Generalized symmetries of remarkable (1+2)-dimensional Fokker-Planck equation

TL;DR

This paper uncovers the generalized symmetry algebra of a special (1+2)-dimensional Fokker-Planck (Kolmogorov) equation, revealing its deep algebraic structure and enabling new methods for solving related differential equations.

Contribution

It introduces a novel method to find the algebra of generalized symmetries of the Kolmogorov equation and links it to the second Weyl algebra, providing new insights and solution techniques.

Findings

Identified the algebra of generalized symmetries for the Kolmogorov equation.

Established an isomorphism with the second Weyl algebra.

Applied symmetry methods to find exact solutions of the equation.

Abstract

Using an original method, we find the algebra of generalized symmetries of a remarkable (1+2)-dimensional ultraparabolic Fokker-Planck equation, which is also called the Kolmogorov equation and is singled out within the entire class of ultraparabolic linear second-order partial differential equations with three independent variables by its wonderful symmetry properties. It turns out that the essential subalgebra of this algebra, which consists of linear generalized symmetries, is generated by the recursion operators associated with the nilradical of the essential Lie invariance algebra of the Kolmogorov equation, and the Casimir operator of the Levi factor of the latter algebra unexpectedly arises in the consideration. We also establish an isomorphism between this algebra and the Lie algebra associated with the second Weyl algebra, which provides a dual perspective for studying their properties. After developing the theoretical background of finding exact solutions of homogeneous linear systems of differential equations using their linear generalized symmetries, we efficiently apply it to the Kolmogorov equation.