Overcoming Non-Commutativity: New Methods for Linear Quaternion Differential Equations

Overcoming Non-Commutativity: New Methods for Linear Quaternion Differential Equations | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Overcoming Non-Commutativity: New Methods for Linear Quaternion Differential Equations Zhenfeng Cai, Kit Ian Kou, Weinian Zhang, Yanglin Zhang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9184333/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 15 You are reading this latest preprint version Abstract The non-commutativity of quaternion multiplication presents a fundamental obstacle in analyzing linear quaternion-valued differential equations (QDEs). While the exponential solution for homogeneous linear QDEs by Campos and Mawhin is a cornerstone of the field, its reliance on a restrictive commutativity condition limits it to a narrow, complex-like subclass of functions. This work overcomes this limitation by introducing a novel algorithmic framework that solves the homogeneous initial value problem without any commutativity assumptions. We first demonstrate that the commutativity condition is equivalent to confining the dynamics to a complex-valued subspace. Our primary contribution is a method that systematically reduces the QDE to a solvable real nonlinear differential equation. We further derive closed-form solutions for key non-commutative cases. These results dramatically expand the solvable landscape of linear QDEs, with direct applications in control theory, quantum mechanics, and hypercomplex signal processing, where non-commutative dynamics are intrinsic. We demonstrate the power of our approach by applying it to Robinson’s Quaternion Kinematical Differential Equations and a problem in medical image communication security. Quaternion-valued differential equations non-commutative dynamics algo- rithmic solution framework homogeneous initial value problem hypercomplex signal pro- cessing Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 09 May, 2026 Reviews received at journal 01 May, 2026 Reviews received at journal 27 Apr, 2026 Reviews received at journal 22 Apr, 2026 Reviews received at journal 09 Apr, 2026 Reviewers agreed at journal 03 Apr, 2026 Reviewers agreed at journal 31 Mar, 2026 Reviewers agreed at journal 31 Mar, 2026 Reviewers agreed at journal 30 Mar, 2026 Reviewers agreed at journal 30 Mar, 2026 Reviewers agreed at journal 29 Mar, 2026 Reviewers invited by journal 29 Mar, 2026 Editor assigned by journal 25 Mar, 2026 Submission checks completed at journal 25 Mar, 2026 First submitted to journal 21 Mar, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9184333","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":615507383,"identity":"462c12e6-235b-4a5b-a3cd-ff567589b60e","order_by":0,"name":"Zhenfeng Cai","email":"","orcid":"","institution":"Hubei University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Zhenfeng","middleName":"","lastName":"Cai","suffix":""},{"id":615507387,"identity":"a46184f9-95b2-4952-9c00-de74ea0ced76","order_by":1,"name":"Kit Ian Kou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAm0lEQVRIiWNgGAWjYBACCWY2IFlBupYzJGlhAGphbCNFi2Q7W+Jj3nmHo/nbD7Bu+ECMFmlmtsPGvNsO5844k8B2cwYxWuSY2dukQVoabjCw3eYhXsucw7nzidYCdNgxad6Gw7kbiNYi2cyWbDjnWHruxjOJbcT5ReL8McMHb2qsc+cdP3zsBlEhhgQYG0jUMApGwSgYBaMAJwAAHGsvpikRcO0AAAAASUVORK5CYII=","orcid":"","institution":"University of Macau","correspondingAuthor":true,"prefix":"","firstName":"Kit","middleName":"Ian","lastName":"Kou","suffix":""},{"id":615507390,"identity":"121ae054-7ce6-4d20-b8bd-a3a071cf62f4","order_by":2,"name":"Weinian Zhang","email":"","orcid":"","institution":"Sichuan University","correspondingAuthor":false,"prefix":"","firstName":"Weinian","middleName":"","lastName":"Zhang","suffix":""},{"id":615507394,"identity":"2a1d2b40-3711-437f-ab6e-e2b078752ea7","order_by":3,"name":"Yanglin Zhang","email":"","orcid":"","institution":"Guangxi University","correspondingAuthor":false,"prefix":"","firstName":"Yanglin","middleName":"","lastName":"Zhang","suffix":""}],"badges":[],"createdAt":"2026-03-21 08:23:54","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9184333/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9184333/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106093307,"identity":"7b0f3edc-568c-4c84-b3d4-5ab612a313ff","added_by":"auto","created_at":"2026-04-03 11:36:38","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":8006957,"visible":true,"origin":"","legend":"","description":"","filename":"LaTeXdocuments.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9184333/v1_covered_8b3b101f-6b57-47d1-b125-c2f46929b5d5.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Overcoming Non-Commutativity: New Methods for Linear Quaternion Differential Equations","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"qualitative-theory-of-dynamical-systems","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"qtds","sideBox":"Learn more about [Qualitative Theory of Dynamical Systems](http://link.springer.com/journal/12306)","snPcode":"12346","submissionUrl":"https://submission.nature.com/new-submission/12346/3","title":"Qualitative Theory of Dynamical Systems","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Quaternion-valued differential equations, non-commutative dynamics, algo- rithmic solution framework, homogeneous initial value problem, hypercomplex signal pro- cessing","lastPublishedDoi":"10.21203/rs.3.rs-9184333/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9184333/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The non-commutativity of quaternion multiplication presents a fundamental obstacle in analyzing linear quaternion-valued differential equations (QDEs). 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