Soliton solutions of logarithmic wave equation and their application for polycrystalline metals

Wave equations with logarithmic nonlinearity are applied to Korteweg-type materials which can undergo liquid-solid or liquid-gas phase transitions. One of predictions of the theory is a periodical pattern for inhomogeneities of density, which can occur in the form of bubbles or cells. Such inhomogeneities are described by soliton and solitary wave solutions of the logarithmic wave equation in the vicinity of a liquid-solid phase transition. During the solidification process, these inhomogeneities become centers of nucleation of grains. Previous works were dealing with generic natural silicate materials in geophysics, such as magmas in volcanic conduits, where the (approximately) periodical flows and structures were observed. Here we report an experimental evidence of a large-scale periodicity in structure of grains in the structural steel S235/A570 Grade 36, copper C-Cu/C14200, stainless steel X10CrNiTi18-10/AISI 321, and aluminium-magnesium alloy 5083/5056.