{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T17:17:58Z","timestamp":1772644678591,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T00:00:00Z","timestamp":1772236800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University","award":["IMSIU-DDRSP 2602"],"award-info":[{"award-number":["IMSIU-DDRSP 2602"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper investigates the optical solitons to the M-truncated fractional (1+1)-dimensional nonlinear generalized Bretherton model with arbitrary constants. It is employed to forecast the movement of liquid droplets or gas bubbles in microchannels, which is crucial for drug delivery systems, biomedical diagnostics, and lab-on-a-chip technologies. We obtain optical soliton solutions using the extended hyperbolic function method (EHFM) and the modified extended tanh method (METM). Numerous solutions, such as singular, periodic\u2013singular, bright, and dark optical solitons, are obtained from our investigation. The 2D graphical depiction of the solutions shows a variety of wave patterns that change with varied values of \u03b1 and t. The wave\u2019s amplitude forms become more apparent as \u03b1 and t increase. Using 2D plots, the comparison of fractional effects for the M-truncated fractional derivative is demonstrated by giving specific values to the fractional parameter.<\/jats:p>","DOI":"10.3390\/axioms15030171","type":"journal-article","created":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T10:24:34Z","timestamp":1772447074000},"page":"171","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Wave Structures and Soliton Solutions of the Fractional Bretherton Model for Microchannel Droplet Transport"],"prefix":"10.3390","volume":"15","author":[{"given":"Kiran","family":"Khushi","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Okara, Okara 53600, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-5225-2206","authenticated-orcid":false,"given":"Emad K.","family":"Jaradat","sequence":"additional","affiliation":[{"name":"Department of Physics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia"}]},{"given":"Sayed M.","family":"Abo-Dahab","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7338-6288","authenticated-orcid":false,"given":"Hamood Ur","family":"Rehman","sequence":"additional","affiliation":[{"name":"Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku AZ1096, Azerbaijan"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1007\/s11082-022-04261-y","article-title":"Application of new Kudryashov method to various nonlinear partial differential equations","volume":"5","author":"Malik","year":"2023","journal-title":"Opt. Quantum Electron."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"164489","DOI":"10.1016\/j.ijleo.2020.164489","article-title":"Singular and bright-singular combo optical solitons in birefringent fibers to the biswas-arshed equation","volume":"210","author":"Rehman","year":"2020","journal-title":"Optik"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/j.cjph.2019.10.003","article-title":"Soliton solutions of higher order dispersive cubic-quintic nonlinear Schr\u00f6dinger equation and its applications","volume":"67","author":"Rehman","year":"2020","journal-title":"Chin. J. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"849395","DOI":"10.1155\/2014\/849395","article-title":"Fractional calculus and its applications in applied mathematics and other sciences","volume":"2014","author":"Ray","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.cnsns.2018.04.019","article-title":"A new collection of real world applications of fractional calculus in science and engineering","volume":"64","author":"Sun","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/BF02832299","article-title":"Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution","volume":"24","author":"Jumarie","year":"2007","journal-title":"J. Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kumar, D., and Baleanu, D. (2019). Fractional calculus and its applications in physics. Front. Phys., 7.","DOI":"10.3389\/fphy.2019.00081"},{"key":"ref_8","unstructured":"Liu, F., Zhuang, P., and Liu, Q. (2015). Numerical Methods of Fractional Partial Differential Equations and Applications, Science Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1048","DOI":"10.1080\/00207160.2017.1343941","article-title":"Numerical methods for fractional partial differential equations","volume":"95","author":"Li","year":"2018","journal-title":"Int. J. Comput. Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Alshehry, A.S., Shah, R., Shah, N.A., and Dassios, I. (2022). A reliable technique for solving fractional partial differential equation. Axioms, 11.","DOI":"10.3390\/axioms11100574"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1088\/0253-6102\/58\/5\/02","article-title":"(G\u2032\/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics","volume":"58","author":"Zheng","year":"2012","journal-title":"Commun. Theor. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1016\/j.cjph.2016.10.019","article-title":"Soliton solutions for the space-time nonlinear partial differential equations with fractional-orders","volume":"55","author":"Choi","year":"2017","journal-title":"Chin. J. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"043110","DOI":"10.1063\/5.0259937","article-title":"Fractional sub-equation neural networks (fSENNs) method for exact solutions of space-time fractional partial differential equations","volume":"35","author":"Wang","year":"2025","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.ijleo.2018.01.100","article-title":"Optical solitons for Lakshmanan-Porsezian-Daniel model by modified simple equation method","volume":"160","author":"Biswas","year":"2018","journal-title":"Optik"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1211","DOI":"10.1007\/s11082-023-05453-w","article-title":"Optical soliton solutions to the perturbed Biswas-Milovic equation with Kudryashov\u2019s law of refractive index","volume":"55","author":"Zahran","year":"2023","journal-title":"Opt. Quantum Electron."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/s11082-023-05652-5","article-title":"Retrieval of optical soliton solutions of stochastic perturbed Schr\u00f6dinger-Hirota equation with Kerr law in the presence of spatio-temporal dispersion","volume":"56","author":"Ozisik","year":"2024","journal-title":"Opt. Quantum Electron."