(\[eq:bogos\]) is given by $$\begin{aligned}
        \!\!\!\!\!\!&\begin{pmatrix}-\partial_x^2-\mu+4a_1|\phi|^2+9a_2|\phi|^4 & -2a_1\phi^2-6a_2|\phi|^2\phi^2 \\ 2a_1\phi^{*2}+6a_2|\phi|^2\phi^{*2} & \!\! \partial_x^2+\mu-4a_1|\phi|^2-9a_2|\phi|^4 \end{pmatrix}\begin{pmatrix}u \\ v \end{pmatrix} \nonumber \\
        &=\epsilon\begin{pmatrix}u\\v\end{pmatrix}. \label{eq:bogocqnls}
    \end{aligned}$$ It is known that a bubble and unstable dark solitons appear when $ a_1<0 $ and $ a_2>0 $[@BarashenkovMakhankov; @BarashenkovPanova]. This case is considered in Subsec. \[sec:criticalcase\]. As shown in [@BarashenkovPanova], when the soliton velocity is smaller than the critical value, a small perturbation induces “nucleation dynamics”, and the soliton cannot preserve its shape any more. So, this instability is not convective but absolute.\
The Landau velocity (\[eq:defofpl\]) is given by $$\begin{aligned}
        p_{\text{L}} = \sqrt{\rho_\infty(a_1+3a_2\rho_\infty)}.
    \end{aligned}$$ The necessary condition $ a_1+3a_2\rho_\infty>0 $ follows for a uniform condensate to be stable. The dark soliton solution is given by $$\begin{aligned}
        \phi(x,p,\rho_\infty) =\mathrm{e}^{\mathrm{i}px}\frac{\kappa\rho_0+\mathrm{i}p(\rho_\infty-\rho_0)\tanh\kappa x}{\sqrt{\rho_0(\kappa^2-a_2(\rho_\infty-\rho_0)^2\tanh^2\kappa x)}} \label{eq:dscqnls}
    \end{aligned}$$ with $$\begin{aligned}
        \kappa &=\sqrt{p_{\text{L}}^2-p^2}, \\
        \rho_0 &=\rho(x=0)= \frac{-(2a_2\rho_\infty+a_1)+\sqrt{(2a_2\rho_\infty+a_1)^2+4a_2p^2}}{2a_2}. \label{eq:rho0cqnls}
    \end{aligned}$$ See \[app:cqnls\] for a detailed derivation. Since $ \kappa $ and $ \rho_0 $ are the functions of $ (p,\rho_\infty) $, the dark soliton solution has two parameters $ (p,\rho_\infty) $. From (\[eq:rho0cqnls\]), $$\begin{aligned}
        \lim_{p\rightarrow0}\rho_0=\begin{cases} 0 & (2a_2\rho_\infty+a_1>0) \\ \frac{1}{2a_2}|2a_2\rho_\infty+a_1| & (2a_2\rho_\infty+a_1<0), \end{cases}
    \end{aligned}$$ so the bubble (= a non-topological dark soliton) appears when $ \rho_\infty<-a_1/(2a_2) $. A particle number of soliton $ N 