Solving the Exact Solutions of the Fifth-order Dispersive Caudrey-Dodd-Gibbon Equation Based on the Feature-enhanced Direct Symbolic Computation Neural Network Algorithm

Solving the Exact Solutions of the Fifth-order Dispersive Caudrey-Dodd-Gibbon Equation Based on the Feature-enhanced Direct Symbolic Computation Neural Network Algorithm | 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 Solving the Exact Solutions of the Fifth-order Dispersive Caudrey-Dodd-Gibbon Equation Based on the Feature-enhanced Direct Symbolic Computation Neural Network Algorithm Jiang-Long Shen, Xia Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9350197/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 13 You are reading this latest preprint version Abstract The Caudrey-Dodd-Gibbon (CDG) equation, a core constituent of the fifth-order Korteweg-de Vries (KdV)-type nonlinear wave equations, possesses exact solutions that are of pivotal significance for elucidating the mechanisms of wave propagation under the coupling effect of strong dispersion and intense nonlinearity. To mitigate the limitations inherent in traditional analytical methods for solving such high-order equations—encompassing narrow applicability, high computational complexity, and expression explosion a Feature-Enhanced Direct Symbolic Computation Neural Network(FEDSCNN) algorithm is proposed. This algorithm integrates the adaptive feature extraction capability of neural networks, the rigor of symbolic computation, and the advantages of feature enhancement. It constructs single-hidden-layer "2-5-3-1" and double-hidden-layer "2-5-2-2-1" architectures, and obtains exact solutions through a sequence of steps, namely symbolic derivation, equation mapping transformation, and constraint system solving.Experimental results validate that the proposed algorithm successfully acquires multiple categories of solitary wave solutions and periodic solutions for the CDG equation, with a zero error margin corroborated by the Maple symbolic computation engine. Furthermore, this algorithm circumvents the necessity of presetting solution forms, attains a higher convergence rate, and exhibits enhanced capability in capturing high-order terms. The present study not only furnishes an efficient and innovative approach for solving the CDG equation but also provides a general framework for the analytical solution of analogous high-order nonlinear wave equations, thereby facilitating the interdisciplinary integration of neural networks and symbolic computation. Fifth-order dispersion nonlinear wave equation Feature enhancement Direct symbolic computation Neural networks Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 10 May, 2026 Reviews received at journal 06 May, 2026 Reviews received at journal 26 Apr, 2026 Reviewers agreed at journal 21 Apr, 2026 Reviewers agreed at journal 18 Apr, 2026 Reviews received at journal 13 Apr, 2026 Reviewers agreed at journal 13 Apr, 2026 Reviewers agreed at journal 13 Apr, 2026 Reviewers agreed at journal 13 Apr, 2026 Reviewers invited by journal 13 Apr, 2026 Editor assigned by journal 08 Apr, 2026 Submission checks completed at journal 08 Apr, 2026 First submitted to journal 07 Apr, 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. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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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-9350197","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":625402624,"identity":"0aba12b1-f544-4ab3-8d8c-8080916a6292","order_by":0,"name":"Jiang-Long Shen","email":"","orcid":"","institution":"Yibin University","correspondingAuthor":false,"prefix":"","firstName":"Jiang-Long","middleName":"","lastName":"Shen","suffix":""},{"id":625402634,"identity":"386feb6c-7f0d-439d-9c25-e2bb46213b98","order_by":1,"name":"Xia Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1ElEQVRIiWNgGAWjYDACZgbGAwkVDDwGzFAuMVoYDiScIUkLEBxgbGNgMICbQAgYHGd+cODhvMMy5uy8xyQYKqwTG9jPHsCrRbKZzeBA4rbDPJbNfGkSDGfSExt48hLwauFnZoBoMTjMYybB2HY4sUGCxwCvFjZm9g8HEufAtPwjQgs/Mw/QlgaYlgYitEg28xQcSDiWDvQLj7EFkGHcxpODX4vB+eMbH/6osbY35z9jeONDjbVsP/sZ/FqgoBlCJYB8R4x6IKgjUt0oGAWjYBSMSAAAQ08/OUvlvK8AAAAASUVORK5CYII=","orcid":"","institution":"Yibin University","correspondingAuthor":true,"prefix":"","firstName":"Xia","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2026-04-08 01:38:37","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9350197/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9350197/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107488303,"identity":"35d345f4-daa0-400d-baf1-b65ccefb7969","added_by":"auto","created_at":"2026-04-22 02:44:12","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2794324,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptCDGCSF.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9350197/v1_covered_eb69b0d0-10af-45e0-b86e-dc61f2d45c00.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Solving the Exact Solutions of the Fifth-order Dispersive Caudrey-Dodd-Gibbon Equation Based on the Feature-enhanced Direct Symbolic Computation Neural Network Algorithm","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":"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":"Fifth-order dispersion nonlinear wave equation, Feature enhancement, Direct symbolic computation, Neural networks","lastPublishedDoi":"10.21203/rs.3.rs-9350197/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9350197/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The Caudrey-Dodd-Gibbon (CDG) equation, a core constituent of the fifth-order Korteweg-de Vries (KdV)-type nonlinear wave equations, possesses exact solutions that are of pivotal significance for elucidating the mechanisms of wave propagation under the coupling effect of strong dispersion and intense nonlinearity. To mitigate the limitations inherent in traditional analytical methods for solving such high-order equations—encompassing narrow applicability, high computational complexity, and expression explosion a Feature-Enhanced Direct Symbolic Computation Neural Network(FEDSCNN) algorithm is proposed. This algorithm integrates the adaptive feature extraction capability of neural networks, the rigor of symbolic computation, and the advantages of feature enhancement. It constructs single-hidden-layer \"2-5-3-1\" and double-hidden-layer \"2-5-2-2-1\" architectures, and obtains exact solutions through a sequence of steps, namely symbolic derivation, equation mapping transformation, and constraint system solving.Experimental results validate that the proposed algorithm successfully acquires multiple categories of solitary wave solutions and periodic solutions for the CDG equation, with a zero error margin corroborated by the Maple symbolic computation engine. Furthermore, this algorithm circumvents the necessity of presetting solution forms, attains a higher convergence rate, and exhibits enhanced capability in capturing high-order terms. 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