[Paper Review] Symmetries of Differential Equations via Cartan's Method of Equivalence

Oleg I. Morozov|arXiv (Cornell University)|May 16, 2001

Nonlinear Waves and Solitons10 references18 citations

TL;DR

This paper presents an algorithmic method for computing invariant 1-forms and structure equations of symmetry pseudo-groups of differential equations using Cartan’s method of equivalence and the moving coframe technique. By avoiding the need to compute, analyze, or integrate infinitesimal defining systems, the approach relies solely on differentiation and linear algebra, enabling direct derivation of symmetry structure. The method is demonstrated on a second-order PDE system, confirming a symmetry pseudo-group depending on two arbitrary functions of one variable, consistent with Liouville’s result.

ABSTRACT

We formulate a method of computing invariant 1-forms and structure equations of symmetry pseudo-groups of differential equations based on Cartan's method of equivalence and the moving coframe method introduced by Fels and Olver. Our apparoach does not require a preliminary computation of infinitesimal defining systems, their analysis and integration, and uses differentiation and linear algebra operations only. Examples of its applications are given.

Motivation & Objective

• To develop a direct, algorithmic method for computing symmetry pseudo-group invariants without integrating infinitesimal defining systems.
• To apply Cartan’s method of equivalence and the moving coframe method to differential equations in jet space.
• To derive structure equations and invariant 1-forms of symmetry pseudo-groups using only differentiation and linear algebra operations.
• To demonstrate the method on a second-order PDE system, recovering known symmetry dimension and structure.

Proposed method

• The method begins by lifting the system to first-order jet space, treating it as a subbundle in the 1-jet bundle of a fibered manifold.
• It applies Cartan’s method of equivalence to compute a Cartan moving coframe for the contact transformation pseudo-group on the 1-jet bundle.
• The invariant 1-forms of the full contact pseudo-group are restricted to the subbundle defined by the differential equation system.
• Normalization procedures are applied to the resulting linear dependence conditions among the restricted forms to extract symmetry invariants.
• The structure equations of the symmetry pseudo-group are derived by prolonging the coframe and computing exterior derivatives, leading to a system of Cartan structure equations.
• The process is validated by checking Cartan’s test for involution and identifying the number of arbitrary functions in the solution, indicating symmetry dimension.

Experimental results

Research questions

• RQ1 Can symmetry pseudo-groups of differential equations be computed without solving or integrating infinitesimal defining systems?
• RQ2 How can Cartan’s method of equivalence and the moving coframe method be systematically applied to compute invariant 1-forms of symmetry pseudo-groups?
• RQ3 What is the structure of the symmetry pseudo-group for a given system of differential equations, particularly in terms of its Cartan structure equations?
• RQ4 How can the number of arbitrary functions in the symmetry pseudo-group be determined directly from the structure equations?
• RQ5 Does the method yield results consistent with known results, such as Liouville’s classification of the symmetry group for the system $u_t = v$, $v_x = e^u$?

Key findings

• The method successfully computes the structure equations of the symmetry pseudo-group for the system $u_t = v$, $v_x = e^u$ using only differentiation and linear algebra, bypassing integration of defining systems.
• The resulting coframe is found to be involutive after prolongation, satisfying Cartan’s test with reduced characters $s'_1 = 2$, $s'_2 = ext{...} = s'_9 = 0$, indicating two arbitrary functions of one variable.
• The symmetry pseudo-group is confirmed to depend on two arbitrary functions of one variable, consistent with Liouville’s classical result.
• The structure equations (23) and (25) are torsion-free and closed under the exterior derivative, confirming the integrability of the symmetry structure.
• The procedure of absorption and normalization effectively reduces the system to a canonical form, enabling direct extraction of symmetry invariants without prior analysis of the infinitesimal system.
• The method is algorithmic in principle, though computationally complex, and suggests future optimization via canonical contact forms on higher-order jet bundles.