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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.

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.