low-energy expansion of the solution of the linearized equation. Since the exact zero-energy solutions given in this paper are quite general, we believe that our method will also be useful to derive other low-energy physical properties.\
The organization of the paper is as follows. In Sec. \[sec:fundamental\], we introduce fundamental equations and see the fundamental properties. The definition of the transmission-reflection problem is also given. In Sec. \[sec:mainresult\], we give a main result and verify it by numerical study of CQNLS equation. Sections \[sec:proof1\] and \[sec:proof2\] are devoted to the proof of main results. Discussions, future perspectives, and conclusions are given in Sec. \[sec:summary\]. Some mathematically technical formulae are treated in Appendices.

![\[introfigure\]A schematic picture of the problem that we consider in this paper. $ p $ represents a half of the velocity of the dark soliton. The problem is always considered in the comoving frame of the soliton. $ \mathrm{e}^{\mathrm{i}k_1x} $ is an incident wave of a linearized excitation, $ t\mathrm{e}^{\mathrm{i}k_1x} $ is a transmitted wave, and $ r\mathrm{e}^{\mathrm{i}k_2x} $ is a reflected wave. For more detailed definitions of each quantity, see Sec. \[sec:fundamental\].](introfiguretex2img.eps)

Fundamental Equations and Definition of the Problem {#sec:fundamental}
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Fundamental equations
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We begin with 