{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T11:01:00Z","timestamp":1781780460571,"version":"3.54.5"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T00:00:00Z","timestamp":1693526400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the ion-acoustic wave structures in fluid ions for the Benjamin\u2013Bona\u2013Mahony\u2013Peregrine\u2013Burgers (BBMPB) equation. The various types of wave structures are extracted including the three-wave hypothesis, breather wave, lump periodic, mixed-type wave, periodic cross-kink, cross-kink rational wave, M-shaped rational wave, M-shaped rational wave solution with one kink wave, and M-shaped rational wave with two kink wave solutions. The Hirota bilinear transformation is a powerful tool that allows us to accurately find solutions and predict the behaviour of these wave structures. Through our analysis, we gain a better understanding of the complex dynamics of ion-acoustic waves and their potential applications in various fields. Moreover, our findings contribute to the ongoing research in plasma physics that utilize ion-acoustic wave phenomena. To show the physical behaviour of the solutions, some 3D plots and their respective contour level are shown, choosing different values of the parameters.<\/jats:p>","DOI":"10.3390\/sym15091682","type":"journal-article","created":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T08:19:39Z","timestamp":1693556379000},"page":"1682","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Modelling Symmetric Ion-Acoustic Wave Structures for the BBMPB Equation in Fluid Ions Using Hirota\u2019s Bilinear Technique"],"prefix":"10.3390","volume":"15","author":[{"given":"Baboucarr","family":"Ceesay","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, The University of Lahore, Lahore 54000, Pakistan"},{"name":"Department of Mathematics, The University of The Gambia, Serrekunda P.O. Box 3530, The Gambia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad Zafarullah","family":"Baber","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, The University of Lahore, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1742-585X","authenticated-orcid":false,"given":"Nauman","family":"Ahmed","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, The University of Lahore, Lahore 54000, Pakistan"},{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut P.O. Box 13-5053, Lebanon"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9832-1424","authenticated-orcid":false,"given":"Ali","family":"Akg\u00fcl","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut P.O. Box 13-5053, Lebanon"},{"name":"Department of Mathematics, Art and Science Faculty, Siirt University, 56100 Siirt, Turkey"},{"name":"Department of Mathematics, Mathematics Research Center, Near East University, Near East Boulevard, Mersin 10, 99138 Nicosia, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7462-9173","authenticated-orcid":false,"given":"Alicia","family":"Cordero","sequence":"additional","affiliation":[{"name":"Multidisciplinary Institute of Mathematics, Universitat Polit\u00e8nica de Val\u00e8ncia, 46022 Val\u00e8ncia, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9893-0761","authenticated-orcid":false,"given":"Juan R.","family":"Torregrosa","sequence":"additional","affiliation":[{"name":"Multidisciplinary Institute of Mathematics, Universitat Polit\u00e8nica de Val\u00e8ncia, 46022 Val\u00e8ncia, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,1]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Ion-acoustic wave structures in the fluid ions modeled by higher dimensional generalized Korteweg-de Vries\u2013Zakharov\u2013Kuznetsov equation","volume":"7","author":"Younas","year":"2022","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.joes.2019.01.003","article-title":"An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma","volume":"4","author":"Goswami","year":"2019","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1016\/j.asr.2022.08.062","article-title":"Gas-dynamic approach to the theory of non-linear ion-acoustic waves in plasma with Kaniadakis\u2019 distributed species","volume":"71","author":"Dubinov","year":"2023","journal-title":"Adv. Space Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"100490","DOI":"10.1016\/j.padiff.2023.100490","article-title":"Reciprocal B\u00e4cklund transformations and travelling wave structures of some nonlinear pseudo-parabolic equations","volume":"7","author":"Usman","year":"2023","journal-title":"Partial Differ. Equ. Appl. Math."},{"key":"ref_5","unstructured":"Lipatov, A.S. (2002). The Hybrid Multiscale Simulation Technology: An Introduction with Application to Astrophysical and Laboratory Plasmas, Springer Science & Business Media."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"543","DOI":"10.4236\/ojapps.2020.108038","article-title":"New Exact Solutions for Benjamin-Bona-Mahony-Burgers Equation","volume":"10","author":"Xiang","year":"2020","journal-title":"Open J. Appl. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1432","DOI":"10.4236\/am.2014.510135","article-title":"Application of classification of traveling wave solutions to the Zakhrov-Kuznetsov-Benjamin-Bona-Mahony equation","volume":"5","author":"Yang","year":"2014","journal-title":"Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"51","DOI":"10.20852\/ntmsci.2016422120","article-title":"Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G\u2032G Expansion Method","volume":"4","author":"Akcagil","year":"2016","journal-title":"New Trends Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"103114","DOI":"10.1063\/1.4966020","article-title":"Exact periodic cross-kink wave solutions for the new (2+1)-dimensional KdV equation in fluid flows and plasma physics","volume":"26","author":"Liu","year":"2016","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"106114","DOI":"10.1016\/j.aml.2019.106114","article-title":"Coherent superposition propagation of Laguerre\u2013Gaussian and Hermite\u2013Gaussian solitons","volume":"102","author":"Song","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"107755","DOI":"10.1016\/j.aml.2021.107755","article-title":"The complex-valued astigmatic cosine-Gaussian soliton solution of the nonlocal nonlinear Schr\u00f6dinger equation and its transmission characteristics","volume":"125","author":"Shen","year":"2022","journal-title":"Appl. Math. