On Exact and Approximate Symmetries of Algebraic and Ordinary Differential Equations with a Small Parameter

TL;DR

This paper investigates the behavior of exact and approximate symmetries in algebraic and differential equations with small perturbations, revealing how symmetries evolve or become unstable, and providing methods to compute higher-order approximate symmetries for solutions.

Contribution

It establishes a systematic approach to identify and compute approximate symmetries in perturbed equations, especially highlighting differences between first-order and higher-order ODEs.

Findings

Every point symmetry of unperturbed algebraic and first-order ODEs has an approximate counterpart in the perturbed equations.

Some symmetries of higher-order ODEs are unstable and do not appear as approximate symmetries, but relate to higher-order approximate symmetries.

The paper demonstrates the use of higher-order approximate symmetries and integrating factors to derive approximate solutions, including a nonlinear Boussinesq equation.

Abstract

The framework of Baikov-Gazizov-Ibragimov approximate symmetries has proven useful for many examples where a small perturbation of an ordinary differential equation (ODE) destroys its local symmetry group. For the perturbed model, some of the local symmetries of the unperturbed equation may (or may not) re-appear as approximate symmetries, and new approximate symmetries can appear. Approximate symmetries are useful as a tool for the construction of approximate solutions. We show that for algebraic and first-order differential equations, to every point symmetry of the unperturbed equation, there corresponds an approximate point symmetry of the perturbed equation. For second and higher-order ODEs, this is not the case: some point symmetries of the original ODE may be unstable, that is, they do not arise in the approximate point symmetry classification of the perturbed ODE. We show that such unstable point symmetries correspond to higher-order approximate symmetries of the perturbed ODE, and can be systematically computed. Two detailed examples, including a fourth-order nonlinear Boussinesq equation, are presented. Examples of the use of higher-order approximate symmetries and approximate integrating factors to obtain approximate solutions of higher-order ODEs are provided.