velocity (or a half of the sound wave velocity) is given by $$\begin{aligned}
        p_{\text{L}}=\sqrt{\frac{\rho_\infty F'(\rho_\infty)}{2}}. \label{eq:defofpl}
    \end{aligned}$$ (Note that the sound wave velocity is not $ p_{\text{L}} $ but $ 2p_{\text{L}} $; see the next subsection.) Since $ p_{\text{L}} $ must be real, in order for the uniform condensate to be stable, $ F'(\rho_\infty)>0 $ must hold. The coefficients $ \bar{u}_i $ and $ \bar{v}_i $ can be, e.g., chosen as follows: $$\begin{aligned}
        \bar{u}_{i}&=\sqrt{1+\frac{\epsilon-2p k_i^{}+k_i^2}{2p_{\text{L}}^2}}, \label{eq:ucoeff}\\
        \bar{v}_{i}&=\sqrt{1-\frac{\epsilon-2p k_i^{}-k_i^2}{2p_{\text{L}}^2}}. \label{eq:vcoeff}
    \end{aligned}$$ When $ \epsilon>0 $ and $ -p_{\text{L}}<p<p_{\text{L}} $, the quartic equation (\[eq:uniformdisp\]) has one real positive root, one real negative root, and two complex roots conjugate to each other. We call a real positive (negative) root $ k_1 \ (k_2) $, and a complex root with positive (negative) imaginary part $ k_3 \ (k_4) $. The low-energy expansions of them are given by $$\begin{aligned}
        k_1 &= \frac{\epsilon}{2(p+p_{\text{L}})}+O(\epsilon^3), \label{eq:kexpand1} \\
        k_2 &= \frac{\epsilon}{2(p-p_{\text{L}})}+O(\epsilon^3), \label{eq:kexpand2} \\
        k_3 &= 2\mathrm{i}\sqrt{p_{\text{L}}^2-p^2}+\frac{p\epsilon}{2(p_{\text{L}}^2-p^2)}+O(\epsilon^2),  \label{eq:kexpand3} \\
        k_4 &= -2\mathrm{i}\sqrt{p_{\text{L}}^2-p^2}+\frac{p\epsilon}{2(p_{\text{L}}^2-p^2)}+O(\epsilon^2).  \label{eq:kexpand4}
    \end{aligned}$$ $ w_1 $ and $ w_2 $ are plane wave solutions propagating in the positive and negative directions, respectively. $ w_3 $ and $ w_4 $ are exponentially divergent unphysical solutions.

Dark soliton solution in comoving 