Pattern Localization in the Swift–Hohenberg Equation via Slowly Varying Spatial Heterogeneity

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F25%3A00386167" target="_blank" >RIV/68407700:21340/25:00386167 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1137/24M1695245" target="_blank" >https://doi.org/10.1137/24M1695245</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1137/24M1695245" target="_blank" >10.1137/24M1695245</a>

Result language

angličtina

Original language name

Pattern Localization in the Swift–Hohenberg Equation via Slowly Varying Spatial Heterogeneity

Original language description

Theories of localized pattern formation are important to understand a broad range of natural patterns, but they are less well understood than more established mechanisms of domain-filling pattern formation. Here, we extend recent work on pattern localization via slow spatial heterogeneity in reaction-diffusion systems to the Swift--Hohenberg equation. We use a WKB asymptotic approach to show that, in the limit of a large domain and slowly varying heterogeneity, conditions for Turing-type linear instability localize in a simple way, with the spatial variable playing the role of a parameter. For nonlinearities locally corresponding to supercritical bifurcations in the spatially homogeneous system, this analysis asymptotically predicts regions where patterned states are confined, which we confirm numerically. We resolve the inner region of this asymptotic approach, finding excellent agreement with the tails of these confined pattern regions. In the locally subcritical case, however, this theory is insufficient to fully predict such confined regions, and so we propose an approach based on numerical continuation of a local homogeneous analog system. Pattern localization in the heterogeneous system can then be determined based on the Maxwell point of this system, with the spatial variable parameterizing this point. We compare this theory of localization via spatial heterogeneity to localized patterns arising from homoclinic snaking, and suggest a way to distinguish between different localization mechanisms in natural systems based on how these structures decay to the background state (i.e., how their tails decay). We also explore cases where both of these local theories of pattern formation fail to capture the interaction between spatial heterogeneity and underlying pattern-forming mechanisms, suggesting that more work needs to be done to fully disentangle exogenous and intrinsic heterogeneity.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10102 - Applied mathematics

Project

—

Continuities

S - Specificky vyzkum na vysokych skolach

Publication year

2025

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Siam Journal on Applied Dynamical Systems

ISSN

1536-0040

e-ISSN

—

Volume of the periodical

24

Issue of the periodical within the volume

4

Country of publishing house

US - UNITED STATES

Number of pages

44

Pages from-to

2804-2847

UT code for WoS article

001619487000003

EID of the result in the Scopus database

2-s2.0-105025198349