Point and generalized symmetries of the heat equation revisited

TL;DR

This paper revisits the symmetries of the heat equation, revealing its discrete symmetry structure, introducing pseudo-discrete elements, and extending the symmetry classification to related equations like Burgers.

Contribution

It provides a detailed classification of point symmetries of the heat equation, clarifies the nature of its discrete symmetries, and introduces the concept of pseudo-discrete elements, extending symmetry analysis methods.

Findings

The heat equation has exactly one independent discrete symmetry, sign change of the dependent variable.

Alternating the sign of the space variable is a pseudo-discrete element, not a true discrete symmetry.

The approach can be extended to other linear and nonlinear differential equations.

Abstract

We derive a nice representation for point symmetry transformations of the (1+1)-dimensional linear heat equation and properly interpret them. This allows us to prove that the pseudogroup of these transformations has exactly two connected components. That is, the heat equation admits a single independent discrete symmetry, which can be chosen to be alternating the sign of the dependent variable. We introduce the notion of pseudo-discrete elements of a Lie group and show that alternating the sign of the space variable, which was for a long time misinterpreted as a discrete symmetry of the heat equation, is in fact a pseudo-discrete element of its essential point symmetry group. The classification of subalgebras of the essential Lie invariance algebra of the heat equation is enhanced and the description of generalized symmetries of this equation is refined as well. We also consider the Burgers equation because of its relation to the heat equation and prove that it admits no discrete point symmetries. The developed approach to point-symmetry groups whose elements have components that are linear fractional in some variables can directly be extended to many other linear and nonlinear differential equations.