Principal bifurcations and symmetries in the emergence of reaction-diffusion-advection patterns on finite domains

TL;DR

This paper investigates how boundary conditions and symmetries influence pattern formation in a reaction-diffusion-advection system, revealing new mechanisms for pattern selection and wave propagation on finite domains.

Contribution

It introduces a detailed analysis of pattern selection mechanisms considering boundary effects and symmetry breaking in reaction-diffusion-advection systems.

Findings

Boundary conditions determine the preservation or destruction of translational symmetry.

Stationary periodic states are characterized by specific criteria related to boundary conditions.

Nonlinear wave propagation against flow emerges from oscillatory Hopf instabilities.

Abstract

Pattern formation mechanisms of a reaction-diffusion-advection system, with one diffusivity, differential advection, and (Robin) boundary conditions of Danckwerts type, are being studied. Pattern selection requires mapping the domains of coexistence and stability of propagating or stationary nonuniform solutions, which for the general case of far from instability onsets, is conducted using spatial dynamics and numerical continuations. The selection is determined by the boundary conditions which either preserve or destroy the translational symmetry of the model. Accordingly, we explain the criterion and the properties of stationary periodic states if the system is bounded and show that propagation of nonlinear waves (including solitary) against the advective flow corresponds to coexisting family that emerges nonlinearly from a distinct oscillatory Hopf instability. Consequently, the resulting pattern selection is qualitatively different from the symmetric finite wavenumber Turing or Hopf instabilities.