{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T09:11:25Z","timestamp":1789549885696,"version":"build-2803163510"},"reference-count":26,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,6]],"date-time":"2022-12-06T00:00:00Z","timestamp":1670284800000},"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 the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp(\u2212\u03a6(\u03b7))-function method is used in this article to show how well it can be applied to extract travelling and solitary wave solutions from higher-order nonlinear evolution equations (NLEEs) using the equations mentioned above. Trigonometric, hyperbolic, and exponential functions solitary wave solutions can be extracted using the above-mentioned technique. By changing specific values of the embedded parameters, we can obtain bell-form soliton, consolidated bell-shape soliton, compacton, singular kink soliton, flat kink shape soliton, smooth singular soliton, and other sorts of soliton solutions. The solutions are graphically illustrated in 3D and 2D for the accuracy of the outcome by using the Wolfram Mathematica 10. The verification of numerical solvers on the stability analysis of the solution is substantially aided by the analytic solutions.<\/jats:p>","DOI":"10.3390\/sym14122574","type":"journal-article","created":{"date-parts":[[2022,12,6]],"date-time":"2022-12-06T01:43:48Z","timestamp":1670291028000},"page":"2574","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Solitons Solution of Riemann Wave Equation via Modified Exp Function Method"],"prefix":"10.3390","volume":"14","author":[{"family":"Attaullah","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Basic Sciences, University of Wah, Wah Cantt 47040, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Shakeel","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Basic Sciences, University of Wah, Wah Cantt 47040, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bilal","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Basic Sciences, University of Wah, Wah Cantt 47040, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nehad Ali","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Sejong University, Seoul 05006, Republic of Korea"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0862-0648","authenticated-orcid":false,"given":"Jae Dong","family":"Chung","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Sejong University, Seoul 05006, Republic of Korea"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3557","DOI":"10.1016\/j.camwa.2012.09.006","article-title":"Benjamin-Feir-instability in nonlinear dispersive waves","volume":"64","author":"Helal","year":"2012","journal-title":"Comput. 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