Linearization Problem for Third-Order ODEs with Four- and Five-Dimensional Lie Symmetry Algebras under Contact Transformations

Ahmad Y. Al-Dweik, F. M. Mahomed, Marwan Aloqeili, Omar A. Abuloha

Pith reviewed 2026-05-11 01:00 UTC · model grok-4.3

classification 🧮 math.GM

keywords third-order ODEsLie symmetry algebrascontact transformationslinearizationCartan equivalence methodinvariant coframesdifferential equations

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The pith

The Cartan equivalence method produces invariant coframes that characterize which third-order ODEs with four- and five-dimensional Lie symmetry algebras can be linearized under contact transformations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies the Cartan equivalence method to classify linearizable third-order ordinary differential equations under contact transformations when the equations admit four- or five-dimensional Lie symmetry algebras. It constructs invariant coframes separately for the rank-one branch associated with four-dimensional algebras and the rank-zero branch associated with five-dimensional algebras. These coframes function as the characterizing objects that distinguish the linearizable cases. The authors also supply an explicit procedure for recovering the contact transformations from the coframe data and demonstrate it with examples. A reader cares because the result supplies a concrete test and a transformation recipe that reduces certain nonlinear third-order equations to linear form.

Core claim

Using the Cartan equivalence method, invariant coframes are constructed for two branches of rank one and zero, which characterize linearizable third-order ODEs under contact transformations with four- and five-dimensional Lie symmetry algebras, respectively. A procedure for deriving the corresponding contact transformations is also presented, along with illustrative examples.

What carries the argument

Invariant coframes constructed via the Cartan equivalence method for the rank-one and rank-zero branches of the symmetry algebra.

Load-bearing premise

The third-order ODEs possess exactly four- or five-dimensional Lie symmetry algebras and the Cartan equivalence method applies directly without additional obstructions to their linearization under contact transformations.

What would settle it

An explicit third-order ODE possessing a four-dimensional Lie symmetry algebra that satisfies the rank-one invariant coframe conditions yet admits no contact transformation to a linear equation.

read the original abstract

Using Cartan equivalence method, invariant coframes are constructed for two branches of rank one and zero, which characterize linearizable third-order ODEs under contact transformations with four- and five-dimensional Lie symmetry algebras, respectively. A procedure for deriving the corresponding contact transformations is also presented, along with illustrative examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper constructs explicit invariant coframes via the Cartan method to characterize linearizable third-order ODEs with 4- and 5-dimensional symmetry algebras under contact transformations.

read the letter

The main point is that the authors use the Cartan equivalence method to build invariant coframes for the rank-one and rank-zero cases. These coframes identify when a third-order ODE can be linearized by a contact transformation given a four- or five-dimensional Lie symmetry algebra. They also describe a procedure to recover the transformation and include examples to show how it works in practice. This gives a concrete way to check linearizability for ODEs that fit those symmetry dimensions. The construction is new in its focus on exactly these dimensions and ranks for third-order equations under contact transformations. Earlier work has handled linearization problems and equivalence methods more broadly, but the targeted coframes and recovery procedure here fill a specific gap. The examples make the abstract setup usable. The approach follows standard Lie symmetry and Cartan techniques without signs of circular reasoning or unsupported steps. The paper sticks to the given symmetry dimensions rather than re-deriving the full classification, which keeps the scope manageable. The math looks grounded in the usual prolonged contact structure on jet space. One soft spot is the narrow focus: it covers only third-order ODEs and contact transformations, so it does not address point transformations or higher-order cases that might be more common in applications. The abstract and description give little detail on how complete the branch analysis is or whether all possible coframe equations are solved explicitly. Without seeing the full calculations, it is hard to judge if minor cases were overlooked. The work is aimed at specialists already using geometric methods for differential equations. Someone classifying or solving third-order ODEs with known symmetries could apply the coframes directly to test linearizability. It deserves a serious referee because the contribution is technically precise and builds on established tools with explicit output that others can check or extend.

Referee Report

0 major / 2 minor

Summary. The manuscript applies the Cartan equivalence method to third-order ODEs admitting four- and five-dimensional Lie symmetry algebras under contact transformations. It constructs invariant coframes for the rank-one branch (characterizing the four-dimensional linearizable case) and the rank-zero branch (characterizing the five-dimensional case), supplies an explicit procedure for recovering the contact transformations from the coframes, and illustrates the results with examples.

Significance. If the coframe constructions and equivalence procedure are correct, the work supplies a concrete, algorithmic route to linearization for a subclass of third-order ODEs with high-dimensional symmetry algebras. This extends the Cartan method to contact equivalence problems in a setting where the symmetry dimension is fixed in advance, and the provision of explicit examples strengthens its utility for classification and solution techniques in the theory of integrable ODEs.

minor comments (2)

[§2] §2 (or the section introducing the prolonged contact structure): the branching into rank-one and rank-zero cases is introduced without an explicit statement of how the rank is computed from the structure equations; adding a short paragraph or diagram would improve readability for readers outside the immediate Cartan-equivalence literature.
[Examples section] The illustrative examples in the final section would benefit from a brief verification that the recovered contact transformation indeed maps the given nonlinear ODE to a linear one, e.g., by substituting back and checking the resulting equation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report accurately captures the manuscript's focus on constructing invariant coframes via the Cartan method for linearizable third-order ODEs admitting 4D and 5D Lie symmetry algebras under contact transformations, along with the recovery procedure and examples.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies the established Cartan equivalence method to the prolonged contact structure on the third-order jet space, constructing invariant coframes for the rank-1 and rank-0 branches that characterize linearizable ODEs possessing 4- and 5-dimensional Lie symmetry algebras under contact transformations. The central claims rest on the standard completeness of the symmetry algebra classification and the explicit solvability of the resulting coframe equations, with a procedure for recovering the transformations presented as a direct application. No step reduces a result to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain; the derivation is self-contained against external mathematical benchmarks in Lie symmetry theory and Cartan geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on established mathematical frameworks in differential geometry and symmetry theory without introducing new free parameters or entities.

axioms (2)

standard math Standard properties of Lie algebras and their actions on differential equations
The paper assumes the existence and properties of 4D and 5D Lie symmetry algebras for the ODEs.
domain assumption Applicability of the Cartan equivalence method to contact transformations for ODEs
The method is used to construct invariant coframes.

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discussion (0)

Reference graph

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