Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation

TL;DR

This paper performs an extended symmetry analysis of a (1+2)-dimensional Fokker-Planck equation, classifies its symmetries, and constructs wide families of exact solutions, linking them to solutions of the heat and Kramers equations.

Contribution

It provides a comprehensive symmetry classification and solution construction for the Fokker-Planck equation, revealing its maximal symmetry properties and connections to other equations.

Findings

Maximal eight-dimensional Lie invariance algebra identified.

Complete classification of Lie reductions and solutions achieved.

Established similarity between the Fokker-Planck and Kramers equations.

Abstract

We carry out the extended symmetry analysis of an ultraparabolic Fokker-Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker-Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker-Planck equation using the direct method, analyze its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalized reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker-Planck equation, in particular, those parameterized by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker-Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.