  Schrodinger-type wave equations with logarithmic nonlinearity occur in
hydrodynamic models of Korteweg-type materials with capillarity and surface
tension, which can undergo liquid-solid or liquid-gas phase transitions. One of
the predictions of the theory is a periodic pattern of density inhomogeneities
occurring in the form of either bubbles (topological phase), or cells
(non-topological phase). Such inhomogeneities are described by solitonic
solutions of a logarithmic wave equation, gaussons and kinks, in the vicinity
of the liquid-solid phase transition. During the solidification process, these
inhomogeneities become centers of nucleation, thus shaping the polycrystalline
structure of the metal grains. The theory predicts a Gaussian profile of
material density inside such a cell, which should manifest in a Gaussian-like
profile of microhardness inside a grain. We report experimental evidence of
large-scale periodicity in the structure of grains in the ferrite steel
S235/A570, copper C-Cu/C14200, austenite in steel X10CrNiTi18-10/AISI 321, and
aluminium-magnesium alloy 5083/5056; and also Gaussian-like profiles of
microhardness inside an averaged grain in these materials.
