Rotation of a scroll wave reveals the metric tensor for the associated geodesic filament

Area of Science:

• Mathematical modeling
• Theoretical physics
• Chemical kinetics

Background:

• Reaction-diffusion systems can exhibit complex spatiotemporal patterns, such as rotating scroll waves.
• The filament of these scroll waves is a key topological feature, often exhibiting complex dynamics.
• Previous work suggested a geometric interpretation of scroll wave filaments but lacked a rigorous derivation.

Purpose of the Study:

• To deductively derive the relationship between the scroll wave filament and its geometric properties.
• To determine the metric tensor associated with the filament's geodesic path in three-space.
• To establish a direct link between the diffusivity matrix and the emergent geometry.

Main Methods:

• Starting from the fundamental reaction-diffusion equations.
• Employing a sequence of rigorous mathematical steps to analyze the filament's behavior.
• Deriving the metric tensor in terms of the diffusivity matrix D and its determinant (detD).

Main Results:

• The filament of a stationary scroll wave in a reaction-diffusion medium is shown to be a geodesic curve.
• The metric tensor g associated with this geodesic is derived as g = (detD)D⁻¹.
• The factor (detD) is explicitly derived and shown not to be necessarily constant in space.

Conclusions:

• This study provides the first deductive derivation of the geometric properties of scroll wave filaments.
• The findings establish a direct mathematical link between the medium's diffusivity and the filament's geometry.
• The results offer a deeper understanding of pattern formation in reaction-diffusion systems.