Corner defects in almost planar interface propagation

Abstract

We study existence and stability of interfaces in reaction-diffusion systems which are asymptotically planar. The problem of existence of corners is reduced to an ordinary differential equation that can be viewed as the travelling-wave equation to a viscous conservation law or variants of the Kuramoto-Sivashinsky equation. The corner typically, but not always, points in the direction opposite to the direction of propagation. For the existence and stability problem, we rely on a spatial dynamics formulation with an appropriate equivariant parameterization for relative equilibria.  2005 Elsevier SAS. All rights reserved. RésuméNous étudions l'existence et la stabilité des interfaces asymptotiquement planes dans des systèmes de réaction-diffusion. Le problème de l'existence des défauts est réduit à l'étude d'une équation différentielle ordinaire qui est, selon le cas, approchée par l'équation stationnaire d'une loi de conservation scalaire ou d'une variante de l'équation de Kuramoto-Sivashinsky. Typiquement, les défauts pointent dans la direction opposée à la direction de propagation. Pour l'analyse des problèmes d'existence et de stabilité, nous utilisons une formulation de type dynamique spatiale combinée avec une paramétrisation adéquate d'équilibres relatifs.Hypothesis 2.1 (Existence). We assume that there exists c * > 0 and asymptotic states q ± such that there exists an x-independent planar travelling-wave solution q * (y) of (2.2)(2.5)We emphasize that we allow for the possibility of pulses, q + = q − . The second assumption in this section is concerned with stability of the above travelling wave solution. Therefore, consider the linearized operator( 2.6) Notice that under suitable decay assumptions, q * belongs to the kernel of L * due to the translation invariance in y.Hypothesis 2.2 (Zero-stability). We assume that L * − λ id is invertible for all λ < 0 and that λ = 0 is an isolated eigenvalue with algebraic multiplicity one.Although this might not seem obvious, Hypothesis 2.2 is intimately related to stability properties of the travelling wave q * (·). Consider the linearized operatorand its Fourier conjugates(2.8) Hypothesis 2.3 (Transverse asymptotic stability). Assume that the travelling wave is asymptotically stable in one space dimension, that is, the essential spectrum of M 0 is strictly contained in the left complex half plane and zero is the only eigenvalue in the closed right half plane, with algebraic multiplicity one. Furthermore, assume that the spectra of M k , for k = 0 are strictly contained in the left half plane and that the unique eigenvalue λ d (k), k ∼ 0 with λ d (0) = λ d (0) = 0 satisfies λ d (0) < 0; see Fig. 1. Remark 2.4.(i) This hypothesis and, in particular, the quadratic tangency of the dispersion relation λ d (0) < 0 implies asymptotic stability of the travelling wave with respect to perturbations that are sufficiently localized in the transverse, x-direction [28,35,36]. Fig. 1. To the left the spectrum of M 0 and to the right the critical spectra of the M k parameterized by k.

Interfaces between rolls in the Swift-Hohenberg equation

“…In Section 2, we simplify the existence problem using an equivariant spatial center-manifold reduction, inspired by the general approach to almost planar waves developed in [9] for reaction-diffusion systems; see also [10,11]. In Section 3, we then analyze the reduced system of ordinary differential equations.…”

“…Following the general approach in [9] we start by rewriting this equation as a first order system in which x is the time-like variable. This is easily achieved by taking u = (u,…”

“…Following the general approach to almost planar waves in [9,11], we set u = U ξ * + η 1 e ξ 1 + η 2 e ξ 2 + η 3 e ξ 3 + w ξ , with P ξ w ξ = P w = 0, (2.5)…”

Almost Planar Waves in Anisotropic Media

“…All these nonplanar fronts share the property that the asymptotic planar part of the front propagates towards the middle, center part of the interface. The results in [8] show that, at least for small angles ε, the existence and asymptotic stability of these interior corners is a universal feature in isotropic reaction-diffusion systems, u ∈ R N , that exhibit stable planar fronts. We point out that unstable, or pulsating, fronts may lead to different types of interfaces such as exterior corners, steps, or holes; see Figure 2.1.…”

“…Planar fronts connecting the stable homogeneous equilibria u = 0 and u = 1 may propagate in either direction n = (sin ϕ, cos ϕ), depending upon the initial condition. In a recent article [8], we showed that almost planar fronts exist for a = 1/2: the front position, traced for example by the level set u −1 (1/2), is located on a curve that approaches two straight lines y = ε|x| for ε > 0 small, and the interface propagates in the direction of increasing y.…”