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

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.

33 pages, 2 tables, corrected version

References in corpus (4)

• Conservation Laws and Potential Symmetries of Linear Parabolic Equations
• Enhanced preliminary group classification of a class of generalized diffusion equations
• Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations
• Mapping method of group classification

Cited by in corpus (8)

• Point and generalized symmetries of the heat equation revisited
• Admissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
• Extended symmetry analysis of (1+2)-dimensional fine Kolmogorov backward equation
• Lie reductions and exact solutions of dispersionless Nizhnik equation
• Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equation
• Generalized symmetries of remarkable (1+2)-dimensional Fokker-Planck equation
• Surprising symmetry properties and exact solutions of Kolmogorov backward equations with power diffusivity
• Hidden symmetries, hidden conservation laws and exact solutions of dispersionless Nyzhnyk equation