The one-dimensional Fisher-Kolmogorov equation with a nonlocal nonlinearity in a semiclassical approximation

A model of the evolution of a bacterium population based on the Fisher-Kolmogorov equation is considered. For a one-dimensional equation of the Fisher-Kolmogorov type that contains quadratically nonlinear nonlocal kinetics and weak diffusion terms, a general scheme of semiclassically concentrated asymptotic solutions is developed based on the complex WKB-Maslov method. The solution of the Cauchy problem is constructed in the class of semiclassically concentrated functions. In constructing the solutions, an essential part is played by the dynamic set of Einstein-Ehrenfest equations (a set of equations in average and centered moments) derived in this work. The symmetry operators of the equation, the nonlinear evolution operator, and the class of particular asymptotic semiclassical solutions are found.