The dynamic equations of the proton transport along the hydrogen bonded molecular systems have been obtainedby using completely quantum-mechanical method to be based on new Hamiltonian and model we proposed. Somequantum-mechanical features of the proton-solitons have also been given in such a case. The alteate motion of twodefects resulting from proton transfer occurred in the systems can be explained by the results. The results obtainedshow that the proton-soliton has corpuscle feature and obey classical equations of motion, while the free soliton movesin uniform velocity along the hydrogen bonded chains.