  The self-consistent propagation of generalized $D_{1}$ [coherent-product]
states and of a class of gaussian density matrix generalizations is examined,
at both zero and finite-temperature, for arbitrary interactions between the
localized lattice (electronic or vibronic) excitations and the phonon modes. It
is shown that in all legitimate cases, the evolution of $D_{1}$ states reduces
to the disentangled evolution of the component $D_{2}$ states. The
self-consistency conditions for the latter amount to conditions for
decoherence-free propagation, which complement the $D_{2}$ Davydov soliton
equations in such a way as to lift the nonlinearity of the evolution for the
on-site degrees of freedom. Although it cannot support Davydov solitons, the
coherent-product ansatz does provide a wide class of exact density-matrix
solutions for the joint evolution of the lattice and phonon bath in compatible
systems. Included are solutions for initial states given as a product of a
[largely arbitrary] lattice state and a thermal equilibrium state of the
phonons. It is also shown that external pumping can produce self-consistent
Frohlich-like effects. A few sample cases of coherent, albeit not solitonic,
propagation are briefly discussed.
