  We investigate the existence of closed $G_2$-structures which are solitons
for the Laplacian flow on nilpotent Lie groups. We obtain that seven of the
twelve Lie algebras admitting a closed $G_2$-structure do admit a Laplacian
soliton. Moreover, one of them admits a continuous family of Laplacian solitons
which are pairwise non-homothetic and the Laplacian flow evolution of four of
them is not diagonal.
