{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T07:34:12Z","timestamp":1770881652280,"version":"3.50.1"},"reference-count":54,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T00:00:00Z","timestamp":1687305600000},"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 study, the Jacobi elliptic function method (JEFM) and modified auxiliary equation method (MAEM) are used to investigate the solitary wave solutions of the nonlinear coupled Riemann wave (RW) equation. Nonlinear coupled partial differential equations (NLPDEs) can be transformed into a collection of algebraic equations by utilising a travelling wave transformation. This study\u2019s objective is to learn more about the non-linear coupled RW equation, which accounts for tidal waves, tsunamis, and static uniform media. The variance in the governing model\u2019s travelling wave behavior is investigated using the conformable, beta, and M-truncated derivatives (M-TD). The aforementioned methods can be used to derive solitary wave solutions for trigonometric, hyperbolic, and jacobi functions. We may produce periodic solutions, bell-form soliton, anti-bell-shape soliton, M-shaped, and W-shaped solitons by altering specific parameter values. The mathematical form of each pair of travelling wave solutions is symmetric. Lastly, in order to emphasise the impact of conformable, beta, and M-TD on the behaviour and symmetric solutions for the presented problem, the 2D and 3D representations of the analytical soliton solutions can be produced using Mathematica 10.<\/jats:p>","DOI":"10.3390\/sym15071293","type":"journal-article","created":{"date-parts":[[2023,6,22]],"date-time":"2023-06-22T01:49:32Z","timestamp":1687398572000},"page":"1293","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Dynamical Study of Coupled Riemann Wave Equation Involving Conformable, Beta, and M-Truncated Derivatives via Two Efficient Analytical Methods"],"prefix":"10.3390","volume":"15","author":[{"given":"Rimsha","family":"Ansar","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0491-1528","authenticated-orcid":false,"given":"Muhammad","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0223-4711","authenticated-orcid":false,"given":"Eman","family":"Al-Sarairah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Khalifa University, Abu Dhabi P.O. Box 127788, United Arab Emirates"},{"name":"Department of Mathematics, Al-Hussein Bin Talal University, P.O. Box 20, Ma\u2019an 71111, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5542-8694","authenticated-orcid":false,"given":"Khaled A.","family":"Gepreel","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9431-4195","authenticated-orcid":false,"given":"Mohamed S.","family":"Soliman","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/BF02824479","article-title":"Nonlinear evolution equations and ordinary differential equations of Painleve\u2019type","volume":"23","author":"Ablowitz","year":"1978","journal-title":"Lett. Nuovo Cim."},{"key":"ref_2","unstructured":"Tikhonov, A.N., and Samarskiy, A.A. (1999). Equations of Mathematical Physics, Moscow University Press."},{"key":"ref_3","unstructured":"Renardy, M., and Rogers, R.C. (2006). An Introduction to Partial Differential Equations, Springer Science & Business Media."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"264101","DOI":"10.1103\/PhysRevLett.125.264101","article-title":"Nonlinear spectral synthesis of soliton gas in deep-water surface gravity waves","volume":"125","author":"Suret","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/0032-0633(95)00109-3","article-title":"Experiments on ion-acoustic waves in dusty plasmas","volume":"44","author":"Barkan","year":"1996","journal-title":"Planet. Space Sci."},{"key":"ref_6","first-page":"1021","article-title":"Applications of fractional calculus","volume":"4","author":"Dalir","year":"2010","journal-title":"Appl. Math. Sci."},{"key":"ref_7","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_8","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_9","doi-asserted-by":"crossref","unstructured":"Arnol\u2019d, V.I. (1983). Geometrical Methods in the Theory of Ordinary Differential Equations, Springer.","DOI":"10.1007\/978-1-4684-0147-9"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1016\/j.cnsns.2016.09.006","article-title":"A Caputo fractional derivative of a function with respect to another function","volume":"44","author":"Almeida","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_11","first-page":"130","article-title":"An improved Grunwald-Letnikov fractional differential mask for image texture enhancement","volume":"3","author":"Garg","year":"2012","journal-title":"Int. J. Adv. Comput. Sci. