Propagation of short pulses in the birefringent single-mode fiber near the zero-dispersion wavelength is discussed. Initial pulses are assumed to be linearly polarized at π/4 angle with respect to the polarization axes. With numerical method,it is found that the third-order dispersion,as a small perturbation,does not affect the threshold of the soliton trapping. But,pulse width of one polarization broadens quickly;another broadens slowly. Meanwhile the shapes and the spectra of the pulses,are not any longer symmetric in the two polarizations. Under the certain condition, the radiation of the soliton is dominantly in one polarization.