solutions. As an important tool, parameter derivatives are introduced.

Parameter derivative {#subsec:paradera}
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As seen in the asymptotic form (\[eq:nlsdsasym\]) or in the example of the CQNLS system in Subsec. \[subsec:cqnls\], the dark soliton solution has two parameters, i.e., $ (p,\rho_\infty) $. So we can consider two kinds of parameter derivatives:[^1] $ \partial_p\phi $ and $ \partial_{\rho_\infty}\phi $. We can use arbitrary coordinates to “label” the two-dimensional parameter space $ (\alpha,\beta)=(\alpha(p,\rho_\infty), \beta(p,\rho_\infty)) $, unless the Jacobian of coordinate transformation is singular. Obviously, the final result must not depend on the choice of coordinates. In order to make the story general, we always use these general parameter derivatives, and henceforth, we write the parameter derivative simply by the subscript, i.e., $ \phi_\alpha := \partial_\alpha\phi $ and $ \phi_\beta:=\partial_\beta\phi $. We also introduce the following symbol: $$\begin{aligned}
        [A,B]_{\alpha\beta}:=A_\alpha B_\beta-A_\beta B_\alpha.
    \end{aligned}$$ Note that the ratio $$\begin{aligned}
        \frac{[A,B]_{\alpha\beta}}{[C,D]_{\alpha\beta}} \label{eq:invratio}
    \end{aligned}$$ has a coordinate-free meaning, in other words, it is invariant under coordinate transformations of parameter space. We often construct coordinate-free solutions in such a ratio form.\
An immediate application of parameter derivatives is that one can obtain a particular solution of zero-energy Bogoliubov equation (\[eq:bogos\]). By differentiation of the stationary NLS eq. (\[eq:nls2\]), 