  Experimental data suggests that, at temperatures below 1 K, the pressure in
liquid helium has a cubic dependence on density. Thus the speed of sound scales
as a cubic root of pressure. Near a critical pressure point, this speed
approaches zero whereby the critical pressure is negative, thus indicating a
cavitation instability regime. We demonstrate that to explain this dependence,
one has to view liquid helium as a mixture of three quantum Bose liquids:
dilute (Gross-Pitaevskii-type) Bose-Einstein condensate, Ginzburg-Sobyanin-type
fluid, and logarithmic superfluid. Therefore, the dynamics of such a mixture is
described by a quantum wave equation, which contains not only the polynomial
(Gross-Pitaevskii and Ginzburg-Sobyanin) nonlinearities with respect to a
condensate wavefunction, but also a non-polynomial logarithmic nonlinearity. We
derive an equation of state and speed of sound in our model, and show their
agreement with experiment.
