Gross–Pitaevskii equation ($i\hslash {\partial }_t\psi =(-{\hslash }^2{\nabla }^2/(2m)+g|\psi {|}^2+V)\psi$)

The Gross–Pitaevskii equation describes a dilute interacting condensate through a complex order parameter $\psi$, normalized so that $|\psi {|}^2$ is number density. Repulsive contact interactions have $g>0$. In a frame rotating at the condensate frequency, the operator includes $-\mu$, with chemical potential $\mu$. It is generated by a Hamiltonian functional; its homogeneous positive-density state has $n_0=\mu /g$ when $g,\mu >0$.

For $i{\psi }_t=-D{\nabla }^2\psi +[V(n)+i\Gamma (n)]\psi$, with real $D,V,\Gamma$ and $\psi =\sqrt{n}{e}^{iS}$, the continuity equation isMultiply the equation by ${\psi }^{\ast }$ and subtract its complex conjugate multiplied by $\psi$ to obtain this identity; real interaction and potential terms cancel. Thus the current velocity is $\mathbf{j}/n=2D\nabla S$. With dimensional kinetic operator $-{\hslash }^2{\nabla }^2/(2m)$ and left side $i\hslash {\psi }_t$, this gives the physical superfluid velocity $(\hslash /m)\nabla S$. In normalized equations with $D=1/2$ or $D=1$, the current velocity is respectively $\nabla S$ or $2\nabla S$. The distinction fixes the radial-flow slope of a driven quantum vortex.

Neglecting density-gradient kinetic energy in the stationary Gross–Pitaevskii equation gives $n=(\mu -V{)}_{+}/g$ for repulsive interactions. For a radial harmonic trap $V=m{\omega }_r^2{r}^2/2$, this is $n=n_0(1-r^2/R_r^2{)}_{+}$ with $R_r=\sqrt{2g{n}_0/(m{\omega }_r^2)}$. Hard boundaries can truncate this natural support. The approximation fails in vortex cores and boundary healing regions and requires a cloud much larger than its healing length.

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.

With $E=\frac{1}{2}\int [|\nabla \psi {|}^2+\frac{1}{2}(1-|\psi {|}^2{)}^2]$ and the renormalized momentum of a condensate, a Gross–Pitaevskii solitary wave is a critical point of $E-v{p}_x$. Along a differentiable solution family, fixed bulk normalization and vanishing variation boundary terms give $dE/d\lambda =vd{p}_x/d\lambda$. Therefore $dE/d{p}_x=v$ wherever $d{p}_x/d\lambda \ne 0$. At a cusp or turning point the parametrized identity is the appropriate statement; one must not assume a globally single-valued dispersion branch.

For a condensate with uniform bulk complex field one, its convergent localized momentum isThe subtraction removes the uniform-background phase-gradient contribution. If $\psi -1=O(r^{-1})$ and $\nabla \psi =O(r^{-2})$, its integrand is $O(r^{-3})$, integrable in two dimensions. Under decaying variations its functional derivative is $\delta p_j/\delta {\psi }^{\ast }=-i{\partial }_j\psi$; the subtraction is a boundary correction and does not change the bulk equation.

The healing length of a repulsive homogeneous condensate is the length at which density-gradient kinetic energy and interaction energy balance. The displayed convention sets the dimensionless Laplacian coefficient to one in the stationary Gross–Pitaevskii equation. A hard-wall number density profile has width $\sqrt{2}{\ell }_h$; definitions without the factor $\sqrt{2}$ occur, so the equation and convention must be specified.

A repulsive condensate next to an impenetrable planar wall has Dirichlet boundary condition $\psi (0)=0$. Writing $\psi =\sqrt{n_0}f$ reduces its stationary Gross–Pitaevskii equation to ${\ell }_h^2{f}^{\prime \prime }+(1-f^2)f=0$. The bulk conditions $f\to 1,f^{\prime }\to 0$ give ${\ell }_h^2(f^{\prime }{)}^2=(1-f^2{)}^2/2$ and hence the displayed number density. Omitting $n_0$ describes normalized number density, not dimensional number density.