{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,28]],"date-time":"2026-05-28T01:42:56Z","timestamp":1779932576487,"version":"3.53.1"},"reference-count":54,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,19]],"date-time":"2022-08-19T00:00:00Z","timestamp":1660867200000},"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>In this manuscript, the novel auxiliary equation methodology (NAEM) is employed to scrutinize various forms of solitary wave solutions for the modified equal-width wave (MEW) equation. M-truncated along with Atangana\u2013Baleanu (AB)-fractional derivatives are employed to study the soliton solutions of the problem. The fractional MEW equations are important for describing hydro-magnetic waves in cold plasma. A comparative analysis is utilized to study the influence of the fractional parameter on the generated solutions. Secured solutions include bright, dark, singular, periodic and many other types of soliton solutions. In compared to other methods, the solutions demonstrate that the proposed technique is particularly effective, straightforward, and trustworthy that contains families of solutions. In addition, the symbolic soft computation is used to verify the obtained solutions. Finally, the system is subjected to a sensitive analysis. Integer-order results calculated by the symmetry method present in the literature can be addressed as limiting cases of the present study.<\/jats:p>","DOI":"10.3390\/sym14081731","type":"journal-article","created":{"date-parts":[[2022,8,22]],"date-time":"2022-08-22T01:56:40Z","timestamp":1661133400000},"page":"1731","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Soliton Solutions and Sensitive Analysis of Modified Equal-Width Equation Using Fractional Operators"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5153-297X","authenticated-orcid":false,"given":"Muhammad Bilal","family":"Riaz","sequence":"first","affiliation":[{"name":"Faculty of Technical Physics, Information Technology and Applied Mathematics, Lodz University of Technology, 90-924 Lodz, Poland"},{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3786-7225","authenticated-orcid":false,"given":"Adam","family":"Wojciechowski","sequence":"additional","affiliation":[{"name":"Faculty of Technical Physics, Information Technology and Applied Mathematics, Lodz University of Technology, 90-924 Lodz, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2902-4455","authenticated-orcid":false,"given":"Georgia Irina","family":"Oros","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Univerisity of Oradea, 410087 Oradea, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9731-7212","authenticated-orcid":false,"given":"Riaz Ur","family":"Rahman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,19]]},"reference":[{"key":"ref_1","first-page":"105","article-title":"The fractional dynamics of a linear triatomic molecule","volume":"73","author":"Baleanu","year":"2021","journal-title":"Rom. Rep. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"167","DOI":"10.3389\/fphy.2020.00167","article-title":"New optical solutions of the fractional Gerdjikov-Ivanov equation with conformable derivative","volume":"8","author":"Ghanbari","year":"2020","journal-title":"Front. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Abouelregal, A.E., Nofal, T.A., and Alsharari, F. (2022). A thermodynamic two-temperature model with distinct fractional derivative operators for an infinite body with a cylindrical cavity and varying properties. J. Ocean. Eng. Sci.","DOI":"10.1016\/j.joes.2022.05.001"},{"key":"ref_4","first-page":"48","article-title":"A generalized exp-function method for fractional Riccati differential equations","volume":"1","author":"Zang","year":"2010","journal-title":"Commun. Fract. Calc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7843","DOI":"10.1016\/j.aej.2022.01.029","article-title":"Soliton theory and modulation instability analysis: The Ivancevic option pricing model in economy","volume":"61","author":"Chen","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1515\/phys-2016-0010","article-title":"Analysis of time fractional Hunter-Saxton equation: A model of neumatic liquid crystal","volume":"14","author":"Atangana","year":"2016","journal-title":"Open Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"105356","DOI":"10.1016\/j.rinp.2022.105356","article-title":"Fractional derivative-based performance analysis to Caudrey-Dodd-Gibbon-Sawada-Kotera equation","volume":"36","author":"Jhangeer","year":"2022","journal-title":"Results Phys."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Atangana, A., and Alqahtani, R.T. (2016). Modelling the Spread of River Blindness Disease via the Caputo Fractional Derivative and the Beta-derivative. Entropy, 18.","DOI":"10.3390\/e18020040"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"103725","DOI":"10.1016\/j.rinp.2020.103725","article-title":"Traveling wave solutions for the fractional Wazwaz-Benjamin-Bona-Mahony model in arising shallow water waves","volume":"20","author":"Akram","year":"2021","journal-title":"Results Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"075209","DOI":"10.1088\/1402-4896\/abf868","article-title":"Fractional approach for analysis of the model describing wind-influenced projectile motion","volume":"96","author":"Veeresha","year":"2021","journal-title":"Phys. Scr."