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Onder, I., Cinar, M., Secer, A., and Bayram, M. (2022). Analytical solutions of simplified modified Camassa-Holm equation with conformable and M-truncated derivatives: A comparative study. J. Ocean. Eng. Sci., in press.","DOI":"10.1016\/j.joes.2022.06.012"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/j.chaos.2016.02.012","article-title":"Chaos in a simple nonlinear system with Atangana\u2013Baleanu derivatives with fractional order","volume":"89","author":"Atangana","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107535","DOI":"10.1155\/2014\/107535","article-title":"Extension of matched asymptotic method to fractional boundary layers problems","volume":"2014","author":"Atangana","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_15","first-page":"1","article-title":"Optical solitons of fractional complex Ginzburg\u2013Landau equation with conformable, beta, and M-truncated derivatives: A comparative study","volume":"2020","author":"Hussain","year":"2020","journal-title":"Adv. Differ. Eq."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.cjph.2018.12.010","article-title":"Physical properties of the projectile motion using the conformable derivative","volume":"58","author":"Alharbi","year":"2019","journal-title":"Chin. J. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Majid, S.Z., Faridi, W.A., Asjad, M.I., Abd El-Rahman, M., and Eldin, S.M. (2023). Explicit Soliton Structure Formation for the Riemann Wave Equation and a Sensitive Demonstration. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7020102"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"103131","DOI":"10.1016\/j.rinp.2020.103131","article-title":"Competent closed form soliton solutions to the Riemann wave equation and the Novikov-Veselov equation","volume":"17","author":"Barman","year":"2020","journal-title":"Results Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s00245-002-0751-5","article-title":"Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot","volume":"46","author":"Triggiani","year":"2002","journal-title":"Appl. Math. Optim."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2574","DOI":"10.3390\/sym14122574","article-title":"Solitons Solution of Riemann Wave Equation via Modified Exp Function Method","volume":"14","author":"Shakeel","year":"2022","journal-title":"Symmetry"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/S0375-9601(02)00180-9","article-title":"The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations","volume":"295","author":"Parkes","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"12539","DOI":"10.1016\/j.aej.2022.06.047","article-title":"Soliton solutions and fractional effects to the time-fractional modified equal width equation","volume":"61","author":"Bashar","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_23","unstructured":"Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press."},{"key":"ref_24","unstructured":"Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus, Academic Press."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1186\/s13662-017-1139-9","article-title":"A new fractional model for giving up smoking dynamics","volume":"2017","author":"Singh","year":"2017","journal-title":"Adv. Differ. Eq."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1007\/BF00879562","article-title":"A new dissipation model based on memory mechanism","volume":"91","author":"Caputo","year":"1971","journal-title":"Pure Appl. Geophys."},{"key":"ref_27","first-page":"73","article-title":"A new definition of fractional derivative without singular kernel","volume":"1","author":"Caputo","year":"2019","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2298\/TSCI160111018A","article-title":"New fractional derivatives with nonlocal and non-singular kernel. Theory and application to heat transfer model","volume":"20","author":"Atangana","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.cam.2014.01.002","article-title":"A new definition of fractional derivative","volume":"264","author":"Khalil","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/math-2015-0081","article-title":"New properties of conformable derivative","volume":"13","author":"Atangana","year":"2015","journal-title":"Open Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"105730","DOI":"10.1016\/j.cnsns.2021.105730","article-title":"An extended Kudryashov technique for solving stochastic nonlinear models with generalized conformable derivatives","volume":"97","author":"Hyder","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"106426","DOI":"10.1016\/j.rinp.2023.106426","article-title":"On nonlinear optical solitons of fractional Biswas-Arshed Model with beta derivative","volume":"48","author":"Arafat","year":"2023","journal-title":"Results Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1515\/phys-2016-0010","article-title":"Analysis of time-fractional Hunter\u2013Saxton equation: A model of neumatic liquid crystal","volume":"14","author":"Atangana","year":"2016","journal-title":"Open Phys."