Negative curvature of zero sound dispersion In the theory of a Landau-Fermi liquid, one of the major predictions is the dispersion of zero sound. From the linearized kinetic equation, we know that the dimensionless dispersion $s$ is given by
$$ 
s=\frac{\omega}{qv_F}=\begin{cases}
1+2e^{-2(1+1/F_0^s)},\quad &F_0^s\ll1\\
\sqrt{F_0^s/3},\quad &F_0^s\gg1
\end{cases}
$$
where $F_0^s$ is the Landau parameter quantifying interactions, $\omega/q$ is the phase velocity of the excitation, and $v_F$ is the Fermi velocity.
However, I read this work (Two-dimensional Fermi liquids sustain surprising roton-like plasmons beyond the particle-hole band, by Sultan et. al.) which gives a schematic representation of the elementary excitations of a Fermi liquid in Fig. 1. The authors then state that

At relatively low wave-vectors, zero-sound is observed as a well-defined mode with a linear dispersion relation, located above the PHB. It displays then a negative curvature, finally entering the PHB.

where PHB means particle-hole band. My question is if there is any in-depth study that talks about this "negative curvature" of the zero-sound dispersion. I would think that this would amount to taking higher-order terms of $q$ in the above expression for $s$, but I have not found any reference that discusses this apparent "plateauing" of the zero sound mode.
 A: To preface the answer below, this is not a subject I have ever actively participated in, so it may be interpreting the history wrong.
In this paper by Cowley here, the distinguishing factor between zero sound and first sound (ordinary acoustic sound) is the relationship between measured frequency $\omega$ and the excitation lifetime $\frac{1}{\tau}$. If the frequency is much slower than the lifetime $\omega \ll \frac{1}{\tau}$, it is called first sound and is the usual vibration mode of a condensed matter phase. If the frequency is much faster than the lifetime, then it is called zero sound $\omega \gg \frac{1}{\tau}$. So, technically, zero and first sound are smoothly connected, but in practice you often need different experimental approaches to measure each one. Let's focus on the zero sound regime (phonon), since that is most relevant to the roton zero-sound mode.
The roton was originally thought to be a completely separate excitation from the zero-sound phonon, as shown in the image below.

But experimentally it was found that the zero-sound phonon and roton are connected and are the same excitation (read this for references), as shown in the image below. Both of these images are from this reference.

In fact this minimum seems to be found in all sorts of non-superfluid liquids, from liquid hydrogen to supercritical liquid argon (see here). So the modern understanding is that the roton has little to do with vortices etc., but represents the fact that density fluctuations with wavelengths close to the interatomic lattice spacing of the nearby solid phase cost relatively lower energy (i.e. energy lowering when $q\sim \frac{2\pi}{a}$, where $a$ is the solid-phase lattice spacing). A discussion of this fact is found here.
Nonetheless, it does seem that some researchers are still  studying theories of the roton that distinguish it from the phonon, but I don't know how they reconcile such theories with the fact that many non-superfluids also exhibit the roton.
