Logarithmic Schrodinger-like equation as a model for magma transport

We show that, under a suitable assumption on the pressure tensor, the mass and momentum balance equations of hydrodynamical theory, introduced in the early 1980s by many authors to describe matrix separation, yield the equations of the Madelung fluid that are equivalent to the Schr ̈ odinger-like equation with logarithmic nonlinearity. This equation has solitary-waves solutions as required by many experimental volcanic models.