= Longitudinal scaling identity for a condensate solitary wave
{title2=$E=\int|\partial_z\psi|^2dV$}
For a finite-energy <Gross–Pitaevskii solitary wave>, write $E=E_z+E_\perp+E_{\rm pot}$ with $E_z=\frac12\int|\psi_z|^2$. 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 $\lambda E_z+\lambda^{-1}(E_\perp+E_{\rm pot})$. Stationarity of $E-Up$ at $\lambda=1$ implies $E_z=E_\perp+E_{\rm pot}$ and hence the displayed identity. This is a directional <Pohozaev identity>, valid when the integrals and variation boundary terms converge.
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