$ is the dark soliton solution in the frame where the surrounding condensate is at rest and the soliton is moving. Writing the soliton velocity $ v $, the stability criterion for the dark solitons is expressed by $ \partial P/\partial v<0 $.\
We can rewrite the above integral by the density profile $ \rho(x) $: $$\begin{aligned}
    \begin{split}
        P &= -p\int_{-\infty}^\infty\mathrm{d}x\left( \rho_\infty-\rho \right)\left( \frac{\rho_\infty}{\rho}-1 \right) \\
        &= -p N-\rho_\infty \delta.
    \end{split}
    \end{aligned}$$ Here remember that the soliton velocity is given by $ v=-2p $, as stated in the preceding subsection. The stability condition is rewritten as $ \partial (-P)/\partial p <0 $.

Definition of the scattering problem {#sec:defofsp}
------------------------------------

In this subsection, we define the transmission and reflection problem of Bogoliubov phonons shown in Fig. \[introfigure\]. Since the linearized equation does not satisfy simple particle number conservation, we must define transmission and reflection coefficients via the conservation of excitation energy. The conservation of excitation energy corresponds to the constancy of the following Wronskian:[@Kagan; @DanshitaYokoshiKurihara]: $$\begin{aligned}
        W = u^*\partial_xu-u\partial_xu^*+v^*\partial_xv-v\partial_xv^*.
    \end{aligned}$$ Let us assume that the asymptotic form of the condensate wavefunction $ \phi(x) $ is given by Eq. (\[eq:nlsdsasym\]). In this situation, sufficiently far from the origin, the Bogoliubov equations have 