Shaping wave patterns in reaction-diffusion systems

TL;DR

This paper introduces an analytical method to control the shape of traveling waves in reaction-diffusion systems, enabling precise manipulation of interfaces and pulses without detailed knowledge of reaction kinetics.

Contribution

The authors develop a novel control approach using nonlinear phase diffusion and eikonal equations, applicable even with unknown reaction kinetics, to shape wave patterns in two-dimensional reaction-diffusion systems.

Findings

Control signals can enforce desired wave shapes perpendicular to propagation.

Wave profile along propagation remains nearly unaffected by control.

Method applicable with measured wave profiles and known diffusion coefficients.

Abstract

We present a method to control the two-dimensional shape of traveling wave solutions to reaction-diffusion systems, as e.g. interfaces and excitation pulses. Control signals that realize a pre-given wave shape are determined analytically from nonlinear evolution equation for isoconcentration lines as the perturbed nonlinear phase diffusion equation or the perturbed linear eikonal equation. While the control enforces a desired wave shape perpendicular to the local propagation direction, the wave profile along the propagation direction itself remains almost unaffected. Provided that the one-dimensional wave profile and its propagation velocity can be measured experimentally, and the diffusion coefficients of the reacting species are given, the new approach can be applied even if the underlying nonlinear reaction kinetics are unknown.