},{"key":"ref_17","first-page":"1791","article-title":"Exact travelling wave solutions of the generalized Bretherton equation","volume":"215","author":"Romeiras","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11071-012-0637-2","article-title":"Similarity reductions and exact solutions of generalized Bretherton equation with time-dependent coefficients","volume":"71","author":"Gupta","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1017\/S0022112064001355","article-title":"Resonant interactions between waves. The case of discrete oscillations","volume":"20","author":"Bretherton","year":"1964","journal-title":"J. Fluid Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/0375-9601(91)90481-M","article-title":"On types of nonlinear nonintegrable equations with exact solutions","volume":"155","author":"Kudryashov","year":"1991","journal-title":"Phys. Lett. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1074","DOI":"10.1016\/j.physleta.2011.01.010","article-title":"Exact solutions of the generalized Bretherton equation","volume":"375","author":"Kudryashov","year":"2011","journal-title":"Phys. Lett. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1111\/1467-9590.00075","article-title":"Nonlinear wave interactions in nonlinear nonintegrable systems","volume":"100","author":"Berloff","year":"1998","journal-title":"Stud. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Qawaqneh, H., Hakami, K.H., Altalbe, A., and Bayram, M. (2024). Discovery of Truncated M-fractional Exact Solitons, and Qualitative Analysis to the Generalized Bretherton Model. Mathematics, 12.","DOI":"10.20944\/preprints202408.0269.v1"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"169800","DOI":"10.1016\/j.ijleo.2022.169800","article-title":"On the examination of optical soliton pulses of Manakov system with auxiliary equation technique","volume":"268","author":"Ozisik","year":"2022","journal-title":"Optik"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s40314-021-01470-1","article-title":"The first integral method and some nonlinear models","volume":"40","author":"Ghosh","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1080\/00207160412331336026","article-title":"The tanh function method for solving some important non-linear partial differential equations","volume":"82","author":"Evans","year":"2005","journal-title":"Int. J. Comput. Math."},{"key":"ref_27","first-page":"598","article-title":"New extended generalized Kudryashov method for solving three nonlinear partial differential equations","volume":"25","author":"Zayed","year":"2020","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"020203","DOI":"10.1088\/1674-1056\/23\/2\/020203","article-title":"A novel (G\u2032\/G)-expansion method and its application to the Boussinesq equation","volume":"23","author":"Alam","year":"2013","journal-title":"Chin. Phys. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s11082-017-1195-0","article-title":"Solitons and other solutions to the nonlinear Bogoyavlenskii equations using the generalized Riccati equation mapping method","volume":"49","author":"Zayed","year":"2017","journal-title":"Opt. Quantum Electron."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"171305","DOI":"10.1016\/j.ijleo.2023.171305","article-title":"Analysis of Brownian motion in stochastic Schr\u00f6dinger wave equation using Sardar sub-equation method","volume":"289","author":"Rehman","year":"2023","journal-title":"Optik"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"163091","DOI":"10.1016\/j.ijleo.2019.163091","article-title":"Optical solitons with Biswas-Arshed model using mapping method","volume":"194","author":"Rehman","year":"2019","journal-title":"Optik"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"171535","DOI":"10.1016\/j.ijleo.2023.171535","article-title":"Unveiling optical solitons: Solving two forms of nonlinear Schr\u00f6dinger equations with unified solver method","volume":"295","author":"Boulaaras","year":"2023","journal-title":"Optik"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1088\/0253-6102\/57\/2\/01","article-title":"Abundant exact traveling wave solutions of generalized bretherton equation via improved (G\u2032\/G)-expansion method","volume":"57","author":"Akbar","year":"2012","journal-title":"Commun. Theor. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Oguz, C., Yildirim, A., and Meherrem, S. (2012). On the solution of the generalized Bretherton equation by the homogeneous balance method. Commun. Numer. Anal., 2012.","DOI":"10.5899\/2012\/cna-00052"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"284865","DOI":"10.1155\/2013\/284865","article-title":"Abundant Exact Solition-Like Solutions to the Generalized Bretherton Equation with Arbitrary Constants","volume":"2013","author":"Yu","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/j.matcom.2022.03.007","article-title":"Application of modified extended tanh method in solving fractional order coupled wave equations","volume":"198","author":"Dubey","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1621","DOI":"10.1007\/s11082-024-07490-5","article-title":"Exact solutions of paraxial equation via extended hyperbolic function method","volume":"56","author":"Akram","year":"2024","journal-title":"Opt. Quantum Electron."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Wang, K. (2023). New perspective to the fractal Konopelchenko-Dubrovsky equations with M-truncated fractional derivative. Int. J. Geom. Methods Mod. Phys., 20.","DOI":"10.1142\/S021988782350072X"},{"key":"ref_39","unstructured":"Sousa, J.V.D.C., and Oliveira, E.C.D. (2017). A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties. arXiv."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/3\/171\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T13:33:01Z","timestamp":1772631181000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/3\/171"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,28]]},"references-count":39,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["axioms15030171"],"URL":"https:\/\/doi.org\/10.3390\/axioms15030171","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,28]]}}}