Symmetry Operators for the Fokker-Plank-Kolmogorov Equation with Nonlocal Quadratic Nonlinearity

TL;DR

This paper explores symmetry operators for a nonlinear Fokker-Planck-Kolmogorov equation with nonlocal quadratic nonlinearity, reducing the nonlinear problem to a linear one and analyzing the relation between their symmetries.

Contribution

It introduces a method to relate symmetry operators of nonlinear and linear Fokker-Planck-Kolmogorov equations with nonlocal nonlinear drift terms.

Findings

Reduction of nonlinear problem to linear form

Identification of symmetry operators for the equations

Examples of one-dimensional symmetry operators

Abstract

The Cauchy problem for the Fokker-Plank-Kolmogorov equation with a nonlocal nonlinear drift term is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Fokker-Plank-Kolmogorov equations is considered. Illustrative examples of the one-dimensional symmetry operators are presented.