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.

For a finite-energy Gross–Pitaevskii solitary wave, write $E=E_z+E_{\perp }+E_{\mathrm{pot}}$ with $E_z=\frac{1}{2}\int |{\psi }_z{|}^2$. The longitudinal dilation ${\psi }_{\lambda }(x_{\perp },z)=\psi (x_{\perp },\lambda z)$ leaves its renormalized momentum of a condensate invariant and changes energy to $\lambda E_z+{\lambda }^{-1}(E_{\perp }+E_{\mathrm{pot}})$. Stationarity of $E-Up$ at $\lambda =1$ implies $E_z=E_{\perp }+E_{\mathrm{pot}}$ and hence the displayed identity. This is a directional Pohozaev identity, valid when the integrals and variation boundary terms converge.

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.