A Geometric Approach to Stationary Defect Solutions in One Space Dimension

Abstract

Abstract. In this manuscript, we consider the impact of a small jump-type spatial heterogeneity on the existence of stationary localized patterns in a system of partial differential equations in one spatial dimension, i.e., defined on R. This problem corresponds to analyzing a discontinuous and non-under the assumption that the unperturbed system, i.e., the ε → 0 limit system, possesses a heteroclinic orbit Γ that connects two hyperbolic equilibrium points (plus several additional nondegeneracy conditions). The unperturbed orbit Γ represents a localized structure in the PDE setting. We define the (pinned) defect solution Γε as a heteroclinic solution to the perturbed system such that limε→0 Γε = Γ (as graphs). We distinguish between three types of defect solutions: trivial, local, and global defect solutions. The main goal of this manuscript is to develop a comprehensive and asymptotically explicit theory of the existence of local defect solutions. We find that both the dimension of the problem as well as the nature of the linearized system near the endpoints of the heteroclinic orbit Γ have a remarkably rich impact on the existence of these local defect solutions. We first introduce the various concepts in the setting of planar systems (n = 2) and-for reasons of transparency of presentation-consider the three-dimensional problem in full detail. Then, we generalize our results to the n-dimensional problem, with special interest for the additional phenomena introduced by having n ≥ 4. We complement the general approach by working out two explicit examples in full detail: (i) the existence of pinned local defect kink solutions in a heterogeneous Fisher-Kolmogorov equation (n = 4) and (ii) the existence of pinned local defect front and pulse solutions in a heterogeneous generalized FitzHugh-Nagumo system (n = 6).

Pattern localization to a domain edge

“…Pattern forming systems with a step-like (also called "jump-type") heterogeneity have a rich history in the mathematical literature (see e.g. [29][30][31][32][33][34][35][36][37][38]) where, they have been studied in the context of front-pinning [36], pulse localization [37], and wavenumber selection [38], to name a few recent examples. These studies predominantly focussed on excitable media and the models studied are not mass-conserving.…”

Wavenumber selection via spatial parameter jump

“…As mentioned above, we have the ε-dependent family of periodic orbits P(ε). Since (10)- (11) are equivariant under the Gauge symmetry R φ , we may assume that the periodic orbits have the form a * (κ; ε) = s(κ; ε)e iκy b * (κ; ε) = (p(κ; ε) + iq(κ; ε)) e iκy for real functions s, p, q :…”

“…The analysis in [11] for the nonlinear wave and Schrdinger equations is global, but relies on rather explicit knowledge of the phase portraits in spatial dynamics. In contrast, the analysis of reaction-diffusion spikes in [11] is global but perturbative in nature. Previous work by Scheel, Goh, and others investigates existence of non-stationary striped wave-trains in the case of a moving parameter jump [16] and also in slowly-growing domains [15], which may be seen as an analogue of a slowly moving jump.…”

Pulse Solutions for an Extended Klausmeier Model with Spatially Varying Coefficients

“…However, these studies are usually limited to models with constant coefficients. Some research has focused on the introduction of localized spatial inhomogeneities [44,34,35,48,49,21]; also (often formal) research has been done on reaction-diffusion equations with (less restricted) spatially varying coefficients [9,8,2,47,46,7]. In this article, we aim to expand the knowledge of such systems, by studying a reaction-diffusion system with fairly generic spatially varying coefficients rigorously; motivated by its use in ecology (see Remark 2), we consider the following extended Klausmeier model with spatially varying coefficients [27,4]:…”