Bryant’s Laplacian flow is an analogue of Ricci flow that seeks to flow an arbitrary initial closed $G_2$-structure on a 7-manifold toward a torsion-free one, to obtain a Ricci-flat metric with holonomy $G_2$. This talk will give an overview of joint work with Mark Haskins and Rowan Juneman about complete self-similar solutions on the anti-self-dual bundles of ${\mathbb CP}^2$ and $S^4$, with cohomogeneity one actions by SU(3) and Sp(2) respectively. We exhibit examples of all three classes of soliton (steady, expander and shrinker) that are asymptotically conical. In the steady case these form a 1-parameter family, with a complete soliton with exponential volume growth at the boundary of the family. All complete Sp(2)-invariant expanders are asymptotically conical, but in the SU(3)-invariant case there appears to be a boundary of complete expanders with doubly exponential volume growth.