Beyond-all-order asymptotics for homoclinic snaking of localised patterns in reaction-transport systems

[2501.02698] Beyond-all-order asymptotics for homoclinic snaking of localised patterns in reaction-transport systems Skip to main content Search Submit Donate Log in Nonlinear Sciences > Pattern Formation and Solitons arXiv:2501.02698 (nlin) [Submitted on 6 Jan 2025 ( v1 ), last revised 24 Aug 2026 (this version, v3)] Title: Beyond-all-order asymptotics for homoclinic snaking of localised patterns in reaction-transport systems Authors: Edgardo Villar-Sepúlveda View PDF HTML (experimental) Abstract: Spatially localised stationary patterns of arbitrary wide spatial extent emerge from subcritical Turing bifurcations in one-dimensional reaction-diffusion systems. They lie on characteristic bifurcation curves that oscillate around a Maxwell point as part of a homoclinic snaking phenomenon. Here, a generalisation of the exponential asymptotics method by Chapman \& Kozyreff is developed to provide leading-order expressions for the width of the snaking region close to a Turing bifurcation's super/sub-critical transition in arbitrary $n$-component reaction-diffusion systems. First, general expressions are provided for the regular asymptotic approximation of the Maxwell point, which depends algebraically on the parametric distance from the codimension-two super/sub-critical Turing bifurcation. Then, expressions are derived for the width of the snaking, which is exponentially small in the same distance. The general theory is developed using a vectorised form of regular and exponential asymptotic expansions for localised patterns, which are matched at a Stokes' line. Detailed calculations for a particular example are algebraically cumbersome, depend sensitively on the form of the reaction kinetics, and rely on the summation of a weakly convergent series obtained via optimal truncation. Nevertheless, the process can be automated, and code is provided that carries out the calculations automatically, only requiring the user to input the model and the values of parameters at a codimension-two bifurcation point. The general theory also applies to higher-order equations, including those that can be recast as a reaction-diffusion system. The theory is illustrated by comparing numerical computations of localised patterns in two versions of the Swift-Hohenberg equation with different nonlinearities, and versions of some classic activator-inhibitor reaction-diffusion systems. Comments: 73 pages, 14 figures Subjects: Pattern Formation and Solitons (nlin.PS) Cite as: arXiv:2501.02698 [nlin.PS] (or arXiv:2501.02698v3 [nlin.PS] for this version) https://doi.org/10.48550/arXiv.2501.02698 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Edgardo Villar-Sepúlveda 2026 Nonlinearity 39 095025 Related DOI : https://doi.org/10.1088/1361-6544/ae9fe4 Focus to learn more DOI(s) linking to related resources Submission history From: Edgardo Villar-Sepúlveda [ view email ] [v1] Mon, 6 Jan 2025 00:28:47 UTC (2,953 KB) [v2] Fri, 17 Jan 2025 04:37:15 UTC (2,931 KB) [v3] Mon, 24 Aug 2026 21:17:45 UTC (2,806 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: nlin.PS < prev | next > new | recent | 2025-01 Change to browse by: nlin References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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