A note on a Korteweg fluid and the hydrodynamic form of the logarithmic Schrodinger equation

It is shown that the hydrodynamic form of nonlinear logarithmic Schrodinger equation (NLSE), in 3 + 1 dimensions, describes a solitonic flow of an isothermal, inviscid, compressible, capillary (Korteweg) fluid. This formulation corresponds to the so-called diffuse interface models used in hydrodynamics for two-phase fluids. The negative thermodynamic pressure furnished by the model indicates a possible application of the model to the description of a gas-saturated magma melt, rising in a volcanic conduit, in presence of cavitation processes.