0
$\begingroup$

I know the purpose of the NLSE (Evolution of a complex field envelope in a nonlinear dispersive medium). Usually I am solving the 1d-GNLSE when simulating the propagation of a light pulse through a material. But I also found several mentions of higher-dimensional NLSE (1d, 2d, n-d). Why do I want to solve the GNLSE/NLSE in higher dimensions, when do I want to use it, and how can I imagine that?

$\endgroup$
1
$\begingroup$

One classic example of the $d=1,2,3,$ NLSE is in nonlinear optics. Given a paraxial laser beam propogating in Kerr medium, a classical descriptive PDE is the following version of the NLSE: $$i\psi _z ({\bf x},z) + \Delta _{\perp} \psi + |\psi |^2\psi = 0 \,, \quad {\bf x} \in \mathbb{R} ^d, \quad z\geq0 \, , $$ $$ \psi ({\bf x},z=0) = \psi_0 ({\bf x}) \in H^1 \,. $$ The latter $H^1$ space is the Soboloev space with the norm $\|f\|_{H^1} = \sqrt{\| f \|_2 ^2 + \|\nabla f \|_2^2 }$.

So, there's a localized initial profile which propagates with the dimension $z$ and has a dispersion along the perpendicular one-dimensional or two-dimensional wave.

$d=3$ corresponds to ultra-short pulses, in which there is a temporal dispersion. In that case, with an appropriate scaling, you have the same equation with ${\bf x} = (t,x,y)$.

Inasmuch as I know, if you replace $z$ with $t$, then ${\bf x} = (x,y)$ accounts for the waves on the surface of a deep sea, but I don't know much about it.

For more information, take a look at The Nonlinear Schrodinger Equation, which deals with the kerr nonlinearity model in details. Chapters (1,2,5) will be sufficient, I think.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.