Novel Insights into the Exact Solutions of the Modified (3+1) Dimensional Fractional KS Equation with Variable Coefficients

Novel Insights into the Exact Solutions of the Modified (3+1) Dimensional Fractional KS Equation with Variable Coefficients | 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 Novel Insights into the Exact Solutions of the Modified (3+1) Dimensional Fractional KS Equation with Variable Coefficients Jisha CR, Bongsoo Jang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3592046/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 19 Mar, 2024 Read the published version in Nonlinear Dynamics → Version 1 posted 5 You are reading this latest preprint version Abstract This study comprehensively explores a unified methodology for deriving exact solutions to the fractional modified (3+1) dimensional Kudryashov-Sinelshchikov (KS) equation featuring variable coefficients. The fractional KS equation, which incorporates fractional local M-derivatives, presents a significant challenge because of its inherent nonlinearity and the introduction of spatially varying coefficients. Of paramount importance are these variable coefficients, as they introduce spatial dependence and intricately capture spatial variations within the equation. This complexity in spatial variation poses a formidable challenge in obtaining exact solutions. Our analytical examination offers profound insights into the intricate interplay between diverse variable coefficients and fractional parameters. This comprehensive analysis greatly enhances our capacity to interpret solutions across various scenarios, enriching our understanding of the nuanced behavior exhibited by the fractional Kudryashov-Sinelshchikov equation. Our investigation encompasses a diverse range of solution forms, including fractional, polynomial, exponential, and others. The outcomes of this study hold profound implications for an extensive array of scientific domains, spanning mathematical physics, fluid dynamics, and nonlinear optics. Furthermore, this research employs advanced data visualization techniques, comprising 3D plots, contour plots, and stream plots, to facilitate a deep comprehension of intricate physical phenomena. These visual aids concurrently illustrate how analytical solutions are influenced by varying conditions. Ams classification 2010: 45K05, 45G10, 83C15, 35Q35, 35Q68 , 35Q80 Polynomial solutions Unified method KS equation Korteweg-de Vries equation Multi-rational solutions Full Text Cite Share Download PDF Status: Published Journal Publication published 19 Mar, 2024 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Major revisions 28 Dec, 2023 Reviewers agreed at journal 19 Nov, 2023 Reviewers invited by journal 19 Nov, 2023 Editor assigned by journal 11 Nov, 2023 First submitted to journal 10 Nov, 2023 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. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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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-3592046","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":251283838,"identity":"063b969b-38dc-4aaa-88af-9829b7d7d3e5","order_by":0,"name":"Jisha CR","email":"","orcid":"","institution":"Ulsan National Institute of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Jisha","middleName":"","lastName":"CR","suffix":""},{"id":251283839,"identity":"baa79993-6801-408e-8c40-1c5f208e0825","order_by":1,"name":"Bongsoo Jang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4UlEQVRIie3PMQrCMBiG4a8IdkntqgQ9g8VBROlZUgKdPYBDQHAqnRUv4SZugYIuYlehi06ugiCiIqbgbDIK5h3SJOSBv4DN9oO1AaY+A6DyuakbkrgkjjAlZVm5GJKuL4+X4T2Plq57OmAUojGX30lvyjidpUW0GpNAYM1Ba0wz2B6MekkRLTKiBqtKtIhmsHYu+cNLdoq4R4GXCZEspuQmFUEgnIkE1RH1L3HfE7yjBgumUcpJI9GQrr/lBXmGzUW+OZzP17BV32oIQBicyWfP1FELAFcCT4N3NpvN9r+9Aeu1QrpnB2hHAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0001-8535-8906","institution":"Ulsan National Institute of Science and Technology","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Bongsoo","middleName":"","lastName":"Jang","suffix":""}],"badges":[],"createdAt":"2023-11-10 16:46:52","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3592046/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3592046/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11071-024-09386-1","type":"published","date":"2024-03-19T15:00:32+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":53403421,"identity":"656c79b9-f17a-49f2-ad7d-2ebdc5b3c0a8","added_by":"auto","created_at":"2024-03-25 15:05:37","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":5329119,"visible":true,"origin":"","legend":"","description":"","filename":"FractionalKSequationwithvariablecoefficientND.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3592046/v1_covered_397bf240-ba3e-424d-94c2-e7823cf36982.pdf"}],"financialInterests":"","formattedTitle":"Novel Insights into the Exact Solutions of the Modified (3+1) Dimensional Fractional KS Equation with Variable Coefficients","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Polynomial solutions, Unified method, KS equation, Korteweg-de Vries equation, Multi-rational solutions","lastPublishedDoi":"10.21203/rs.3.rs-3592046/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3592046/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study comprehensively explores a unified methodology for deriving exact solutions to the fractional modified (3+1) dimensional \u0026nbsp;Kudryashov-Sinelshchikov (KS) equation featuring variable coefficients. The fractional KS equation, which incorporates fractional local M-derivatives, presents a significant challenge because of its inherent nonlinearity and the introduction of spatially varying coefficients. Of paramount importance are these variable coefficients, as they introduce spatial dependence and intricately capture spatial variations within the equation. This complexity in spatial variation poses a formidable challenge in obtaining exact solutions. Our analytical examination offers profound insights into the intricate interplay between diverse variable coefficients and fractional parameters. This comprehensive analysis greatly enhances our capacity to interpret solutions across various scenarios, enriching our understanding of the nuanced behavior exhibited by the fractional Kudryashov-Sinelshchikov equation. Our investigation encompasses a diverse range of solution forms, including fractional, polynomial, exponential, and others. The outcomes of this study hold profound implications for an extensive array of scientific domains, spanning mathematical physics, fluid dynamics, and nonlinear optics. Furthermore, this research employs advanced data visualization techniques, comprising 3D plots, contour plots, and stream plots, to facilitate a deep comprehension of intricate physical phenomena. 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