Symmetries of linear and nonlinear partial differential equations

TL;DR

This paper explores the structure of symmetries in linear and nonlinear partial differential equations, revealing their algebraic properties and how linear symmetries can generate nonlinear ones, with new findings in gas dynamics.

Contribution

It establishes the relationship between higher and operator symmetries and discovers new symmetries in two-dimensional stationary gas dynamics equations.

Findings

Higher symmetries form a Lie algebra

Operator symmetries form an associative algebra

New symmetries identified in gas dynamics equations

Abstract

We consider higher symmetries and operator symmetries of linear partial differential equations. The higher symmetries form a Lie algebra, and operator ones form an associative algebra. The relationship between these symmetries is established. We show that symmetries of linear equations sometimes generate symmetries of nonlinear ones. New symmetries of two-dimensional stationary equations of gas dynamics are found.