Rigid transition to the stationary structure and imposed convective instability in a reaction-diffusion system with flow (Q1888053)

The work addresses competition of different instabilities and patterns induced by them in reaction-diffusion systems with convective terms, a paradigm example being the Brusselator model which includes the convection part: \[ \begin{aligned} U_t+vU_x & =A-(B+1)V+ U^2V +\sigma U_{xx},\\ V_t+vV_x & =BU- U^2V +V_{xx}.\end{aligned} \] These equations are set for real fields \(U\) and \(V\) and positive control parameters \(v\) (the convection rate), \(A\), \(B\), and \(\sigma\). Recently (in 1997), a new pattern-forming mechanism was proposed, and soon thereafter investigated in detail for systems of this type: a so-called flow and diffusion distributed structure (FDS) appears if a permanent perturbation is applied at the flow's inlet. This mechanism was considered before in a parameter region where the system gives rise to a convective Hopf instability. The present work is focused on the case when the Hopf instability has an absolute character. As a result, the FDS and Hopf-instability-induced patterns compete. By means of direct numerical simulations, and with the help of some analytical considerations (an amplitude equation for the pattern competition is derived), it is concluded that the Hopf-instability-induced structure dominates if the flow rate is large and/or the inlet perturbation, that gives rise to the FDS, is small. In the opposite case, the space is occupied by the FDS.