Instability of Planar Interfaces in Reaction-Diffusion Systems

Instability of planar front solutions to reaction-diffusion systems in two space dimensions is studied. Let ε denote the width of interface. Then the planar front solution—or a solution having an internal transition layer which is flat—loses its stability when the length of interface along the tangential direction exceeds O(ε ^1 /2 ). The wavelength of the fastest growth is of O(ε ^1 /3 ) which is inherent in the system and determined by the nonlinearity and diffusion coefficients. Complete asymptotic characterization of these quantities as ε → 0 is given by the analysis of what is called the singular dispersion relation derived from the linearized eigenvalue problem. The numerical computations also confirm that the theoretically predicted fastest growth wavy pattern actually arises from a randomly perturbed planar front.