  We present a numerical and theoretical study of the zero-dispersion limit of
the focusing Zakharov-Shabat hierarchy, which includes NLS and mKdV flows as
its second and third members. All the odd flows in the hierachy are shown to
preserve real-valued data. We establish the zero-dispersion limit of all the
nontrivial conserved densities and associated fluxes for these odd flows for a
large class of real-valued initial data which includes all ``single hump''
initial data. In particular, it is done for the ``focusing'' mKdV flow. The
method is based on the Lax-Levermore KdV strategy, but here it is carried out
in the context of a nonselfadjoint spectral problem. We find that after an
algebraic transformation the limiting dynamics of the mKdV equation is
identical to that of the KdV equation.
