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Let us consider the dark soliton solution of stationary NLS Eq. (\[eq:nls2\]) in the comoving frame of the soliton. In this coordinate, the soliton is static but the surrounding condensate is flowing. Let us seek the solution with the asymptotic form $$\begin{aligned}
        \phi(x\rightarrow \pm\infty)=\sqrt{\rho_\infty}\mathrm{e}^{\mathrm{i}(px\pm\frac{\delta}{2})}. \label{eq:nlsdsasym}
    \end{aligned}$$ It should be noted that the velocity of the soliton is not $ -p $ but $ -2p $, because the Galilean covariance of NLS equation leads to the following property: $$\begin{aligned}
    \begin{split}
        &\text{$ \phi(x,t) $ is a solution.} \\
        \leftrightarrow \quad & \text{$\tilde{\phi}(x,t,\alpha)=\phi(x+2\alpha t,t)\mathrm{e}^{-\mathrm{i}\alpha x}\mathrm{e}^{-\mathrm{i}\alpha^2t}$ is a solution.}
    \end{split}\label{eq:galilei}
    \end{aligned}$$ So, if one has the solution in the form $ \phi(x,t)=\mathrm{e}^{-\mathrm{i}\mu t}\mathrm{e}^{\mathrm{i}px}f(x) $, the corresponding soliton-moving solution is given by $ \tilde{\phi}(x,t,p)=\mathrm{e}^{-\mathrm{i}(\mu-p^2)t}f(x+2pt) $. However, for brevity, we sometimes call $ p $ “velocity”, ignoring the difference of twice factor.\
From the conservation laws of mass and momentum, one can immediately find two integration constants: $$\begin{aligned}
        j&=\frac{\phi^*\phi_x-\phi\phi_x^*}{2\mathrm{i}}, \\
        j_m &= |\phi_x|^2+\mu|\phi|^2-U(|\phi|^2). \label{eq:genjm}
    \end{aligned}$$ Here $ j $ is a mass current density and $ j_m $ is a momentum current density. Let us write the density and the phase of the condensate as $ \phi=\sqrt{\rho}\mathrm{e}^{\mathrm{i}S} $. Taking account of the asymptotic form 