Taking advantage of the dispersive Fourier transformation technique, the decaying evolution processes of double-pulse mode-locking in a single-walled carbon-nanotube-based Er-doped fiber laser are observed in detail for the first time to our knowledge. The decaying dynamics of the double-pulse mode-locking state is analyzed in the spectral and temporal domains. We reveal that the two pulses in one cluster disappear either simultaneously or one by one during the decaying processes of double-pulse mode-locking states. In addition, the spectral evolution patterns of the special double-pulse states (i.e., bound states) are extremely distinct at different decline rates of the pump power. (C) 2018 Chinese Laser Press