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?

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    $\begingroup$ 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). $\endgroup$ Jul 5, 2020 at 19:44
  • $\begingroup$ Good idea @spiridon_the_sun_rotator. I have edited the question to reflect that $\endgroup$
    – Paddy
    Jul 5, 2020 at 19:59

1 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.


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