Stability of patterns on thin curved surfaces

Area of Science:

• Mathematical modeling
• Chemical kinetics
• Surface physics

Background:

• Reaction-diffusion (R-D) equations model complex spatio-temporal patterns.
• Analyzing R-D systems on curved surfaces is crucial for understanding various phenomena.
• Previous studies often simplified surface geometry or thickness effects.

Purpose of the Study:

• To derive and analyze effective reaction-diffusion equations for thin curved surfaces.
• To investigate the influence of surface thickness (ε) and curvature on R-D systems.
• To study pattern formation and stability in specific geometries like spheres and cylinders.

Main Methods:

• Derivation of effective R-D equations up to O(ε^{2}) for thin surfaces.
• Linear stability analysis applied to the Schnakenberg model.
• Investigation of steady-state dependence on surface thickness for spherical and cylindrical geometries.

Main Results:

• Effective R-D equations exhibit space-dependent reaction kinetics on curved surfaces.
• Surface thickness significantly affects the stability of reaction-diffusion systems.
• A transition in surface thickness can alter the stability of modes, stabilizing or destabilizing them.

Conclusions:

• The interplay between surface thickness and curvature is critical for pattern rearrangement on thin curved surfaces.
• Effective R-D equations provide a valuable tool for studying chemical kinetics on complex geometries.
• This work highlights the importance of geometric factors in reaction-diffusion dynamics.