},{"key":"ref_34","first-page":"312","article-title":"On the mean-value theorem corresponding to a given linear homogeneous differential equation","volume":"24","year":"1922","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1080\/10652460310001600717","article-title":"Generalized Mittag-Leffler function and generalized fractional calculus operators","volume":"15","author":"Kilbas","year":"2004","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4502","DOI":"10.1103\/PhysRevLett.85.4502","article-title":"Novel soliton solutions of the nonlinear Schr\u00f6dinger equation model","volume":"85","author":"Serkin","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/S0375-9601(02)01516-5","article-title":"Applications of the Jacobi elliptic function method to special-type nonlinear equations","volume":"305","author":"Fan","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1042","DOI":"10.1016\/j.chaos.2005.04.071","article-title":"Jacobian elliptic function method for nonlinear differential-difference equations","volume":"27","author":"Dai","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/S0375-9601(01)00580-1","article-title":"Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations","volume":"289","author":"Liu","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/S0375-9601(02)01802-9","article-title":"A note on the Jacobi elliptic function expansion method","volume":"308","author":"Shen","year":"2003","journal-title":"Phys. Lett. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.physleta.2005.07.034","article-title":"New applications of developed Jacobi elliptic function expansion methods","volume":"345","author":"Liu","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Islam, S.R., and Wang, H. (2022). Some analytical soliton solutions of the nonlinear evolution equations. J. Ocean. Eng. Sci., in press.","DOI":"10.1016\/j.joes.2022.05.013"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0375-9601(96)00283-6","article-title":"Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics","volume":"216","author":"Wang","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Wang, Z.L., Sun, L.J., Hua, R., Zhang, L.H., and Wang, H.F. (2022). Lie Symmetry Analysis, Particular Solutions and Conservation Laws of Benjiamin Ono Equation. Symmetry, 14.","DOI":"10.3390\/sym14071315"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Huo, C., and Li, L. (2022). Lie Symmetry Analysis, Particular Solutions and Conservation Laws of a New Extended (3+1)-Dimensional Shallow Water Wave Equation. Symmetry, 14.","DOI":"10.3390\/sym14091855"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"168614","DOI":"10.1016\/j.ijleo.2022.168614","article-title":"The dynamical study of Biswas\u2013Arshed equation via modified auxiliary equation method","volume":"255","author":"Akram","year":"2022","journal-title":"Optik"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/j.matcom.2021.11.004","article-title":"Efficient techniques for traveling wave solutions of time-fractional Zakharov\u2013Kuznetsov equation","volume":"193","author":"Akram","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"25003","DOI":"10.1063\/1.5087647","article-title":"Dispersive long wave of nonlinear fractional Wu-Zhang system via a modified auxiliary equation method","volume":"9","author":"Khater","year":"2019","journal-title":"AIP Adv."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Khater, M.M., Attia, R.A., and Lu, D. (2018). Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions. Math. Comput. Appl., 24.","DOI":"10.3390\/mca24010001"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"106411","DOI":"10.1016\/j.rinp.2023.106411","article-title":"Analytical solitary wave solutions of a time-fractional thin-film ferroelectric material equation involving beta-derivative using modified auxiliary equation method","volume":"48","author":"Wang","year":"2023","journal-title":"Results Phys."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Mohammed, W.W., Cesarano, C., and Al-Askar, F.M. (2022). Solutions to the (4+ 1)-Dimensional Time-Fractional Fokas Equation with M-Truncated Derivative. Mathematics, 11.","DOI":"10.3390\/math11010194"},{"key":"ref_52","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_53","doi-asserted-by":"crossref","first-page":"102120","DOI":"10.1016\/j.asej.2023.102120","article-title":"A study of variation in dynamical behavior of fractional complex Ginzburg-Landau model for different fractional operators","volume":"14","author":"Akram","year":"2023","journal-title":"Ain Shams Eng. J."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"e05276","DOI":"10.1016\/j.heliyon.2020.e05276","article-title":"Dynamical analysis of long-wave phenomena for the nonlinear conformable space-time fractional (2+ 1)-dimensional AKNS equation in water wave mechanics","volume":"6","author":"Shahen","year":"2020","journal-title":"Heliyon"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1293\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:58:00Z","timestamp":1760126280000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/7\/1293"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,21]]},"references-count":54,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["sym15071293"],"URL":"https:\/\/doi.org\/10.3390\/sym15071293","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,21]]}}}