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"111347","DOI":"10.1016\/j.chaos.2021.111347","article-title":"Fractional approach for a mathematical model of atmospheric dynamics of CO2 gas with an efficient method","volume":"152","author":"Ilhan","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1515\/nleng-2018-0033","article-title":"Hyperbolic rational solutions to a variety of conformable fractional Boussinesqlike equations","volume":"8","author":"Rezazadeh","year":"2019","journal-title":"Nonlinear Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.joes.2018.12.001","article-title":"Boussinesq equations: M-fractional solitary wave solutions and convergence analysis","volume":"4","author":"Sulaiman","year":"2019","journal-title":"J. Ocean. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"252168","DOI":"10.1155\/2014\/252168","article-title":"Exact Solutions for Nonlinear Wave Equations by the Exp-Function Method","volume":"2014","author":"Hu","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Abdeljabbar, A., Roshid, H.O., and Aldurayhim, A. (2022). Bright, Dark, and Rogue Wave Soliton Solutions of the Quadratic Nonlinear Klein-Gordon Equation. Symmetry, 14.","DOI":"10.3390\/sym14061223"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2150157","DOI":"10.1142\/S0217979221501575","article-title":"Computational and bright soliton solutions and sensitivity behavior of Camassa-Holm and nonlinear Schr\u00f6dinger dynamical equation","volume":"35","author":"Raza","year":"2021","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.joes.2021.08.005","article-title":"A variety of physical structures to the generalized equal-width equation derived from Wazwaz-Benjamin-Bona-Mahony model","volume":"7","author":"Jaradat","year":"2021","journal-title":"J. Ocean. Eng. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Abouelregal, A.E., and Alanazi, R. (2022). Fractional Moore-Gibson-Thompson heat transfer model with two-temperature and non-singular kernels for 3D thermoelastic solid. J. Ocean. Eng. Sci.","DOI":"10.1016\/j.joes.2022.04.008"},{"key":"ref_19","first-page":"440","article-title":"Solitary wave and other solutions for nonlinear heat equations","volume":"2","author":"Nikitin","year":"2004","journal-title":"Cent. Eur. J. Math."},{"key":"ref_20","unstructured":"Nguyen, A.T., Nikan, O., and Avazzadeh, Z. (2022). Traveling wave solutions of the nonlinear Gilson-Pickering equation in crystal lattice theory. J. Ocean Eng. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"104735","DOI":"10.1016\/j.rinp.2021.104735","article-title":"Multistability and dynamic behavior of non-linear wave solutions for analytical kink periodic and quasi-periodic wave structures in plasma physics","volume":"29","author":"Jhangeer","year":"2021","journal-title":"Results Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Karpov, P., and Brazovskii, S. (2022). Pattern Formation and Aggregation in Ensembles of Solitons in Quasi One-Dimensional Electronic Systems. Symmetry, 14.","DOI":"10.3390\/sym14050972"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"104151","DOI":"10.1016\/j.rinp.2021.104151","article-title":"Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo,Cu))","volume":"25","author":"Riaz","year":"2021","journal-title":"Results Phys."},{"key":"ref_24","first-page":"1","article-title":"Long Wave Length Soliton Solutions of Navier Stokes Equation","volume":"9","author":"Sinha","year":"2014","journal-title":"Int. J. Differ. Equ."},{"key":"ref_25","first-page":"3962","article-title":"A note on the modified simple equation method applied to Sharma-Tasso-Olver equation","volume":"218","author":"Zayed","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"104533","DOI":"10.1016\/j.rinp.2021.104533","article-title":"Sensitive behavior and optical solitons of complex fractional Ginzburg Landau equation: A comparative paradigm","volume":"28","author":"Arshed","year":"2021","journal-title":"Results Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.joes.2019.12.004","article-title":"Exact solutions of the conformable fractional EW and MEW equations by a new generalized expansion method","volume":"5","author":"Shallal","year":"2019","journal-title":"J. Ocean. Eng. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.cjph.2019.10.025","article-title":"Optical pulses with Kundu Mukherjee-Naskar model in fiber communication systems","volume":"64","author":"Yildirim","year":"2020","journal-title":"Chin. J. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2250002","DOI":"10.1142\/S0217979222500023","article-title":"A variety of fractional soliton solutions for three important coupled models arising in mathematical physics","volume":"36","author":"Arshed","year":"2021","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_30","first-page":"372","article-title":"New analytical solitary and periodic wave solutions for generalized variable-coefficients modified KdV equation with external-force term presenting atmospheric blocking in oceans","volume":"7","author":"Gaballah","year":"2021","journal-title":"J. Ocean. Eng. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"111495","DOI":"10.1016\/j.chaos.2021.111495","article-title":"On the solitary wave solution of the viscosity capillarity van der Waals p-system along with Painleve analysis","volume":"153","author":"Akbar","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2964","DOI":"10.1002\/mma.7967","article-title":"New and more dual mode solitary wave solutions for the Kraenkel-Manna-Merle system incorporating fractal effects","volume":"45","author":"Raza","year":"2022","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"164133","DOI":"10.1016\/j.ijleo.2019.164133","article-title":"Exploring the dark and singular soliton solutions of Biswas-Arshed model with full non-linear form","volume":"204","author":"Zafar","year":"2020","journal-title":"Optik"},{"key":"ref_34","first-page":"878","article-title":"New extended rational expansion method and exact solutions of Boussinesq equation and Jimbo-Miwa equations","volume":"189","author":"Wang","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_35","unstructured":"Kumar, R., and Verma, R.S. (2022). Dynamics of some new solutions to the coupled DSW equations traveling horizontally on the seabed. J. Ocean. