(\[eq:nlsdsasym\]), the chemical potential $ \mu $ becomes the same as (\[eq:cp\]), and the above constants are determined as $$\begin{aligned}
        j&=\rho_\infty p, \label{eq:dsj} \\
        j_m&=2\rho_\infty p^2+\rho_\infty F(\rho_\infty)-U(\rho_\infty). \label{eq:dsjm}
    \end{aligned}$$ The conservation laws are then rewritten as $$\begin{aligned}
        S_x &= \frac{j}{\rho} =\frac{\rho_\infty p}{\rho} \ \leftrightarrow \ S = p\int_0^x\frac{\rho_\infty\mathrm{d}x}{\rho}, \label{eq:phasecond} \\
        \frac{(\rho_x)^2}{4} &= -p^2(\rho_\infty-\rho)^2+\rho\left[ U(\rho)-U(\rho_\infty)-(\rho-\rho_\infty)U'(\rho_\infty) \right]. \label{eq:momconsrv}
    \end{aligned}$$ Thus, one can at least obtain the formal solution $$\begin{aligned}
        \pm 2(x-x_0) = \int\!\!\frac{\mathrm{d}\rho}{\sqrt{\text{R.H.S. of Eq. (\ref{eq:momconsrv})}}},
    \end{aligned}$$ even though it is not easy in general to carry out this integration and obtain the solution in closed form “$ \rho(x)=\dots $”. Henceforth we do not need this formal solution, but we assume the existence of a dark soliton solution which has no singularity and satisfies the asymptotic condition (\[eq:nlsdsasym\]).\
From Eq. (\[eq:phasecond\]), The phase shift $ \delta $ in Eq. (\[eq:nlsdsasym\]) can be written down explicitly: $$\begin{aligned}
        \delta = p\int_{-\infty}^\infty\!\!\mathrm{d}x\left( \frac{\rho_\infty}{\rho}-1 \right).
    \end{aligned}$$ We also introduce the symbol for the particle number of the dark soliton for later convenience: $$\begin{aligned}
        N := \int_{-\infty}^\infty\!\!\mathrm{d}x\left( \rho-\rho_\infty \right)<0.
    \end{aligned}$$

Barashenkov’s criterion
-----------------------

The stability of the soliton is described by the following renormalized momentum[@Barashenkov]: $$\begin{aligned}
        P = \int_{-\infty}^\infty\mathrm{d}x\left( \frac{\tilde{\phi}^*\tilde{\phi}_x-\tilde{\phi}\tilde{\phi}^*_x}{2\mathrm{i}} \right)\left( 1-\frac{\rho_\infty}{|\tilde{\phi}|^2} \right)
    \end{aligned}$$ Here $ \tilde{\phi}(x,t) 