What are phonons, rotons, and maxons, and what does their dispersion curve have to do with superfluidity? I understand that they are quasiparticles, but I'm not entirely sure what that implies. Are these concepts purely notational and/or mathematical? Or are these three quasiparticles fundamentally different from each other?
Superfluid Helium-4 has a very well studied excitation structure -- at very low momenta, there is a low energy excitation, the phonon, that corresponds to a periodic density fluctuation in the superfluid with well defined wave-number and an energy $E = c \hbar k$ (c being the speed of sound in the superfluid). Though others might quibble with me over vocabulary, I prefer to call phonons "collective excitations" and reserve "quasiparticle" for excitations that correspond to renormalized single-particle excitations (like an electron in a Fermi Liquid).
In either case though, what is meant is simply that the excitation is long-lived, or, by the uncertainty principle, that it has a sharply defined energy, and here the phonon does.
This linear relation breaks down at higher momenta, where the $E,k$ curve turns down, then turns back up. Near the local maximum, there are sharp excitations with an inverse-parabolic dispersion. These are the maxons. Near the local minimum we have a sharp excitation with a parabolic dispersion, and these are the rotons.
Historically rotons were introduced by Landau, with a guess for their dispersion ($E=\Delta+(p-p_0)^2/2m$), not merely as a mathematical device, but because a superfluid described only by the phonon dispersion fails to capture the actual thermodynamics observed in Helium-4. On the other hand, Landau wasn't concerned about a qualitative, microscopic picture of what rotons are, in the sense that we know what phonons mean for the local density of the system.
So, I am actually not too sure what rotons are in that same microscopic sense, and the same for maxons, although it is apparent that a gas of phonons and a gas of rotons are very different things. On the other hand, this is somewhat paradoxical since all three of these excitations are just different parts of the same dispersion curve, so to say they are fundamentally different is a phrase I wouldn't use without a great deal of caution...
Lastly, knowing all of the excitations present in superfluid helium is very important for calculating its thermodynamic properties, etc., however I think (and hopefully someone will correct me if I'm wrong!) the only excitation that is intrinsic to superfluidity is the phonon mode - this is because the superfluid state breaks a continuous gauge symmetry of the normal fluid, and thus phonons are the massless Goldstone modes of the superfluid state.