  We obtain exact periodic solutions of the positive and negative modified
Kortweg-de Vries (mKdV) equations. We examine the dynamical stability of these
solitary wave lattices through direct numerical simulations. While the positive
mKdV breather lattice solutions are found to be unstable, the two-soliton
lattice solution of the same equation is found to be stable. Similarly, a
negative mKdV lattice solution is found to be stable. We also touch upon the
implications of these results for the KdV equation.
