= Solution
A neutral superfluid's broken global $U(1)$ phase symmetry produces a <Goldstone mode>. Coupling phase to density fluctuations generally gives a long-wavelength <phonon> with $E(p)\simeq cp$, rather than the quadratic dispersion of a noninteracting condensate. In a dilute interacting Bose gas, the <Bogoliubov quasiparticle dispersion> is
$$
E(p)=\sqrt{\varepsilon_p(\varepsilon_p+2gn)},\qquad
\varepsilon_p=\frac{p^2}{2m},\qquad c=\sqrt{\frac{gn}{m}}.
$$
For a heavy weak obstacle moving at speed $v$, emitting an excitation of <momentum> $\mathbf p$ changes the available <energy> by $E(p)-\mathbf v\cdot\mathbf p$. The <Landau criterion> therefore gives
$$
\boxed{v_L=\inf_{p>0}\frac{E(p)}p.}
$$
The dilute-gas spectrum has $v_L=c$. In liquid helium, a <roton> minimum can make the minimum ratio smaller than the sound speed; the slope of a line from the origin tangent to the full dispersion determines the kinematic bound. A neutral paired Fermi superfluid also has a fermionic pair-breaking bound of order $\Delta/p_F$ in weak coupling. The smallest relevant branch bound must be considered. Actual flow can fail earlier through vortex nucleation, surface defects or other instabilities: the excitation-energy criterion is not a complete dynamical guarantee of dissipation-free flow.
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