= Solution
Suppress tildes in the normalized equation and let $X=x-vt$. A <Gross–Pitaevskii solitary wave> stationary in these translating coordinates obeys
$$
\boxed{\nabla^2\psi+(1-|\psi|^2)\psi-2iv\partial_X\psi=0,\qquad
\psi(X,y)\to1\ \text{as }X^2+y^2\to\infty.}
$$
The 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> are
$$
\boxed{E=\frac12\int_{\mathbb R^2}\left[|\nabla\psi|^2+\frac12(1-|\psi|^2)^2\right]dX\,dy,}
$$
$$
\boxed{\mathbf p=\frac1{2i}\int_{\mathbb R^2}\left[(\psi^*-1)\nabla\psi-(\psi-1)\nabla\psi^*\right]dX\,dy.}
$$
The 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 $\psi\to1$ 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> $1/\sqrt2$.
<Integration by parts>, with decaying variations and their boundary terms vanishing, gives
$$
\frac{\delta E}{\delta\psi^*}=-\frac12\nabla^2\psi+\frac12(|\psi|^2-1)\psi,
\qquad \frac{\delta p_x}{\delta\psi^*}=-i\partial_X\psi.
$$
Thus the translating equation is precisely $\delta(E-vp_x)=0$. Along a differentiable family $\psi_\lambda$ with fixed bulk normalization, evaluate this stationarity on $\partial_\lambda\psi$ to obtain
$$
\frac{dE}{d\lambda}=v\frac{dp_x}{d\lambda},\qquad
\boxed{\frac{dE}{dp_x}=v}
$$
where $dp_x/d\lambda\ne0$. 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 $i\psi_t=\delta E/\delta\psi^*$, consistent with the normalized equation; changing the normalization of $E$ alone would change the claimed slope.
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