Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system

The initiation and propagation of reaction-diffusion travellingwaves in two regions coupled together by the linear diffusiveinterchange of the autocatalytic species is considered via aninitial-value problem in which amounts of the autocatalyst areintroduced locally into otherwise uniform concentrations ofthe other species. The reaction in one region is given by quadraticautocatalysis, while the reaction in the other is given by quadraticautocatalysis together with the linear decay of the autocatalyst.A priori bounds for the initial-value problem are obtained first.These, together with the solution valid for small inputs ofthe autocatalyst, enable conditions to be derived under whichtravelling waves can be initiated giving a wave for all withk<2, or if k<(2–1)/(–1), where k and aredimensionless groups corresponding to the rate of chemical decayof the autocatalyst and to the strength of coupling respectively.The global asymptotic stability of the unreacted state is thendiscussed. A solution valid for strong coupling between thetwo regions is then derived. The equations governing the permanent-formtravelling waves are treated in some detail, general propertiesof their solution and a solution valid for weak coupling beingderived. Finally, the large-time solution of the initial-valueproblem is considered. This shows that, when travelling wavesare initiated, they travel with their minimum possible speed: