Soliton solution of the NLS equation

Question

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?

Answer 1

The usual soliton for the NLS is $$ \psi(x,t)=e^{ikx-i\omega t}\sqrt{\frac{\alpha}{m\lambda}}{\rm sech}(\sqrt{\alpha}(x-Ut) $$ where $m$ is the mass and $\lambda$ is the coeficient of the $|\psi|^2\psi$ term. The parameters $\alpha$ and $U$ are arbitrary. Your book has interchanged the role of $x$ and $t$. I suspect that it deals with optical fibres in which one has electic field modes $$ E(x,t)= A(x,t)e^{ikx-\omega t} $$ where $ A $ obeys a NLS with the role of $x$ and $t$ reversed. For details see here page 287.