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

Technical Summary: Linearization of Third-Order ODEs with Four- and Five-Dimensional Lie Symmetry Algebras under Contact Transformations

Problem Statement
The paper addresses the linearization problem for scalar third-order ordinary differential equations (ODEs) of the form $u''' = f(x, u, u', u'')$. Specifically, it investigates the conditions under which such equations can be transformed into linear canonical forms via contact transformations (transformations involving the independent variable, dependent variable, and the first derivative). The study focuses on two specific classes of linearizable equations characterized by their Lie symmetry algebras: those admitting a four-dimensional Lie algebra and those admitting a five-dimensional Lie algebra.

While previous works (e.g., Chern, Neut, Petitot, Ibragimov, Meleshko, and Al-Dweik et al.) have established criteria for linearization under point transformations or provided conditions for equivalence to linear forms, this work aims to:

1. Generalize these results to the broader class of contact transformations.
2. Distinguish clearly between the canonical forms admitting four versus five symmetries.
3. Provide an explicit, algorithmic procedure for constructing the contact transformations that reduce a given nonlinear ODE to its linear canonical form, rather than merely stating existence conditions.

Methodology
The authors employ Cartan's equivalence method to solve the problem. The methodology proceeds through the following technical steps:

1. Geometric Setup: The problem is formulated on the second-order jet space $J^2$ with local coordinates $(x, u, p, q)$ where $p=u'$ and $q=u''$. A base coframe is defined, and the equivalence of two ODEs under contact transformations is expressed as a fundamental equivalence condition involving a pullback of this coframe by a nine-dimensional structure group $G$.
2. Reduction Process: The authors perform a systematic reduction of the structure group by normalizing torsion coefficients in the structure equations. This involves:
   • Absorbing torsion terms into the group parameters.
   • Identifying essential torsion coefficients which serve as relative invariants ($I_1, I_2, \dots, I_{15}$).
   • Branching the analysis based on the vanishing or non-vanishing of these invariants.
3. Branching and Classification: The reduction process leads to two primary branches based on the invariant $I_3$ (related to the non-vanishing of a Wünschmann-type invariant).
   • Branch $I_3 \neq 0$: Further reduction leads to a sub-branch defined by the vanishing of a set of invariants ($I_6, I_8, I_{10}, I_{11}, I_{12}$). Within this, a further distinction is made based on the invariant $K = I_9/J_3^2$:
     • Case $K$ is constant: Corresponds to the canonical form $\bar{u}''' = s\bar{u}' + \bar{u}$ (five symmetries).
     • Case $K$ is not constant: Corresponds to the canonical form $\bar{u}''' = \bar{a}_3(\bar{x})\bar{u}$ (four symmetries), subject to additional vanishing conditions on derivatives of $K$.
4. Construction of Transformations: Once the invariant coframes are established for the canonical forms, the authors derive systems of partial differential equations (PDEs) to construct the explicit contact transformations. These systems are linear or Riccati in nature, offering a constructive alternative to solving nonlinear PDE systems directly.

Key Results

• Theorem 3.1 (Five Symmetries): A third-order ODE is equivalent to the canonical form $\bar{u}''' = s\bar{u}' + \bar{u}$ (where $s$ is a constant) via contact transformations if and only if the relative invariants $I_6, I_8, I_{10}, I_{11}, I_{12}$ vanish identically, and the invariant $K$ is constant (specifically $K=s$).
• Theorem 3.2 (Four Symmetries): A third-order ODE is equivalent to the canonical form $\bar{u}''' = \bar{a}_3(\bar{x})\bar{u}$ (where $\bar{a}_3$ is non-constant) via contact transformations if and only if the relative invariants $I_6, I_8, I_{10}, I_{11}, I_{12}$ vanish, and the additional invariants $I_{13}, I_{14}, I_{15}$ (involving derivatives of $K$) vanish, with the condition that $K$ is not constant ($\hat{D}_x K \neq 0$).
• Constructive Procedures (Propositions 4.1 and 4.2): The paper provides explicit algorithms to find the contact transformations $(\bar{x}, \bar{u}, \bar{p}) = (\phi, \psi, \chi)$:
  • For the five-symmetry case, the transformation is found by solving a system of first-order linear PDEs for auxiliary functions $a_1, \phi, \eta, \chi, \psi$.
  • For the four-symmetry case, the procedure involves an additional step of solving a Riccati-type equation for an auxiliary function $H$, followed by linear systems for $b, a_1, \phi, \eta, \chi, \psi$.
• Illustrative Examples:
  • The equation $u''' = \alpha (u'')^2 / u'$ is shown to be linearizable to the five-symmetry form under contact transformations, but not under point transformations (demonstrating the necessity of the contact framework).
  • The equation $u''' = -x(u')^4(u'')^3 + u(u')^3(u'')^3$ is shown to be linearizable to the four-symmetry form under contact transformations, again failing the criteria for point transformation linearization.

Significance and Claims
The authors claim that their work provides a novel framework for implementing Cartan's method that effectively manages the complexity and expression growth inherent in the equivalence problem. The primary contributions are:

1. Generalization: Extending linearization criteria from point transformations to the more general contact transformations, thereby capturing a wider class of linearizable ODEs.
2. Distinction: Clearly separating the canonical forms based on the dimension of their Lie symmetry algebras (four vs. five), which were previously treated less distinctly in the context of contact transformations.
3. Algorithmic Construction: Unlike previous approaches that often required solving nonlinear systems of PDEs or did not address the construction of transformations, this paper offers a constructive, algorithmic procedure. By reducing the problem to systems of linear or Riccati PDEs derived from invariant coframes, the method provides a practical pathway for computing the explicit transformations.
4. Geometric Clarity: The use of invariant coframes provides a complete geometric characterization of the linearizable equations, defining the necessary and sufficient conditions for equivalence.

The paper concludes that this framework offers an effective computational pathway for constructing contact transformations, extending the utility of Cartan's equivalence method to the linearization of third-order ODEs in a broader context than previously available.