[Paper Review] Group Classification of Generalised Eikonal Equations

Roman O. Popovych, Irina Yehorchenko|ArXiv.org|Dec 22, 2001

Nonlinear Waves and Solitons3 citations

TL;DR

This paper presents a complete group classification of generalized eikonal equations of the form $ u_a u_a = F(t, u, u_t) $ using a combined method involving equivalence groups, symmetry algebra structure analysis, and classification of determining equations. It identifies all non-equivalent equations with extended symmetry algebras, including new nonlinear first-order PDEs with wide symmetry groups, and proves non-equivalence via algebraic invariants such as dimension and isomorphism types of symmetry algebras.

ABSTRACT

A new approach to the problem of group classification is applied to the class of first-order non-linear equations of the form $u_a u_a=F(t,u,u_t)$. It allowed complete solution of the group classification problem for a class of equations for functions depending on multiple independent variables, where highest derivatives enter nonlinearly. Equivalence groups of the class under consideration and algebraic properties of the symmetry algebra are studied. The class of equations considered presents generalisation of the eikonal and Hamilton-Jacobi equations. The paper contains the list of all non-equivalent equations from this class with symmetry extensions, and proofs of such non- equivalence. New first order non-linear equations possessing wide symmetry groups were constructed.

Motivation & Objective

• To solve the group classification problem for a class of first-order nonlinear PDEs with nonlinearity in derivatives, generalizing the eikonal and Hamilton-Jacobi equations.
• To identify all equations within this class that admit symmetry extensions beyond the general case.
• To establish a complete list of non-equivalent equations with extended symmetry algebras, using algebraic and transformational criteria.
• To demonstrate the effectiveness of a combined method—integrating equivalence groups, symmetry algebra structure, and solving determining equations—for solving complex group classification problems in PDEs.

Proposed method

• Application of a combined method integrating equivalence group analysis, structure analysis of symmetry algebras, and solving determining equations for Lie symmetries.
• Use of equivalence transformations to reduce the number of cases to be analyzed, including transformations that simplify the arbitrary element $ A(u) $.
• Classification of functions $ A(u) $ satisfying specific differential conditions that lead to symmetry extensions, such as $ (A_u/A)_u = 0 $ or $ A_u/A = \nu A + \mu $.
• Proof of non-equivalence of resulting equations via non-isomorphism of their maximal symmetry algebras, particularly by comparing dimensions and algebraic structure.
• Systematic reduction of cases using equivalence transformations, such as $ \widetilde{t} = t, \widetilde{u} = |u + \mu|^{1 - \nu/2} $, to unify similar cases.
• Leveraging known results from Lie group theory and symmetry classification, including the use of canonical forms for special functions like $ \text{sech}^2 u $, $ \text{csch}^2 u $, and trigonometric functions.

Experimental results

Research questions

• RQ1 Which equations in the class $ u_a u_a = F(t, u, u_t) $ admit symmetry extensions beyond the general case?
• RQ2 What are the complete sets of non-equivalent equations with extended symmetry algebras in this class?
• RQ3 How can equivalence transformations be used to reduce the number of cases in group classification of PDEs with multi-variable arbitrary functions?
• RQ4 What algebraic properties (e.g., dimension, isomorphism type) distinguish the symmetry algebras of non-equivalent equations in this class?
• RQ5 Can the combined method of symmetry algebra structure analysis and equivalence group techniques solve group classification problems that are intractable with direct integration of determining equations?

Key findings

• The paper provides a complete classification of all non-equivalent equations from the class $ u_a u_a = F(t, u, u_t) $ that possess symmetry extensions, with the list fully enumerated in Table 1.
• The maximal symmetry algebras of the classified equations are non-isomorphic, with dimensions ranging from 3 to 6, and their structure determines non-equivalence.
• New first-order nonlinear PDEs with wide symmetry groups were constructed, including equations with $ A(u) = \varepsilon_2 / \cosh^2 u $, $ \varepsilon_2 / \sinh^2 u $, and $ \varepsilon_2 / \cos^2 u $, under specific conditions on $ \varepsilon_1, \varepsilon_2 $.
• Equations with $ A(u) $ satisfying $ (A_u/A)_u = 0 $, $ A_u \neq 0 $, yield a ninth case of symmetry extension, and are non-equivalent to other cases.
• The case $ A_u/A = \nu A + \mu $ leads to three canonical forms: $ \varepsilon_2 / \cosh^2 u $, $ \varepsilon_2 / \sinh^2 u $, and $ \varepsilon_2 / \cos^2 u $, with $ \varepsilon_0 \widetilde{\varepsilon}_0 = \varepsilon_1 \widetilde{\varepsilon}_1 = \varepsilon_2 \widetilde{\varepsilon}_2 $ as the condition for equivalence.
• For all other functions $ A(u) $, no symmetry extension occurs, confirming the completeness of the classification.