# Soliton solution of the NLS equation

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?

• I think you should provide a bit more information : NLS -denotes non-linear Schroedinger equation, as far as I understand, it would be more convenient for the audience to see it written explicitly, or reference(extract from the aforementioned book). Jul 5, 2020 at 19:44
• Good idea @spiridon_the_sun_rotator. I have edited the question to reflect that Jul 5, 2020 at 19:59

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.