{"paper_id":"2541b5aa-4cf5-4f14-b5c3-6002448d588f","body_text":"Solitary wave solutions, periodic and superposition solutions to the system of first-order (2+1)-dimensional Boussinesq's equations derived from the Euler equations for an ideal fluid model | 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 Solitary wave solutions, periodic and superposition solutions to the system of first-order (2+1)-dimensional Boussinesq's equations derived from the Euler equations for an ideal fluid model Piotr Rozmej, Anna Karczewska This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7081945/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 18 Apr, 2026 Read the published version in Journal of Nonlinear Science → Version 1 posted 11 You are reading this latest preprint version Abstract This article concludes the study of (2+1)-dimensional nonlinear wave equations that can be derived in a model of an ideal fluid with irrotational motion. In the considered case of identical scaling of the $x,y$ variables, obtaining a (2+1)-dimensional wave equation analogous to the KdV equation is impossible. Instead, from a system of two first-order Boussinesq equations, a non-linear wave equation for the auxiliary function $f(x,y,t)$ defining the velocity potential can be obtained, and only from its solutions can the surface wave form $\\eta(x,y,t)$ be obtained. We demonstrate the existence of families of (2+1)-dimensional traveling wave solutions, including solitary and periodic solutions, of both cnoidal and superposition types. MSC Classification: 02.30.Jr , 05.45.-a , 47.35.B , 47.35.Fg Boussinesq’s equations nonlinear wave equations solitary wave solutions periodic solutions superposition solutions Full Text Additional Declarations No competing interests reported. Supplementary Files GraphAbstr.pdf Cite Share Download PDF Status: Published Journal Publication published 18 Apr, 2026 Read the published version in Journal of Nonlinear Science → Version 1 posted Editorial decision: Accepted 18 Mar, 2026 Reviews received at journal 17 Mar, 2026 Reviewers agreed at journal 04 Feb, 2026 Reviewers agreed at journal 03 Feb, 2026 Reviews received at journal 07 Sep, 2025 Reviewers agreed at journal 12 Aug, 2025 Reviewers agreed at journal 30 Jul, 2025 Reviewers invited by journal 27 Jul, 2025 Editor assigned by journal 11 Jul, 2025 Submission checks completed at journal 11 Jul, 2025 First submitted to journal 09 Jul, 2025 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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