Contributions to the Algorithmic Foundations of Approximate Lie Symmetry Algebras of Differential Equations

TL;DR

This paper develops a new algorithmic framework for defining and computing approximate Lie symmetry algebras of differential equations, addressing stability and reliability issues inherent in symbolic methods.

Contribution

It introduces a local, numerical approach using SVD and geometric involutive forms to identify approximate Lie symmetry algebras near exact ones, with methods to evaluate their reliability.

Findings

The algorithm effectively finds nearby symmetry algebras using SVD.

The approach can distinguish stable from unstable symmetry regions.

It provides a way to analyze the reliability of approximate symmetries.

Abstract

Lie symmetry transformations that leave a differential equation invariant play a fundamental role in science and mathematics. Such Lie symmetry groups uniquely determine their Lie symmetry algebras. Exact differential elimination algorithms have been developed to determine the dimension and structure constants of the Lie symmetry algebra of an exact polynomially nonlinear differential equation. Directly applying these symbolic algorithms to approximate models is prone to instability since these algorithms strongly depend on the orderings of the variables involved. This motivates the need to address questions at the algorithmic foundation of approximate Lie symmetry algebras of differential equations. How do we define approximate Lie symmetry? How do we compute and apply approximate Lie symmetry algebras of differential equations? How reliable are the results? To address such questions, we define approximate Lie symmetry algebras in terms of exact Lie symmetry algebras of a nearby differential equation. Our algorithm for identifying these hidden approximate Lie symmetry algebras uses the SVD to find nearby rank-deficient systems combined with approximate geometric involutive forms of the approximate symmetry-defining systems. Our approach is local and depends on an input base point in the space of independent and dependent variables. In general, our local approach can yield many different nearby Lie symmetry algebras. We also outline a numerical approach to determining the approximate isomorphic of such Lie symmetry algebras as the local base point varies across a grid in the base space. We also provide a method for evaluating the reliability of our results. This enables the base space to be partitioned into regions where different local Lie symmetry algebras are admitted, separated by unstable transition regions.