  We use the bracket flow/algebraic soliton approach to study the Laplacian
flow of $G_2$-structures and its solitons in the homogeneous case. We prove
that any homogeneous Laplacian soliton is equivalent to a semi-algebraic
soliton (i.e.\ a $G$-invariant $G_2$-structure on a homogeneous space $G/K$
that flows by pull-back of automorphisms of $G$ up to scaling). Algebraic
solitons are geometrically characterized among Laplacian solitons as those with
a `diagonal' evolution. Unlike the Ricci flow case, where any homogeneous Ricci
soliton is isometric to an algebraic soliton, we have found, as an application
of the above characterization, an example of a left-invariant closed
semi-algebraic soliton on a nilpotent Lie group which is not equivalent to any
algebraic soliton. The (normalized) bracket flow evolution of such a soliton is
periodic. In the context of solvable Lie groups with a codimension-one abelian
normal subgroup, we obtain long time existence for any closed Laplacian flow
solution; furthermore, the norm of the torsion is strictly decreasing and
converges to zero. We also classify algebraic solitons in this class and
exhibit several explicit examples of closed expanding Laplacian solitons.
