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Notes on Lie symmetry group methods for differential equations

F. Güngör

TL;DR

This paper surveys Lie symmetry group methods for differential equations, unifying flows, prolongations, invariants, and reduction into a coherent framework. It develops the infinitesimal invariance criterion, the construction of differential invariants, and invariant differentiation to build higher order invariants and invariant equations. It then explains how to reduce ODEs by symmetry, perform group classification for equations with arbitrary terms, and construct group-invariant solutions via optimal subalgebras. The approach is inherently algorithmic and applicable to a broad class of equations, with implications for conservation laws, discretizations, and higher order symmetries.

Abstract

Fundamentals on Lie group methods and applications to differential equations are surveyed. Many examples are included to elucidate their extensive applicability for analytically solving both ordinary and partial differential equations.

Notes on Lie symmetry group methods for differential equations

TL;DR

This paper surveys Lie symmetry group methods for differential equations, unifying flows, prolongations, invariants, and reduction into a coherent framework. It develops the infinitesimal invariance criterion, the construction of differential invariants, and invariant differentiation to build higher order invariants and invariant equations. It then explains how to reduce ODEs by symmetry, perform group classification for equations with arbitrary terms, and construct group-invariant solutions via optimal subalgebras. The approach is inherently algorithmic and applicable to a broad class of equations, with implications for conservation laws, discretizations, and higher order symmetries.

Abstract

Fundamentals on Lie group methods and applications to differential equations are surveyed. Many examples are included to elucidate their extensive applicability for analytically solving both ordinary and partial differential equations.

Paper Structure

This paper contains 10 sections, 9 theorems, 613 equations, 2 tables.

Table of Contents

  1. Vector fields and integral curves
  2. Differential equations and their symmetry group
  3. Differential invariants
  4. Invariant differentiation
  5. Reduction of order for ordinary differential equations
  6. Group classification problem
  7. Group-invariant solutions
  8. Linearization by symmetry structure
  9. Lie's Linearization Theorem
  10. Concluding remarks

Key Result

Theorem 1.1

If $x_0$ is a regular point of $\mathbf{v}$, then there exist local rectifying (or straightening out) coordinates $y=(y_1,\ldots,y_n)$ near $x_0$ such that $y=\partial_{y_1}$ generates the translational flow $\exp{(t \mathbf{v})}y=(y_1+t,y_2,\ldots,y_n)$.

Theorems & Definitions (50)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Theorem 2.4
  • Example 2.5
  • Remark 2.6
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • ...and 40 more