A new concept of self-Fourier soliton is proposed. The evolution and transmission of self-Fourier Gauss signal and self-Fourier soliton in optical fibers are studied numerically. The characteristics of signal evolution and transmission are studied analytically in both time domain and frequency domain. The results show that a class of non-self-Fourier soliton is obtained from inputting self-Fourier Gauss signals, and the amplitude of the soliton from the self-Fourier Gauss signal could be influenced by the effect of third-order dispersion on propagation. The effect of third-order dispersion also results in the time delay of the soliton and spectrum asymmetry of the soliton.