single third order highly nonlinear ordinary differential equation.