 Gross–Pitaevskii solitary wave

For the normalized conservative equation $-2i{\psi }_t={\nabla }^2\psi +(1-|\psi {|}^2)\psi$, a localized disturbance translating at speed $v$ obeys ${\nabla }^2\psi +(1-|\psi {|}^2)\psi -2iv{\psi }_x=0$ in translating coordinates. Its complex-field boundary is $\psi \to 1$. This permits zero-net-winding vortex pairs, but excludes a lone quantum vortex. The usual subsonic far field has $\psi -1=O(r^{-1})$ in two dimensions.