equation ,saddle-node bifurcation ,universal scaling laws ,cubic-quintic nonlinear Schrödinger equation

Introduction
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In this paper, we solve a scattering problem of linearized excitations in the presence of a dark soliton in one-dimensional(1D) nonlinear Schrödinger (NLS) equation with a general nonlinearity: $$\begin{aligned}
        \mathrm{i}\partial_t\phi = -\partial_x^2\phi+F(|\phi|^2)\phi, \label{eq:intro001}
    \end{aligned}$$ and discuss the physical and mathematical significance of our results. For a schematic picture, see Fig. \[introfigure\]. The precise mathematical definition of the problem will be given in Sec. \[sec:fundamental\]. If we regard the system as a Bose-Einstein condensate(BEC), the linearized excitation is a Bogoliubov phonon, so the problem can be also called a soliton-phonon scattering problem, as this paper entitled.\
The NLS equation (\[eq:intro001\]) has a great number of applications in nonlinear optics, superconductors, magnetism, BECs, and so on. Particularly, much attention has been focused on the experimental realizations of BECs in laser-cooled ultracold atoms for more than a decade, because of high-controllability of the system parameters. By using elongated laser beams, low-dimensional systems are realized, and a dark soliton can be created via the phase imprinting method[@BurgerPhaseImprinting]. The Bogoliubov theory is also well confirmed[@Andrews; @Stamper; @Steinhauer].\
It is known that 1D NLS with a *cubic* nonlinearity is completely integrable[@ZakharovShabat; @ZakharovShabat2]. Because of 