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106005","DOI":"10.1016\/j.cnsns.2021.106005","article-title":"Periodic propagation of complex-valued hyperbolic-cosine-Gaussian solitons and breathers with complicated light field structure in strongly nonlocal nonlinear media","volume":"103","author":"Shen","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2200472","DOI":"10.1002\/andp.202200472","article-title":"Interactions of Localized Wave Structures on Periodic Backgrounds for the Coupled Lakshmanan\u2013Porsezian\u2013Daniel Equations in Birefringent Optical Fibers","volume":"535","author":"Li","year":"2023","journal-title":"Ann. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107316","DOI":"10.1016\/j.cnsns.2023.107316","article-title":"The Riemann\u2013Hilbert approach for the higher-order Gerdjikov\u2013Ivanov equation, soliton interactions and position shift","volume":"124","author":"Zou","year":"2023","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhang, S., Zhu, F., and Xu, B. (2023). Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schr\u00f6dinger Equations. Symmetry, 15.","DOI":"10.3390\/sym15061211"},{"key":"ref_16","first-page":"12","article-title":"The direct algebraic method to complex nonlinear partial differential equations","volume":"5","author":"Taghizadeh","year":"2013","journal-title":"Int. J. Appl. Math. Comput."},{"key":"ref_17","first-page":"179","article-title":"Modulation instability analysis, optical solitons and other solutions to the (2+ 1)-dimensional hyperbolic nonlinear Schrodinger\u2019s equation","volume":"10","author":"Sulaiman","year":"2022","journal-title":"Comput. Methods Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"115004","DOI":"10.1088\/1572-9494\/ac18bb","article-title":"Diverse acoustic wave propagation to confirmable time\u2013space fractional KP equation arising in dusty plasma","volume":"73","author":"Seadawy","year":"2021","journal-title":"Commun. Theor. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"106677","DOI":"10.1016\/j.rinp.2023.106677","article-title":"On the analytical study of predator\u2013prey model with Holling-II by using the new modified extended direct algebraic technique and its stability analysis","volume":"51","author":"Shahzad","year":"2023","journal-title":"Results Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2350155","DOI":"10.1142\/S0217979223501552","article-title":"Comparative analysis of numerical and newly constructed soliton solutions of stochastic Fisher-type equations in a sufficiently long habitat","volume":"37","author":"Baber","year":"2023","journal-title":"Int. J. Mod. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"688","DOI":"10.1007\/s11082-022-04104-w","article-title":"On the optical soliton structures in the magneto electro-elastic circular rod modeled by nonlinear dynamical longitudinal wave equation","volume":"54","author":"Younas","year":"2022","journal-title":"Opt. Quantum Electron."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2817","DOI":"10.1016\/j.cjph.2018.08.006","article-title":"A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation","volume":"56","author":"Feng","year":"2018","journal-title":"Chin. J. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Hirota, R. (2004). The Direct Method in Soliton Theory (No. 155), Cambridge University Press.","DOI":"10.1017\/CBO9780511543043"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s10509-008-9748-0","article-title":"Ion acoustic solitons and double layers in electron\u2013positron\u2013ion plasmas with dust particulates","volume":"314","author":"Ghosh","year":"2008","journal-title":"Astrophys. Space Sci."},{"key":"ref_25","unstructured":"Petviashvili, V.I., and Pokhotelov, O.A. (1992). Solitary Waves in Plasmas and in the Atmosphere, Taylor & Francis."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1029-242X-2014-500","article-title":"The general traveling wave solutions of the Fisher type equations and some related problems","volume":"2014","author":"Yuan","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1088\/0031-8949\/90\/10\/105201","article-title":"Higher-order rogue wave solutions of the three-wave resonant interaction equation via the generalized Darboux transformation","volume":"90","author":"Wang","year":"2015","journal-title":"Phys. Scr."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1007\/s11071-016-3267-2","article-title":"New three-wave solutions for the (3+1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation","volume":"88","author":"Liu","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_29","first-page":"403","article-title":"Localized structures in optical and matter-wave media: A selection of recent studies","volume":"73","author":"Mihalache","year":"2021","journal-title":"Rom. Rep. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4270906","DOI":"10.1155\/2020\/4270906","article-title":"New periodic wave, cross-kink wave, breather, and the interaction phenomenon for the (2+ 1)-dimensional Sharmo\u2013Tasso\u2013Olver equation","volume":"2020","author":"Ma","year":"2020","journal-title":"Complexity"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Alsallami, S.A., Rizvi, S.T., and Seadawy, A.R. (2023). Study of stochastic\u2013fractional Drinfel\u2019d\u2013Sokolov\u2013Wilson equation for M-shaped rational, homoclinic breather, periodic and Kink-Cross rational solutions. Mathematics, 11.","DOI":"10.3390\/math11061504"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11082-021-03276-1","article-title":"Breather, multi-wave, periodic-cross kink, M-shaped and interactions solutions for perturbed NLSE with quadratic cubic nonlinearity","volume":"53","author":"Seadawy","year":"2021","journal-title":"Opt. Quantum Electron."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1682\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:44:42Z","timestamp":1760129082000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1682"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,1]]},"references-count":32,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["sym15091682"],"URL":"https:\/\/doi.org\/10.3390\/sym15091682","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,1]]}}}