We present the derivation of an exact special case solution (for a classical lattice) for the Su-Schrieffer-Heeger model for the calculation of soliton dynamics in trans-polyacetylene. Our solution is exact, in the sense that the ansatz state yields an exact solution provided that the equations of motion for its parameters are obeyed. However, these equations can be solved only numerically (in principle to any desired accuracy), not analytically. The model is applied to time simulations of neutral solitons as a function of temperature. We find agreement of the results of our time simulations with experimental data on the mobility of neutral solitons in the system. Comparative calculations using the completely adiabatic model indicate that the results of this model are at variance both with experiment and with those of our model. A simple consideration of the potential barriers for soliton displacement leads to an overestimation of the soliton mobility for low temperatures and an underestimation for higher ones. In an appendix we discuss in some detail the relationship of this exact solution with the state ansatz as presented in our previous paper. We find that the ansatz state and the exact solution yield identical results for lattice momenta, displacements and site occupancies, but differ in a time dependent phase factor. Thus spectra computed with the dynamics resulting from the exact solution for the classical lattice on one hand and from the ansatz state on the other would differ from each other.