Generalized symmetries of remarkable (1+2)-dimensional Fokker-Planck equation
arXiv:2409.10348 · doi:10.1017/S0956792525100107
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.
arXiv comment30 pages, 2 figures, published version, minor corrections
References in corpus (14)
- Invariant Variation Problems
- A Simple Construction of Recursion Operators for Multidimensional Dispersionless Integrable Systems
- Conservation laws and normal forms of evolution equations
- Reducibility of zero curvature representations with application to recursion operators
- Higher Symmetries of the Laplacian via Quantization
- Recursion operator for stationary Nizhnik--Veselov--Novikov equation
- Point and generalized symmetries of the heat equation revisited
- Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation
- Generalized symmetries, conservation laws and Hamiltonian structures of an isothermal no-slip drift flux model
- Generalized symmetries and conservation laws of (1+1)-dimensional Klein-Gordon equation
- Representations of sl(2,C) in the BGG category O and master symmetries
- Extended symmetry analysis of (1+2)-dimensional fine Kolmogorov backward equation
- Higher symmetries of powers of the Laplacian and rings of differential operators
- Generalized symmetries of Burgers equation