{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:35:28Z","timestamp":1760060128132,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,8]],"date-time":"2025-08-08T00:00:00Z","timestamp":1754611200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["KFU252697"],"award-info":[{"award-number":["KFU252697"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The paper consists of various types of wave solutions for the truncated M-fractional Bateman\u2013Burgers equation, a significant mathematical physics equation. This model describes the nonlinear waves and solitons in different physical fields such as optical fibers, plasma physics, fluid dynamics, traffic flow, etc. Through the application of the expa function method and the modified simplest equation method, we are able to obtain exact series of soliton solutions. The results differ from the current solutions of the Bateman\u2013Burgers model because of the fractional derivative. The achieved results could be helpful in various engineering and scientific domains. The Mathematica software is used to assist in obtaining and verifying the exact solutions and to obtain contour plots of the solutions in two and three dimensions. To ensure that the model in question is stable, a stability analysis is also carried out using the modulation instability method. Future research on the system in question and related systems will benefit from the findings. The methods used are simple and effective.<\/jats:p>","DOI":"10.3390\/axioms14080617","type":"journal-article","created":{"date-parts":[[2025,8,8]],"date-time":"2025-08-08T09:56:42Z","timestamp":1754647002000},"page":"617","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modulation Instability and Abundant Exact Solitons to the Fractional Mathematical Physics Model Through Two Distinct Methods"],"prefix":"10.3390","volume":"14","author":[{"given":"Abdulaziz Khalid","family":"Alsharidi","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Hasa 31982, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9394-4681","authenticated-orcid":false,"given":"Ahmet","family":"Bekir","sequence":"additional","affiliation":[{"name":"Neighbourhood of Akcaglan, Imarli Street, Number: 28\/4, 26030 Eskisehir, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2957","DOI":"10.1016\/j.aej.2020.03.032","article-title":"Novel exact solutions of the fractional Bogoyavlensky\u2013Konopelchenko equation involving the Atangana-Baleanu-Riemann derivative","volume":"59","author":"Khater","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ali, H.M., Ali, A.S., Mahmoud, M., and Abdel-Aty, A.-H. (2022). Analytical approximate solutions of fractional nonlinear Drinfeld\u2013Sokolov\u2013Wilson model using modified Mittag\u2013Leffler function. J. Ocean. Eng. Sci.","DOI":"10.1016\/j.joes.2022.06.006"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Zafar, A., Raheel, M., Ali, M.R., Myrzakulova, Z., Bekir, A., and Myrzakulov, R. (2023). Exact Solutions of M-Fractional Kuralay Equation via Three Analytical Schemes. Symmetry, 15.","DOI":"10.3390\/sym15101862"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1007\/s11766-021-4145-3","article-title":"Exact solutions of conformable time fractional Zoomeron equation via IBSEFM","volume":"36","author":"Demirbilek","year":"2021","journal-title":"Appl. Math. J. Chin. Univ."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hong, B., and Wang, J. (2022). Exact solutions for the generalized Atangana-Baleanu-Riemann fractional (3 + 1)-Dimensional Kadomtsev\u2013Petviashvili equation. Symmetry, 15.","DOI":"10.3390\/sym15010003"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Almatrafi, M.B. (2024). Solitary Wave Solutions to a Fractional-Order Fokas Equation via the Improved Modified Extended Tanh-Function Approach. Mathematics, 13.","DOI":"10.3390\/math13010109"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Fadhal, E., Akbulut, A., Kaplan, M., Awadalla, M., and Abuasbeh, K. (2022). Extraction of exact solutions of higher order Sasa-Satsuma equation in the sense of beta derivative. Symmetry, 14.","DOI":"10.3390\/sym14112390"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Sahoo, S., Saha Ray, S., Abdou, M.A.M., Inc, M., and Chu, Y.-M. (2020). New soliton solutions of fractional Jaulent-Miodek system with symmetry analysis. Symmetry, 12.","DOI":"10.3390\/sym12061001"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"108148","DOI":"10.1016\/j.rinp.2025.108148","article-title":"Investigation of a novel exact wave solution structure in nonlinear thermoelasticity using modern techniques","volume":"70","author":"Rabie","year":"2025","journal-title":"Results Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1007\/s11082-024-06907-5","article-title":"Exact solutions of cubic-quintic-septimal nonlinear Schr\u00f6dinger wave equation","volume":"56","author":"Mahmood","year":"2024","journal-title":"Opt. Quantum Electron."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"085252","DOI":"10.1088\/1402-4896\/ad62a6","article-title":"Various exact solutions to the time-fractional nonlinear Schr\u00f6dinger equation via the new modified Sardar sub-equation method","volume":"99","author":"Murad","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_12","first-page":"145","article-title":"New Exact Traveling Wave Solutions of the (3 + 1)-Dimensional Chiral Nonlinear Schrodinger Equation Using Two Reliable Techniques: Annual Meeting in Mathematics 2023","volume":"22","author":"Torvattanabun","year":"2024","journal-title":"Thai J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s11082-023-04565-7","article-title":"Optical solitons to time-fractional Sasa-Satsuma higher-order non-linear Schr\u00f6dinger equation via three analytical techniques","volume":"55","author":"Raheel","year":"2023","journal-title":"Opt. Quantum Electron."