Towards nonlinear selection of reaction-diffusion patterns in presence of advection: a spatial dynamics approach

Abstract

We present a theoretical study of nonlinear pattern selection mechanisms in a model of bounded reaction-diffusion-advection system. The model describes the activator-inhibitor type dynamics of a membrane reactor characterized by a differential advection and a single diffusion; the latter excludes any finite wave number instability in the absence of advection. The focus is on three types of different behaviors, and the respective sensitivity to boundary and initial conditions: traveling waves, stationary periodic states, and excitable pulses. The theoretical methodology centers on the spatial dynamics approach, i.e. bifurcation theory of nonuniform solutions. These solutions coexist in overlapping parameter regimes, and multiple solutions of each type may be simultaneously stable. The results provide an efficient understanding of the pattern selection mechanisms that operate under realistic boundary conditions, such as Danckwerts type. The applicability of the results to broader reaction-diffusion-advection contexts is also discussed.

Defect-based local error estimators for high-order splitting methods involving three linear operators

“…we aim for a suitable representation for W [1] . Due to (4.1) and Lemma 5 (Appendix A), W [1] satisfies the inhomogeneous Sylvester equation…”

“…Approach We proceed from the local error integral (2.12) involving the defect D = S [1] , and S [1] in turn is represented by an integral which is derived from a differential equation of Sylvester type.…”

“…Relevant examples are for instance given by reaction-diffusion-advection equations. Applications fields where this type of problems appears are, for instance, chemical reactor theory [1], fishery [2], or population dynamics [3]. Furthermore, dimensional splitting into three operators may promise computational advantages in the numerical solution of nonlinear Schrödinger equations.…”

Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection

“…We keep the first kernel asymmetric and we change both amplitudes of the nonlocal couplings σ 1 and σ 2 . and (25). In this case, the third variable v simply follows the variable u adiabatically and we obtain v = uh 2 /f 2 .…”

“…RDA systems have been intensively explored as models for heterogeneous catalysis with in-and outflow of the chemical species [17,23,24]. Their generic features have been subject to detailed mathematical analysis recently [25][26][27]. Particularly interesting are reactiondiffusion systems with differential flows of the chemical species which can cause novel spatial instabilities known as differential flow-induced chemical instabilities (DIFICIs) [28][29][30][31].…”