{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T15:35:12Z","timestamp":1746804912869,"version":"3.40.5"},"reference-count":16,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100005273","name":"Natural Science Foundation of Yunnan Province","doi-asserted-by":"publisher","award":["2013FZ117","11161020"],"award-info":[{"award-number":["2013FZ117","11161020"]}],"id":[{"id":"10.13039\/501100005273","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2013FZ117","11161020"],"award-info":[{"award-number":["2013FZ117","11161020"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>We consider a modification of the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>K<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>2,2<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>equation<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:msub><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><mml:mi>u<\/mml:mi><mml:msub><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><mml:mi>x<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mn>2<\/mml:mn><mml:mi>k<\/mml:mi><mml:msub><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msub><mml:msub><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mn>2<\/mml:mn><mml:mi>u<\/mml:mi><mml:msub><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math>using the bifurcation method of dynamical systems and the method of phase portraits analysis. From dynamic point of view, some peakons, solitary, and smooth periodic waves are found and their exact parametric representations are presented. Also, the coexistence of peakon and solitary wave solutions is investigated.<\/jats:p>","DOI":"10.1155\/2014\/130520","type":"journal-article","created":{"date-parts":[[2014,2,23]],"date-time":"2014-02-23T16:01:53Z","timestamp":1393171313000},"page":"1-12","source":"Crossref","is-referenced-by-count":0,"title":["Peakon, Solitary and Smooth Periodic Wave Solutions for a Modification of the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>K<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>2,2<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>Equation"],"prefix":"10.1155","volume":"2014","author":[{"given":"Qing","family":"Meng","sequence":"first","affiliation":[{"name":"Department of Physics, Honghe University, Mengzi, Yunnan 661100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bin","family":"He","sequence":"additional","affiliation":[{"name":"College of Mathematics, Honghe University, Mengzi, Yunnan 661100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"doi-asserted-by":"publisher","key":"1","DOI":"10.1103\/PhysRevLett.70.564"},{"doi-asserted-by":"publisher","key":"2","DOI":"10.1016\/S0375-9601(97)00241-7"},{"doi-asserted-by":"publisher","key":"3","DOI":"10.1103\/PhysRevLett.73.1737"},{"doi-asserted-by":"publisher","key":"5","DOI":"10.1016\/S0167-2789(98)00148-1"},{"doi-asserted-by":"publisher","key":"6","DOI":"10.1016\/S0096-3003(01)00233-8"},{"doi-asserted-by":"publisher","key":"7","DOI":"10.1016\/S0096-3003(01)00234-X"},{"doi-asserted-by":"publisher","key":"8","DOI":"10.1016\/j.amc.2004.06.004"},{"doi-asserted-by":"publisher","key":"9","DOI":"10.1016\/j.cam.2006.07.026"},{"doi-asserted-by":"publisher","key":"10","DOI":"10.1016\/j.camwa.2010.01.013"},{"doi-asserted-by":"publisher","key":"11","DOI":"10.1016\/j.physleta.2008.05.002"},{"doi-asserted-by":"publisher","key":"12","DOI":"10.1016\/j.cnsns.2010.09.009"},{"key":"13","doi-asserted-by":"crossref","first-page":"1825","DOI":"10.5506\/APhysPolB.44.1825","volume":"44","year":"2013","journal-title":"Acta Physica Polonica B"},{"year":"2007","key":"15"},{"doi-asserted-by":"publisher","key":"16","DOI":"10.1016\/S0307-904X(00)00031-7"},{"doi-asserted-by":"publisher","key":"17","DOI":"10.12785\/amis\/070409"},{"doi-asserted-by":"publisher","key":"18","DOI":"10.1016\/j.nonrwa.2008.11.009"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/130520.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/130520.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/130520.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,22]],"date-time":"2017-06-22T04:26:13Z","timestamp":1498105573000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/130520\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":16,"alternative-id":["130520","130520"],"URL":"https:\/\/doi.org\/10.1155\/2014\/130520","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2014]]}}}