  We formulate the mapping between a large class of nonlinear wave equations
and flow equations for barotropic fluid with internal surface tension and
capillary effects. Motivated by statistical mechanics and multi-channel physics
arguments, we focus on wave equations with logarithmic nonlinearity, and
further generalize them to matrix equations. We map the resulting equation to
flow equations of multi-channel or multi-component Korteweg-type materials. For
some special cases, we analytically derive Gaussian-type matrix solutions and
study them in the context of fluid mechanics.
