The dynamical properties of a phase soliton with a fractional charge \(\pm e/2\) in the bond-charge-density-wave (BCDW) background of one-dimensional quarter-filled electron–lattice systems are studied by numerical and semi-phenomenological methods using the Su–Schrieffer–Heeger (SSH) model. A special focus is on the time evolutions of the velocity and energy of the charged soliton subject to an electric field. Several interesting properties are obtained: the saturation of the soliton velocity, the divergence of the soliton energy and the propagation of condensed acoustic phonons. The saturation velocity, which is independent of the applied field strength, is less than the phason velocity. These properties are different from those of the sine-Gordon model which was believed to describe the commensurate BCDW systems. The velocity–energy relation is also different from that of the sine-Gordon soliton but is described by the same form as that of the acoustic polaron in the SSH model. These results suggest th...