Constructing a partner diffusion matrix from imposed reaction functions

Develop a general constructive procedure to determine the nonlinear diffusivity matrix D(θ) and associated flux potentials μ(θ) for a specified reaction vector R(θ) in the coupled reaction-diffusion system θ_t = ∇·[D(θ) ∇θ] + R(θ), such that the system admits the nonclassical symmetry with invariant surface condition μ_t = A μ and reduces to the linear Helmholtz system ∇^2F + M F = 0 under the constraint D(θ)^{-1} A μ(θ) = - M μ(θ) + R(θ), where A and M are constant commuting matrices.

Background

The paper develops a nonclassical symmetry reduction for coupled reaction-diffusion systems that transforms certain nonlinear models into linear Helmholtz systems, enabling explicit solutions with nontrivial spatiotemporal structure. This reduction hinges on a compatibility constraint linking the nonlinear diffusivity matrix D(θ), the flux potentials μ(θ), and the reaction vector R(θ). When D(θ) is specified, the authors show how to construct compatible R(θ) straightforwardly to satisfy the constraint.

In scalar cases, prior work provides techniques (including recursive constructions) to derive diffusivity from reaction terms, albeit via difficult Abel-type equations. However, for coupled systems, the authors explicitly state that they lack a simple method to construct D(θ) directly from given R(θ). Solving this inverse problem would broaden the applicability of the nonclassical reduction framework and facilitate modeling across diverse physical and biological settings.