By using nonlinear Schrdinger equation including Raman gain and self-steepening but ignoring fiber loss situation, the propagation characteristics of Airy pulse are simulated and analyzed in the single-mode fiber. Simulations show that Airy pulse can be converted into soliton and its propagation direction is skewed due to the effects of Raman gain and self-steepening under a certain condition. In time domain, the number of small peaks of Airy pulse reduces rapidly. Airy pulse becomes a peak structure containing a main peak and sub-peaks where the energies can be ignored by changing the coefficient a reasonablely, which is approximated as the soliton structure. Therefore, Airy pulse is regarded as transforming into soliton. Meanwhile, in the case of small values b, there exists a significant difference in shape between Airy pulse and soliton. With the value of parameter b increasing slowly, the shape of Airy pulse is very close to soliton's, therefore Airy pulse can transform into soliton by changing value b reasonablely. Compared with by changing b value, Airy pulse convered into the soliton is stable by changing the a value reasonablely. Simultaneously, with the increases of values of coefficient a and amplitude b, the time-shift of Airy pulse increases. However, the time-shift of Airy pulse would decrease when Raman gain and Self-steepening become strong, no matter what the values of a and b are. Further, the acceleration properties of Airy pulse are investigated. It is found that Airy pulse autoacceleration is not a stable value at the beginning but it gradually stabilizes with the increase of transmission distance. The reason is that the energies of secondary peaks exert a tremendous influence on the main lobe of Airy pulse at the beginning, however, secondary peaks diffuse fast with the increase of transmission and then the influence can be ignored to a certain extent. So, the main peak gradually stabilizes with the increase of transmission distance.