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1007\/s11082-024-06518-0","article-title":"Diverse exact soliton solutions for three distinct equations with conformable derivatives via expa function technique","volume":"56","author":"Eslami","year":"2024","journal-title":"Opt. Quantum Electron."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"23410","DOI":"10.3934\/math.20241138","article-title":"Investigation of soliton solutions to the truncated M-fractional (3+ 1)-dimensional Gross-Pitaevskii equation with periodic potential","volume":"9","author":"Qawaqneh","year":"2024","journal-title":"AIMS Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"100527","DOI":"10.1016\/j.padiff.2023.100527","article-title":"Variable coefficient exact solution of Sharma\u2013Tasso\u2013Olver model by enhanced modified simple equation method","volume":"7","author":"Sheikh","year":"2023","journal-title":"Partial. Differ. Equations Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"8705388","DOI":"10.1155\/2022\/8705388","article-title":"Solitary wave solutions of conformable time fractional equations using modified simplest equation method","volume":"1","author":"Razzaq","year":"2022","journal-title":"Complexity"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Murad, M.A.S., Iqbal, M., Arnous, A.H., Yildirim, Y., Jawad, A.J.M., Hussein, L., and Biswas, A. (2024). Optical dromions for Radha\u2013Lakshmanan model with fractional temporal evolution by modified simplest equation. J. Opt., 1\u201310.","DOI":"10.1007\/s12596-024-02201-5"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.ijleo.2017.08.048","article-title":"New solitary solutions of the Gardner equation and Whitham\u2013Broer\u2013Kaup equations by the modified simplest equation method","volume":"147","author":"Kuo","year":"2017","journal-title":"Optik"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1140\/epjp\/s13360-021-02103-6","article-title":"On the dynamical behavior of nonlinear Fitzhugh\u2013Nagumo and Bateman\u2013Burger equations in quantum model using Sinc collocation scheme","volume":"136","author":"Ahmad","year":"2021","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1140\/epjp\/s13360-025-06159-6","article-title":"Explicit wave solutions profile of (3 + 1)-dimensional Bateman\u2013Burgers equation via bilinear neural network method","volume":"140","author":"Qasim","year":"2025","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_22","first-page":"48","article-title":"The analytical solutions of Bateman-Burgers equation","volume":"20","author":"Saengcharoenthaworn","year":"2023","journal-title":"PBRU Sci. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"101102","DOI":"10.1016\/j.padiff.2025.101102","article-title":"Atomic solutions to Bateman-Burgers type equation via tensor products","volume":"13","author":"Alhawatmeh","year":"2025","journal-title":"Partial. Differ. Equations Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2350243","DOI":"10.1142\/S0217984923502433","article-title":"Similarity reductions and new exact solutions for (3 + 1)-dimensional B\u2013B equation","volume":"38","author":"Gaber","year":"2024","journal-title":"Mod. Phys. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s13370-025-01295-9","article-title":"Lie group analysis and conservation laws for the time-fractional 3D Bateman\u2013Burgers equation","volume":"36","author":"Zinat","year":"2025","journal-title":"Afr. Mat."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"271","DOI":"10.47974\/JIM-1474","article-title":"Describing Bateman-Burgers equation in one and two dimensions using Homotopy perturbation method","volume":"26","author":"Akour","year":"2023","journal-title":"J. Interdiscip. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1950052","DOI":"10.1142\/S0217984919500520","article-title":"M-fractional solitons and periodic wave solutions to the Hirota-Maccari system","volume":"33","author":"Sulaiman","year":"2019","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_28","first-page":"83","article-title":"A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties","volume":"16","author":"Sousa","year":"2018","journal-title":"Int. J. Anal. Appl."},{"key":"ref_29","first-page":"451","article-title":"General expa-function method for nonlinear evolution equations","volume":"217","author":"Ali","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_30","first-page":"120","article-title":"Generalized kudryashov method and general expa function method for solving a high order nonlinear schr\u00f6dinger equation","volume":"6","author":"Zayed","year":"2017","journal-title":"J. Space Explor."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1080\/09500340.2017.1407002","article-title":"New exact solutions of the Tzitz\u00e9ica-type equations in non-linear optics using the expa function method","volume":"65","author":"Hosseini","year":"2018","journal-title":"J. Mod. Opt."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1016\/j.aej.2020.10.055","article-title":"Modulation instability analysis and optical solitons in birefringent fibers to RKL equation without four wave mixing","volume":"60","author":"Rehman","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Alomair, A., Naim, A.S.A., and Bekir, A. (2024). Exploration of Soliton Solutions to the Special Korteweg\u2013De Vries Equation with a Stability Analysis and Modulation Instability. Mathematics, 13.","DOI":"10.3390\/math13010054"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/617\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:26:06Z","timestamp":1760034366000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/617"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,8]]},"references-count":33,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080617"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080617","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,8,8]]}}}