By virtue of the method of multiple scales, we have investigated the behavior of waves in a one-dimensional nonlinear monoatomic chain under damping and found that the amplitude of the lattice waves decreases and the group velocity becomes slow under the action of damping of an order of O(ε). At the same time, we have derived PDNLS equation and obtained its analytical solution. Our results show that: there exists propagating envelope soliton in the one-dimensional nonlinear monoatomic chain under the damping, provided the damping is of an order of O(ε). However, the chain supports the propagation of the soliton of bell shape or kink as the damping is of an order of O(ε2).