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?


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.

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