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

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

—

Návaznosti

S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2025

Kód důvěrnosti údajů

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

Název periodika

Siam Journal on Applied Dynamical Systems

ISSN

1536-0040

e-ISSN

—

Svazek periodika

24

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

44

Strana od-do

2804-2847

Kód UT WoS článku

001619487000003

EID výsledku v databázi Scopus

2-s2.0-105025198349