  Due to their explicit construction, Aloff-Wallach spaces are prominent in
flux compactifications. They carry G_2-structures and admit the G_2-instanton
equations, which are natural BPS equations for Yang-Mills instantons on
seven-manifolds and extremize a Chern-Simons-type functional. We consider the
Chern-Simons flow between different G_2-instantons on Aloff-Wallach spaces,
which is equivalent to Spin(7)-instantons on a cylinder over them. For a
general SU(3)-equivariant gauge connection, the generalized instanton equations
turn into gradient-flow equations on C^3 x R^2, with a particular cubic
superpotential. For the simplest member of the Aloff-Wallach family (with
3-Sasakian structure) we present an explicit instanton solution of tanh-like
shape.
