Gross–Pitaevskii solitary wave
= Gross–Pitaevskii solitary wave
{c}
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 $\psi\to1$. 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.