Geometric Integration of Differential Equations

This map of content organizes the framework of ${\mathcal{C}}^{\mathrm{\infty }}$-structures, a geometric approach to integrating ordinary differential equations (ODEs) and involutive distributions. This theory generalizes classical Lie symmetry methods and solvable structures, providing tools to integrate systems that lack sufficient Lie point symmetries.

1. Core Concepts

Fundamental definitions that underpin the theory.

• distribution: The geometric object representing a system of ODEs or PDEs.

• symmetry of a distribution: The classical notion of a vector field that preserves the distribution structure (maps integral manifolds to integral manifolds).

• cinf-symmetry of distribution: A generalization where the vector field preserves the distribution only "directionally" ($[X,\mathcal{Z}]\subseteq \mathcal{Z}\oplus ⟨X⟩$).

• symmetrizing factor: A function that corrects a ${\mathcal{C}}^{\mathrm{\infty }}$-symmetry to become a true symmetry.

• cinf-structure: An ordered set of vector fields (and one distribution) that allows for sequential integration.

• solvable structure: The precursor to ${\mathcal{C}}^{\mathrm{\infty }}$-structures; requires true symmetries.

• generalized Cinf-symmetry ODE: The specific application of these concepts to scalar ODEs.

2. Key Theorems & Properties

Mathematical guarantees and structural results.

• integration by Pfaffian equations: The main theorem stating that a ${\mathcal{C}}^{\mathrm{\infty }}$-structure guarantees integrability via quadratures (Pfaffian equations).

• symmetrizing factor lemma: Guarantee that every ${\mathcal{C}}^{\mathrm{\infty }}$-symmetry can be locally converted into a symmetry.

• cinf-structure-based method: How an m-th order ODE is reduced to m first-order Pfaffian equations.

• integrating factors vs symmetrizing factors: The duality between symmetrizing factors (vector fields) and integrating factors (forms).

• algorithm for determining cinf-structures: Steps to find a ${\mathcal{C}}^{\mathrm{\infty }}$-structure for a given ODE.

3. Applications & Examples

Specific systems analyzed in the literature.

• integration of Lotka-Volterra model: Application to biological population dynamics.

• Korteweg-de Vries equation: Traveling wave solutions found via this method.

• Zakharov-Kuznetsov equation: Application to plasma physics.

• Rikitake system: Analysis of the chaotic dynamo system.

• equations lacking Lie symmetries: Collection of examples where this method succeeds while Lie theory fails.