  We find the general solution to the time-dependent Hartree-Fock problem for
scattering solutions of the Gross-Neveu models, with both discrete (GN_2) and
continuous (NJL_2) chiral symmetry. We find new multi-breather solutions both
for the GN_2 model, generalizing the Dashen-Hasslacher-Neveu breather solution,
and also new twisted breathers for the NJL_2 model. These solutions satisfy the
full TDHF consistency conditions, and only in the special cases of GN_2 kink
scattering do these conditions reduce to the integrable Sinh-Gordon equation.
We also show that all baryons and breathers are composed of constituent twisted
kinks of the NJL_2 model. Our solution depends crucially on a general class of
transparent, time dependent Dirac potentials found recently by algebraic
methods.
