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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 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.

Use decaying variations with the same complex bulk field one, so that integration by parts produces no variation boundary term. The functional derivatives of energy and renormalized momentum of a condensate areFor the second identity, variation of the two momentum terms and integration of the ${\partial }_z\delta {\psi }^{\ast }$ term each supply $(2i{)}^{-1}{\psi }_z$, so together they give $-i{\psi }_z$. It follows thatis exactly the stationary Gross–Pitaevskii solitary wave equation. Thus each solitary wave is a constrained energy critical point with multiplier $U$.

Let $s$ parametrize a differentiable family of these waves. Apply the critical-point identity to the variation ${\partial }_s\psi$, keeping the multiplier fixed at its value for that member of the family. Including the complex conjugate variation givesHence the energy–momentum slope of a solitary wave iswherever $dp/ds\ne 0$. At a momentum turning point the parametrized identity $dE/ds=Udp/ds$ remains the correct statement; a globally single-valued energy as a function of momentum is not needed. Differentiating $E-U(s)p$ as a composite function would include an extra $-p{U}^{\prime }$ term, so that composite derivative must not be set to zero.

Split the solitary-wave energy into longitudinal, transverse and interaction contributions,Consider the longitudinal dilation ${\psi }_{\lambda }(x,y,z)=\psi (x,y,\lambda z)$ for $\lambda >0$. Changing variables to $Z=\lambda z$ givesThe renormalized momentum of a condensate is unchanged: its single $z$ derivative contributes $\lambda$, cancelling the measure factor ${\lambda }^{-1}$. Since the wave is a critical point of $E-Up$, differentiating this admissible scaling variation at $\lambda =1$ givesTherefore the longitudinal scaling identity for a condensate solitary wave yieldsThis directional Pohozaev identity assumes the finite energy and sufficient decay to justify the dilation variation; it can equivalently be obtained with cutoffs followed by a limit. It is an identity for solitary-wave critical points, not for arbitrary localized fields.

Suppress tildes in the normalized equation and let $X=x-vt$. A Gross–Pitaevskii solitary wave stationary in these translating coordinates obeysThe boundary fixes the bulk complex argument and describes a localized disturbance with zero net vortex winding number; a vortex pair is possible. No extra plane-wave factor is needed when only the coordinates are changed. A full Galilean transformation of the field would additionally change the bulk flow and its complex argument convention.

Subtract the uniform background from the grand potential. A convenient dimensionless energy and the renormalized momentum of a condensate areThe subtraction in $\mathbf{p}$ removes the troublesome background complex argument gradient. In particular, for the usual subsonic localized far field $\psi -1=O(r^{-1})$, $\nabla \psi =O(r^{-2})$, and $1-|\psi {|}^2=O(r^{-2})$, the energy density is $O(r^{-4})$ and the momentum integrand is $O(r^{-3})$, both integrable in two dimensions. Merely writing $\psi \to 1$ without sufficient decay is not by itself a convergence proof. The common solitary-wave branch is subsonic: linearizing this normalization gives ${\omega }^2=k^2/2+k^4/4$ and speed of sound $1/\sqrt{2}$.

Integration by parts, with decaying variations and their boundary terms vanishing, givesThus the translating equation is precisely $\delta (E-v{p}_x)=0$. Along a differentiable family ${\psi }_{\lambda }$ with fixed bulk normalization, evaluate this stationarity on ${\partial }_{\lambda }\psi$ to obtainwhere $d{p}_x/d\lambda \ne 0$. This energy–momentum slope of a solitary wave does not require differentiating the Lagrange multiplier $v$ inside the field variation. At a turning point the parametrized identity remains true, while $E$ need not be a single-valued function of $p_x$ locally. The chosen factors make $i{\psi }_t=\delta E/\delta {\psi }^{\ast }$, consistent with the normalized equation; changing the normalization of $E$ alone would change the claimed slope.