Nonlinear Schrödinger equation in a potential

Question

I've recently become interested in the integrability of nonlinear PDEs while reading these lecture notes.

Question 1: Would the equation $i\Psi_t + \Psi_{xx} - (2|\Psi|^2 + V) \Psi = 0$ for a potential function $V(x)$ posses any physical significance?

I was curious as to how the dynamics of the system would change if a potential function were to be introduced. Given that $V(x,t)\Psi(x,t)$ are multiplied together in the time-dependent Schrodinger equation, I wondered if a similar principle could generalize to the nonlinear case as well. My background is primarily in mathematics, and I was unsure if such an addendum would reduce the physical significance of the system.

Answer 1

The equation you've written is known as the Gross-Pitaevskii equation, which is one of the central concepts used to describe Bose-Einstein condensates.

(Indeed, this is mentioned explicitly in the disambiguation text at the top of the Wikipedia page for the NLSE.)