Stability of combustion waves in a simplified gas-solid combustion model in porous media

Traveling waves of an FKPP-type model for self-organized growth

We consider a reaction–diffusion system of densities of two types of particles, introduced by Hannezo et al. (Cell 171(1):242–255.e27, 2017). It is a simple model for a growth process: active, branching particles form the growing boundary layer of an otherwise static tissue, represented by inactive particles. The active particles diffuse, branch and become irreversibly inactive upon collision with a particle of arbitrary type. In absence of active particles, this system is in a steady state, without any a priori restriction on the amount of remaining inactive particles. Thus, while related to the well-studied FKPP-equation, this system features a game-changing continuum of steady state solutions, where each corresponds to a possible outcome of the growth process. However, simulations indicate that this system self-organizes: traveling fronts with fixed shape arise under a wide range of initial data. In the present work, we describe all positive and bounded traveling wave solutions, and obtain necessary and sufficient conditions for their existence. We find a surprisingly simple symmetry in the pairs of steady states which are joined via heteroclinic wave orbits. Our approach is constructive: we first prove the existence of almost constant solutions and then extend our results via a continuity argument along the continuum of limiting points.

Stability of combustion waves in a simplified gas-solid combustion model in porous media

We study the stability of the combustion waves that occur in a simplified model for injection of air into a porous medium that initially contains some solid fuel. We determine the essential spectrum of the linearized system at a travelling wave. For certain waves, we are able to use a weight function to stabilize the essential spectrum. We perform a numerical computation of the Evans function to show that some of these waves have no unstable discrete spectrum. The system is partly parabolic, so the linearized operator is not sectorial, and the weight function decays at one end. We use an extension of a recent result about partly parabolic systems that are stabilized by such weight functions to show nonlinear stability.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Stability of nonlinear waves and patterns and related topics

Periodic and localized travelling waves such as wave trains, pulses, fronts and patterns of more complex structure often occur in natural and experimentally built systems. In mathematics, these objects are realized as solutions of nonlinear partial differential equations. The existence, dynamic properties and bifurcations of those solutions are of interest. In particular, their stability is important for applications, as the waves that are observable are usually stable. When the waves are unstable, further investigation is warranted of the way the instability is exhibited, i.e. the nature of the instability, and also coherent structures that appear as a result of an instability of travelling waves. A variety of analytical, numerical and hybrid techniques are used to study travelling waves and their properties.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.