The effective Hamiltonian of the one-dimensional electron-phonon system nearly at the half filling

We study the effect of acoustic mode phonons around a soliton using the effective Hamiltonian given by the Su-Schrieffer-Heeger model. In the one-dimensional electron-phonon system near the half-filling, the non-linear excitation like a soliton appears as distortions of both the optical and the acoustic component of the order parameter. It is shown that the interaction between the acoustic mode phonon and the moving soliton becomes strong near the phonon velocity. This analytical method is also applicable to phase mode solitons in the case away from the half-filling.