{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T04:53:38Z","timestamp":1762059218048,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,7]],"date-time":"2022-07-07T00:00:00Z","timestamp":1657152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NSF of China","doi-asserted-by":"publisher","award":["11901111","2021KQNCX130","ST202101"],"award-info":[{"award-number":["11901111","2021KQNCX130","ST202101"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Young Innovative Talents Project of Guangdong Universities","award":["11901111","2021KQNCX130","ST202101"],"award-info":[{"award-number":["11901111","2021KQNCX130","ST202101"]}]},{"name":"Science Research Group Project of SEIG","award":["11901111","2021KQNCX130","ST202101"],"award-info":[{"award-number":["11901111","2021KQNCX130","ST202101"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we prove that the \u27e8p,q\u27e9 condition holds, first by using the Fuchs index of the complex Kawahara equation, and then proving that all meromorphic solutions of complex Kawahara equations belong to the class W. Moreover, the complex method is employed to get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Kawahara equation. Our results reveal that all rational solutions ur(x+\u03bdt) and simply periodic solutions us,1(x+\u03bdt) of Kawahara equation are solitary wave solutions, while simply periodic solutions us,2(x+\u03bdt) are not real-valued. Finally, computer simulations are given to demonstrate the main results of this paper. At the same time, we believe that this method is a very effective and powerful method of looking for exact solutions to the mathematical physics equations, and the search process is simpler than other methods.<\/jats:p>","DOI":"10.3390\/axioms11070330","type":"journal-article","created":{"date-parts":[[2022,7,7]],"date-time":"2022-07-07T22:11:47Z","timestamp":1657231907000},"page":"330","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method"],"prefix":"10.3390","volume":"11","author":[{"given":"Feng","family":"Ye","sequence":"first","affiliation":[{"name":"Department of Basic Courses Teaching, Software Engineering Institute of Guangzhou, Guangzhou 510990, China"}]},{"given":"Jian","family":"Tian","sequence":"additional","affiliation":[{"name":"Department of Basic Courses Teaching, Software Engineering Institute of Guangzhou, Guangzhou 510990, China"}]},{"given":"Xiaoting","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Basic Courses Teaching, Software Engineering Institute of Guangzhou, Guangzhou 510990, China"}]},{"given":"Chunling","family":"Jiang","sequence":"additional","affiliation":[{"name":"Department of Basic Courses Teaching, Software Engineering Institute of Guangzhou, Guangzhou 510990, China"}]},{"given":"Tong","family":"Ouyang","sequence":"additional","affiliation":[{"name":"Department of Basic Courses Teaching, Software Engineering Institute of Guangzhou, Guangzhou 510990, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6651-1714","authenticated-orcid":false,"given":"Yongyi","family":"Gu","sequence":"additional","affiliation":[{"name":"School of statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510006, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1143\/JPSJ.33.260","article-title":"Oscillatory solitary waves in dispersive media","volume":"33","author":"Kawahara","year":"1972","journal-title":"J. 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