We compute the relaxations experienced by a superconducting qubit and the simultaneous variation induced on the shape of a microwave pulse during the propagation of the pulse through the qubit. The environmentally affected propagation and the dressed relaxations are accounted by a microscopic-master-Maxwell equation pair. It is shown that the qubit longitudinal relaxation vanishes when the pulse envelope adopts a solitonic shape of nπ area whereas its transverse relaxation vanishes when the pulse phase has a periodic variation that is orthogonal to the spectral density of the environment. The pulse would propagate absorption-free when its area matches 2nπ. Otherwise, the environmental feedback decelerates the velocity of the soliton envelope and induces an monotonic increase of phase in the microwave. A pulse of non-2nπ area thus ramifies into a transparent part that travels absorption-free at incident velocity and a slowing part that decays through space. The ramification explains the environmental origin of pulse splitting observed in self-induced transparency.