# Derivation of the Zakharov-Shabat System and Lax Pair for the Gross–Pitaevskii Equation

Question: In addition to showing that the nonlinear Schrodinger equation $$i \Psi_t + \Psi_{xx} - 2|\Psi|^2 \Psi = 0$$ (without a potential) is integrable and isospectral, the existence of a Lax pair guarantees the capacity to satisfy the zero curvature equation. How was the following Lax pair (which I believe to be the Zakharov-Shabat system expressed differently) derived? If it was by ansatz, how could one generalize it to the case with a potential function? My question is taken from the following paper (pdf). Does a similar lax pair exist for the Gross-Pitaevskii equation $$i \Psi_t + \Psi_{xx} - (2|\Psi|^2 +V) \Psi = 0$$ for a potential function $$V(x)$$?

$$\Psi_x + ik\sigma_3 \Psi = Q \Psi$$ $$\Psi_t + 2ik^3 \sigma_3 \Psi = (2kQ - i Q_x \sigma_3 - i |q|^2 \sigma_3)\Psi$$

Edit 1: If anyone could provide a link to the original paper for the Zakharov-Shabat system, it would be greatly appreciated.

Edit 2: $$V(x)$$ should take a simple form (without consideration for physical plausibility) such as $$V(x) = x$$, $$V(x) = x^2$$, $$V(x) = e^x$$, etc...