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3243","DOI":"10.1088\/0951-7715\/28\/9\/3243","article-title":"Darboux transformation and multi-dark soliton for N-component nonlinear Schr\u00f6dinger equations","volume":"28","author":"Ling","year":"2015","journal-title":"Nonlinearity"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1578","DOI":"10.1016\/j.physleta.2019.02.031","article-title":"An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation","volume":"383","author":"Li","year":"2019","journal-title":"Phys. Lett. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1088\/0253-6102\/63\/5\/549","article-title":"Construction of Soliton-Cnoidal Wave Interaction Solution for the (2+1)-Dimensional Breaking Soliton Equation","volume":"63","author":"Cheng","year":"2015","journal-title":"Commun. Theor. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1512","DOI":"10.1016\/j.chaos.2006.03.010","article-title":"The extended homogeneous balance method and its applications for a class of nonlinear evolution equations","volume":"33","author":"Wakil","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1016\/j.rinp.2018.04.060","article-title":"An efficient numerical scheme for the study of equal width equation","volume":"9","author":"Ghafoor","year":"2018","journal-title":"Results Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.cnsns.2004.07.001","article-title":"The tanh and the sine\u2013cosine methods for a reliable treatment of the modified equal width equation and its variants","volume":"11","author":"Wazwa","year":"2006","journal-title":"Commun. Nonlinear Sci. Numer."},{"key":"ref_42","first-page":"65","article-title":"Application of Galerkin\u2019s method to equal width wave equation","volume":"160","author":"Dogan","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_43","first-page":"619","article-title":"RBF-PS scheme for solving the equal width equation","volume":"222","author":"Uddin","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/0021-9991(92)90054-3","article-title":"Solitary waves of the equal width wave equation","volume":"101","author":"Gardner","year":"1992","journal-title":"J. Comput. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.chaos.2017.06.029","article-title":"The modified extended tanh method with the Riccati equation for solving the space-time fractional EW and MEW equations","volume":"103","author":"Raslan","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"105994","DOI":"10.1016\/j.aml.2019.07.025","article-title":"Diversity of exact solutions to the conformable space-time fractional MEW equation","volume":"99","author":"Shi","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"ref_47","first-page":"1","article-title":"Different soliton solutions to the modified equal-width wave equation with Beta-time fractional derivative via two different methods","volume":"68","author":"Zafar","year":"2022","journal-title":"Rev. Mex. Fis."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"105216","DOI":"10.1016\/j.rinp.2022.105216","article-title":"Exact analytical wave solutions for space-time variable-order fractional modified equal width equation","volume":"33","author":"Ali","year":"2022","journal-title":"Results Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"111645","DOI":"10.1016\/j.chaos.2021.111645","article-title":"Observations of fractional effects of Beta-derivative and M-truncated derivative for space time fractional Phi-4 equation via two analytical techniques","volume":"154","author":"Akram","year":"2022","journal-title":"Chaos Solitonsd Fractals"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"104171","DOI":"10.1016\/j.rinp.2021.104171","article-title":"Sensitive visualization of the fractional Wazwaz-Benjamin-Bona-Mahony equation with fractional derivatives: A comparative analysis","volume":"25","author":"Raza","year":"2021","journal-title":"Results Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.chaos.2018.10.002","article-title":"M-fractional derivative under interval uncertainty: Theory, properties and applications","volume":"117","author":"Salahshour","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Zhu, X., Cheng, J., Chen, Z., and Wu, G. (2022). New Solitary-Wave Solutions of the Van der Waals Normal Form for Granular Materials via New Auxiliary Equation Method. Mathematics, 10.","DOI":"10.3390\/math10152560"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11082-019-1801-4","article-title":"A large family of optical solutions to Kundu-Eckhaus model by a new auxiliary equation method","volume":"51","author":"Rezazadeh","year":"2019","journal-title":"Opt. Quantum Electron."},{"key":"ref_54","first-page":"83","article-title":"A new truncated M-fractional derivative type unifying some fractional derivative type with classical properties","volume":"16","author":"Sousa","year":"2018","journal-title":"Int. J. Anal. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1731\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:12:06Z","timestamp":1760141526000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1731"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,19]]},"references-count":54,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081731"],"URL":"https:\/\/doi.org\/10.3390\/sym14081731","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,19]]}}}