{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:41:26Z","timestamp":1753886486342,"version":"3.41.2"},"reference-count":16,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,10,31]],"date-time":"2012-10-31T00:00:00Z","timestamp":1351641600000},"content-version":"vor","delay-in-days":304,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["40890150","40890153","11271090","2008B080701042","S2012010010121"],"award-info":[{"award-number":["40890150","40890153","11271090","2008B080701042","S2012010010121"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["40890150","40890153","11271090","2008B080701042","S2012010010121"],"award-info":[{"award-number":["40890150","40890153","11271090","2008B080701042","S2012010010121"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["40890150","40890153","11271090","2008B080701042","S2012010010121"],"award-info":[{"award-number":["40890150","40890153","11271090","2008B080701042","S2012010010121"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Program of Guangdong Province","award":["40890150","40890153","11271090","2008B080701042","S2012010010121"],"award-info":[{"award-number":["40890150","40890153","11271090","2008B080701042","S2012010010121"]}]},{"name":"Natural Science foundation of Guangdong Province","award":["40890150","40890153","11271090","2008B080701042","S2012010010121"],"award-info":[{"award-number":["40890150","40890153","11271090","2008B080701042","S2012010010121"]}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>This paper is concerned with the system of Zakharov equations which involves the interactions between Langmuir and ion\u2010acoustic waves in plasma. Abundant explicit and exact solutions of the system of Zakharov equations are derived uniformly by using the first integral method. These exact solutions are include that of the solitary wave solutions of bell\u2010type for <jats:italic>n<\/jats:italic> and <jats:italic>E<\/jats:italic>, the solitary wave solutions of kink\u2010type for <jats:italic>E<\/jats:italic> and bell\u2010type for <jats:italic>n<\/jats:italic>, the singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, and Weierstrass elliptic function doubly periodic wave solutions. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.<\/jats:p>","DOI":"10.1155\/2012\/818345","type":"journal-article","created":{"date-parts":[[2012,10,31]],"date-time":"2012-10-31T21:01:47Z","timestamp":1351717307000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The First\u2010Integral Method and Abundant Explicit Exact Solutions to the Zakharov Equations"],"prefix":"10.1155","volume":"2012","author":[{"given":"Yadong","family":"Shang","sequence":"first","affiliation":[]},{"given":"Xiaoxiao","family":"Zheng","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,10,31]]},"reference":[{"key":"e_1_2_6_1_2","first-page":"908","article-title":"Collapse of Langmuir waves","volume":"35","author":"Zakharov V. 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