Time independence of light in optical fibers

The $A(z,t)$ in the fibre optics equation is the $A$ in $$ E(z,t)=A(z,t) e^{i(kz-omega t)}. $$ It does not include the $e^{-iomega t}$ part of the CW time dependence therefore.

Note that $A$ is the field envelope, which means that $A$ is the average over some duration of time and interval of space. It is closely related to intensity in that respect. For CW excitation, the envelope is constant. There's no mechanism shown that would cause a time-varying envelope within the fiber. In other words the first term on the right serves as a source term, and since the source is CW, the first term on the right is zero.

That's the definition of "CW radiation" in this context.

CW stands for "continuous wave", but here it is being used to mean "not varying with time".

Much like in electronics the term "DC" literally means not changing direction, but has been co-opted in circuit theory to mean not changing at all.