We study localized waves on continuous wave background in an exponential dispersion decreasing fiber with two orthogonal polarization states.We demonstrate that asymmetric W-shaped and M-shaped soliton pulse can be generated from a weak modulation on continuous wave background.The numerical simulation results indicate that the generated asymmetric soliton pulses are robust against small noise or perturbation.In particular,the asymmetric degree of the asymmetric soliton pulse can be effectively controlled by changing the relative frequency of the two components.This character can be used to generate other nonlinear localized waves,such as dark-antidark and antidark-dark soliton pulse pair,symmetric W-shaped and M-shaped soliton pulse.Furthermore,we find that the asymmetric soliton pulse possesses an asymmetric discontinuous spectrum.