A method for the analytical study of autowave processes in weakly conductive and conductive liquids

The article presents mathematical modeling of autowaves in a weakly conductive liquid (ferrocolloid) and in a conductive liquid (salt solution), taking into account ion recharge in the regions of spatial charge in the form of a boundary value problem for a nonstationary system of Nernst-Planck-Poisson equations. Using the proposed mathematical model, a theoretical study of the occurrence of the self-organization process – autowaves in an electromembrane system and in a ferrocolloid (magnetic liquid) was carried out. The main mechanisms in the occurrence of autowaves are the recharging of magnetic particles (in the case of a ferrocolloid) and the recharging of ions (cations and anions) in the case of an electromembrane system. Impurity ions are involved only in the charge transfer process. A numerical analysis of the boundary value problem of the mathematical model is carried out and the main regularities of the phenomenon of recharging on the transfer of magnetic particles are established. The existence of soliton-like autowave solutions for magnetic particles and salt ions is shown. The total concentration of basic ions practically does not change, in contrast, the total concentration of impurity ions decreases. The article proposes a new mathematical method for the approximate analytical solution of a boundary value problem based on the derivation of a nonlinear partial differential equation for a potential in the region of a single wave. It is shown that this equation can be reduced using a number of transformations, including the Hopf–Cole transformation, to a canonical parabolic equation, and in this sense an exact analytical solution is found. In addition, a simple analytical approximation for a single wave was found, a comparison with the numerical solution was carried out and their qualitative and quantitative coincidence was shown (with an accuracy of $\sim 3\%$).