Analytical Exploration of Optical Structures in an Integrable Reduced Spin Hirota–Maxwell–Bloch System

Analytical Exploration of Optical Structures in an Integrable Reduced Spin Hirota–Maxwell–Bloch System | 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 Analytical Exploration of Optical Structures in an Integrable Reduced Spin Hirota–Maxwell–Bloch System Ghulam Hussain Tipu, Kainat Anwar, Fengping Yao, Usman Younas This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6778005/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract An integrable reduced spin Hirota–Maxwell–Bloch (rsHMB) equation is a crucial model for describing femtosecond pulse propagation in erbium-doped fiber, soliton dynamics, and magnetization reversal. The main objective of this research is to derive the exact solution of the reduced integrable spin Hirota–Maxwell–Bloch (rsHMB) equation, as the solutions offer mathematical tools for understanding wave behaviour and have different practical applications. In this research, we use two different analytical methods, the Kumar Malik method, and the modified Sardar sub-equation method, to find the analytical solutions of the equation. The equation is simplified by converting it to a nonlinear ordinary differential equation by the use of a travelling wave transformation. By using these methods, we have obtained different type of solutions to the rsHMB equation, such as Jacobi elliptic functions, exponential, hyperbolic, and trigonometric functions that describe the properties of the model. These analytical solutions are further represent in 3D, 2D and density graphs that show the behaviour of different soliton solutions. We have specifically examined how multiple parameters affect the wave width, velocity, and other crucial characteristics. The advantages of Kumar Malik and Sardar sub-equation method are that these provide many kinds of solitons, such as dark, bright, singular, periodic singular, combined dark–singular and combined dark–bright, singular kink waves, bright solitons, and breather waves. It is important to note that the recommended methods for solving nonlinear partial differential equation are competent, credible, and intriguing tools for analysis. Furthermore, the results are a useful tool for understanding the complicated behaviour of physical systems and providing inspiration for further study. An integrable reduced spin Hirota–Maxwell–Bloch (rsHMB) equation The Kumar-Malik method Sardar sub-equation method Analytical solutions Optical soliton Full Text Additional Declarations No competing interests reported. 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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-6778005","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":463739667,"identity":"0605361c-7aa1-432c-a83c-9bc53b812a0a","order_by":0,"name":"Ghulam Hussain Tipu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVRIiWNgGAWjYDACCSBOYGCQsz/eAGQZWBClhbEBqMWY4cwBkBYJIrUAqUSGGwkwSwkA+egG9gcPdxxOYJz5/OqGHwUSDPzt3Ql4tRjeOcDYkHjmcB6zdE7ZzR6gwyTOnN2AX8uMBKCWtsPFbNI5aTd4gFoMJHKJ05LYI3km7eYfYrTIS0C1zJBgP3abKFsMJBIbZySeSTc24Mlhuy1jIMFD0C/yM5IPfPy5w1rOgP34s5tv/tjI8bf3ErAFGGCQmGHgMQCTeJWDbQGphmhhf0BQ9SgYBaNgFIxMAAAmjUuFM2Jc+wAAAABJRU5ErkJggg==","orcid":"","institution":"Shanghai University","correspondingAuthor":true,"prefix":"","firstName":"Ghulam","middleName":"Hussain","lastName":"Tipu","suffix":""},{"id":463739668,"identity":"1019aceb-6d81-49fb-8edb-2db8ae9dfd56","order_by":1,"name":"Kainat Anwar","email":"","orcid":"","institution":"COMSATS University Islamabad","correspondingAuthor":false,"prefix":"","firstName":"Kainat","middleName":"","lastName":"Anwar","suffix":""},{"id":463739669,"identity":"612c7553-1890-4407-8c6d-334d616fd06a","order_by":2,"name":"Fengping Yao","email":"","orcid":"","institution":"Shanghai University","correspondingAuthor":false,"prefix":"","firstName":"Fengping","middleName":"","lastName":"Yao","suffix":""},{"id":463739670,"identity":"2d73ad8c-f209-42f0-8f6a-b8004c790372","order_by":3,"name":"Usman Younas","email":"","orcid":"","institution":"Shanghai University","correspondingAuthor":false,"prefix":"","firstName":"Usman","middleName":"","lastName":"Younas","suffix":""}],"badges":[],"createdAt":"2025-05-29 15:38:34","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6778005/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6778005/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":83807496,"identity":"dbdaa7aa-5529-4d69-8dc8-bdec42a1e114","added_by":"auto","created_at":"2025-06-03 05:40:00","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1269936,"visible":true,"origin":"","legend":"","description":"","filename":"KP.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6778005/v1_covered_46143d84-99ff-422b-9d1a-b909c3a47ce7.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Analytical Exploration of Optical Structures in an Integrable Reduced Spin Hirota–Maxwell–Bloch System","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"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":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"An integrable reduced spin Hirota–Maxwell–Bloch (rsHMB) equation, The Kumar-Malik method, Sardar sub-equation method, Analytical solutions, Optical soliton","lastPublishedDoi":"10.21203/rs.3.rs-6778005/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6778005/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"An integrable reduced spin Hirota–Maxwell–Bloch (rsHMB) equation is a crucial model for describing femtosecond pulse propagation in erbium-doped fiber, soliton dynamics, and magnetization reversal. 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We have specifically examined how multiple parameters affect the wave width, velocity, and other crucial characteristics. The advantages of Kumar Malik and Sardar sub-equation method are that these provide many kinds of solitons, such as dark, bright, singular, periodic singular, combined dark–singular and combined dark–bright, singular kink waves, bright solitons, and breather waves. It is important to note that the recommended methods for solving nonlinear partial differential equation are competent, credible, and intriguing tools for analysis. 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