My understanding of soliton - it is a moving pulse in a medium which does not change its structure with time. It has other properties like no interaction with other solitons (this could certainly be wrong. Please let me know if I am getting this wrong)
When reading a book on the topic, the author mentions that the NLS (Non-linear Schrödinger) equation has a special solution of the form-
$u(z,t) = sech(t)exp(iz/2)$ which is claimed to be a fundamental soliton.
Taking the modulus of the equation yields a function independent of $z$. Now I get terribly confused. Why is the modulus of $u(z,t)$ independent of $z$? Shouldn't a soliton pulse's form change with $z$ for a fixed $t$?
Edit: Earlier in the book, it is mentioned that $t$ "represents retarded time, i.e., ordinary time, but with the transit time delay of a pulse at the central frequency subtracted off". So is the author saying that $u(z,t)$ is the equation of the profile of a soliton and not the soliton itself?