Surprising symmetry properties and exact solutions of Kolmogorov backward equations with power diffusivity

TL;DR

This paper investigates the symmetry properties and exact solutions of specific Kolmogorov backward equations with power diffusivity, revealing notable symmetries and generating new solutions through Lie reduction and recursion operators.

Contribution

It provides a complete group classification of these equations, extends known results to discrete transformations, and uncovers new exact solutions and hidden symmetries.

Findings

Identified eight key exponents for the diffusion coefficient related to symmetry properties.

Extended symmetry analysis to include discrete transformations and hidden Lie symmetries.

Generated new exact solutions using recursion operators acting on Lie-invariant solutions.

Abstract

Using the original advanced version of the direct method, we efficiently compute the equivalence groupoids and equivalence groups of two peculiar classes of Kolmogorov backward equations with power diffusivity and solve the problems of their complete group classifications. The results on the equivalence groups are double-checked with the algebraic method. Within these classes, the remarkable Fokker-Planck and the fine Kolmogorov backward equations are distinguished by their exceptional symmetry properties. We extend the known results on these two equations to their counterparts with respect to a nontrivial discrete equivalence transformation. Additionally, we carry out Lie reductions of the equations under consideration up to the point equivalence, exhaustively study their hidden Lie symmetries and generate wider families of their new exact solutions via acting by their recursion operators on constructed Lie-invariant solutions. This analysis reveals eight powers of the space variable with exponents -1, 0, 1, 2, 3, 4, 5 and 6 as values of the diffusion coefficient that are prominent due to symmetry properties of the corresponding equations.