 Solution

Introduce the translating coordinate $Z=z-Ut$ and put $\psi (\mathbf{x},t)=\Psi (x,y,Z)$. Then ${\psi }_t=-U{\Psi }_Z$. Substitution in the Gross–Pitaevskii equation gives the stationary Gross–Pitaevskii solitary wave equationThe complex bulk field is fixed by $\Psi \to 1$ far from the localized disturbance. Spatial derivatives in this equation are taken in the translating frame. There is no additional uniform temporal phase here because the normalization has already made the background $\psi =1$ stationary.