A Geometric Approach to Stationary Defect Solutions in One Space Dimension

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 $\mathbb{R}$. This problem corresponds to analyzing a discontinuous and non-autonomous $n$-dimensional system, $\scriptsize\dot{u}=\left\{ \begin{array}{ll} f(u),& t\leq0,\\ f(u)+\varepsilon g(u),& t>0, \end{array}\right.$ under the assumption that the unperturbed system, i.e., the $\varepsilon \to 0$ limit system, possesses a heteroclinic orbit $\Gamma$ that connects two hyperbolic equilibrium points (plus several additional nondegeneracy conditions). The unperturbed orbit $\Gamma$ represents a localized structure in the PDE setting. We define the (pinned) defect solution $\Gamma_\varepsilon$ as a heteroclinic solution to the perturbed system such that $\lim_{\varepsilon \to 0} \Gamma_\varepsilon = \Gamma$ (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 $\Gamma$ 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 $n \geq 4$. 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 $(n=6)$.