the NLS equation with a general nonlinearity $$\begin{aligned}
        \mathrm{i}\partial_t\phi=-\partial_x^2\phi+F(|\phi|^2)\phi. \label{eq:nls}
    \end{aligned}$$ Here, $ F(\rho) $ is a real-valued function such that $ F(0)=0 $. The energy functional (Hamiltonian) which yields this equation is $$\begin{aligned}
        H = \int\!\mathrm{d}x \left(|\partial_x\phi|^2+U(|\phi|^2)\right),
    \end{aligned}$$ where $$\begin{aligned}
        U(\rho) = \int_0^\rho\mathrm{d}\rho' F(\rho').
    \end{aligned}$$ Letting $ \phi=\phi+\delta\phi $ in Eq. (\[eq:nls\]), and discarding higher order terms of $ \delta\phi $, one obtains the following linearized equation: $$\begin{aligned}
        \mathrm{i}\partial_t\delta\phi=\left[-\partial_x^2+ F(|\phi|^2)+|\phi|^2F'(|\phi|^2) \right]\delta\phi+\phi^2F'(|\phi|^2)\delta\phi^*.
    \end{aligned}$$ Writing $ \delta\phi=u $ and $ -\delta\phi^*=v $, one obtains $$\begin{aligned}
        \mathrm{i}\partial_t\begin{pmatrix}u\\v\end{pmatrix}=\mathcal{L}\begin{pmatrix}u\\v\end{pmatrix}, \label{eq:tdbogo}
    \end{aligned}$$ where $ \mathcal{L} $ is a $ 2\times2 $ matrix operator whose components are $$\begin{aligned}
        \mathcal{L}_{11}^{}&=-\mathcal{L}_{22}^{}=-\partial_x^2+F(|\phi|^2)+|\phi|^2F'(|\phi|^2), \\
        \mathcal{L}_{12}^{}&=-\mathcal{L}_{21}^*=-\phi^2F'(|\phi|^2).
    \end{aligned}$$ We use the notation $ (u,v) $ since it is commonly used by condensed matter physicists. If we interpret the system as BEC, this equation is the Bogoliubov equation which describes the Bogoliubov phonon (or Bogoliubov quasiparticle) [@Bogoliubov]. (As a review or a textbook, see, e.g., [@FetterWalecka; @DalfovoGiorginiPitaevskiiStringari; @PethickSmith].) For this reason, henceforth, we call $ \phi $ the condensate wavefunction or the order parameter, and $ (u,v) $ the Bogoliubov (quasiparticle) wavefunction, though the NLS equation itself has more applications in various fields.\
Henceforth we mainly consider the stationary (i.e., time-independent) problem. The 