Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1502.01503

Spectra and stability of spatially periodic pulse patterns: Evans function factorization via Riccati transformation

Björn de Rijk·Arjen Doelman·Jens Rademacher
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the spectral stability analysis of localized patterns to singular perturbed evolution problems, one often encounters that the Evans function respects the scale separation. In such cases the Evans function of the full linear stability problem can be approximated by a product of a slow and a fast reduced Evans function, which correspond to properly scaled slow and fast singular limit problems. This feature has been used in several spectral stability analyses in order to reduce the complexity of the linear stability problem. In these studies the factorization of the Evans function was established via geometric arguments that need to be customized for the specific equations and solutions under consideration. In this paper we develop an alternative factorization method. In this analytic method we use the Riccati transformation and exponential dichotomies to separate slow from fast dynamics. We employ our factorization procedure to study the spectra associated with spatially periodic pulse solutions to a general class of multi-component, singularly perturbed reaction-diffusion equations. Eventually, we obtain expressions of the slow and fast reduced Evans functions, which describe the spectrum in the singular limit. The spectral stability of localized periodic patterns has so far only been investigated in specific models such as the Gierer-Meinhardt equations. Our spectral analysis significantly extends and formalizes these existing results. Moreover, it leads to explicit instability criteria.

Explore related subjects

math.APnlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Björn de Rijk, Arjen Doelman, Jens Rademacher. 2015-02-05. Spectra and stability of spatially periodic pulse patterns: Evans function factorization via Riccati transformation. https://doi.org/10.1137/15m1007264

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Multilayered fluid-structure interactions: existence of weak solutions for time-periodic and initial-value problems

We establish the existence of weak solutions for a class of fully coupled multilayered fluid-structure interaction systems in a three-dimensional spatial setting. The model consists of an incompressible viscous fluid interacting with a thin elastic shell, which is in turn coupled to a three-dimensional elastic solid, yielding a nonstandard $3D/2D/3D$ coupling configuration. The system is driven by time-periodic boundary forcing through Bernoulli-type pressure conditions. For sufficiently small forcing in $L^2$, we prove the existence of at least one time-periodic weak solution. A central analytical difficulty stems from the strong nonlinear coupling across interfaces of different dimensionality and the absence of classical compactness mechanisms. This challenge is overcome through a carefully designed energy framework combined with and new $L^{2}$ compactness arguments adapted to the multilayered geometry. A key structural assumption is the viscoelasticity of the three-dimensional solid, which yields additional diffusion estimates and ensures energy stability. In the purely elastic case, we establish the global-in-time existence of weak solutions to the corresponding initial-value problem, provided that no degeneration (self-contact) of the fluid domain occurs. These results extend existing theories for two-dimensional and reduced-dimensional configurations to a genuinely three-dimensional multilayered setting, providing new analytical insight into complex coupled PDE systems arising in fluid-structure interaction.

math.AP↗
arXiv2026

A linear test approach to global controllability of third- and fifth-order nonlinear dispersive equations

We investigate third- and fifth-order nonlinear dispersive equations of KdV type on the torus and establishes approximate controllability by a fixed four-dimensional control; rather than relying solely on the saturation machinery, the analysis exploits the finite-dimensional controllability of the inviscid Burgers equation linearized around a carefully constructed return trajectory, with the trajectory itself obtained from an observable family. This ``linear test" strategy, yields more information about the structure of the control than the standard approach. In particular, the constructed control is shown to depend continuously on the initial and target states, a property that is by no means automatic in nonlinear control problems, and to decompose as a bounded linear operator applied to the data plus a fixed remainder, with the operator part interestingly independent of the order of dispersion.

math.AP↗
arXiv2026

Global in-time rough large data solution to complex-valued semilinear damped evolution equations

We study the semilinear Cauchy problem for complex-valued damped evolution equations \begin{align*} \partial_t^2u+(-Δ)^σu+(-Δ)^δ\partial_tu=u^p,\ \ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x), \end{align*} with $δ\in[0,σ]$, $σ\in\mathbb{R}_+$ and $p\in\mathbb{N}_+\backslash\{1\}$, where the initial data belong to the rough space $E^α_s$ endowed with the norm \begin{align*} \|f\|_{E^α_s}=\big\|\langleξ\rangle^s\,2^{α|ξ|}\widehat{f}(ξ)\big\|_{L^2}\ \ \mbox{with}\ \ α<0, \ s\in\mathbb{R}. \end{align*} Concerning $(u_0,u_1)\in E^α_{s+\barκ}\times E^α_s$ when $s\geqslant\frac{n}{2}-\frac{2κ+\barκ-2δ}{p-1}-\barκ$ with $κ=\min\{2δ,σ\}$ and $\barκ=\max\{2δ,σ\}$ whose Fourier transforms are supported in a suitable subset of first octant, we prove a global in-time existence result without requiring the smallness of rough initial data.

math.AP↗