# Computing Direct Scattering Transform

I'm working on the Nonlinear Schrodinger equation (NLSE) in 1d: $$i\psi _t (t,x)+ \psi _{xx} + |\psi|^2\psi = 0 \, ,$$ for $t\geq 0$ and $x\in \mathbb{R}$.

This equation is integrable, and so similarly to the KdV equation, there is a way to solve it using the Direct and Inverse Scattering Transform.

What I'm looking for is an algorithm for computing the Direct Scattering Transform (DST), i.e., given $\psi(t,x)$, compute the discrete and continuous spectrum. The help could be in the form of a tutorial, a paper, pseudo-code or even an implementation.

Thank!

• Have you done a literature search? What did you find? – Wolfgang Bangerth Oct 30 '17 at 3:14
• What I've thus far found are either (a) Analytical discussions about how to compute the DST, which weren't helpful. (b) Notes about how to compute it numerically, but never a full description. I imagine that there's work about it out there, but I didn't find it. – Amir Sagiv Oct 30 '17 at 6:06