{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T16:06:50Z","timestamp":1746806810174,"version":"3.40.5"},"reference-count":13,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"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":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"],"award-info":[{"award-number":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"]}],"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":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"],"award-info":[{"award-number":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Guangxi","award":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"],"award-info":[{"award-number":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"]}]},{"name":"Natural Science Foundation of Guangxi","award":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"],"award-info":[{"award-number":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"]}]},{"name":"Innovation Project of GUET Graduate Education","award":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"],"award-info":[{"award-number":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"]}]},{"name":"Innovation Project of GUET Graduate Education","award":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"],"award-info":[{"award-number":["11361017","11161013","2012GXNSFAA053003","2013GXNSFAA019010","XJYC2012021","XJYC2012022"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>This paper is contributed to explore all possible single peak solutions for the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:msup><mml:mrow><mml:mi>K<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>*<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn mathvariant=\"normal\">4,1<\/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:mrow><mml:msub><mml:mi>u<\/mml:mi><mml:mi>t<\/mml:mi><\/mml:msub><mml:mo>=<\/mml:mo><mml:msub><mml:mi>u<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:msub><mml:msup><mml:mi>u<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><mml:mo>+<\/mml:mo><mml:mn>2<\/mml:mn><mml:mi>\u03b1<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>u<\/mml:mi><mml:msub><mml:mi>u<\/mml:mi><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:msub><mml:mi>u<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:msub><mml:msub><mml:mi>u<\/mml:mi><mml:mrow><mml:mi>x<\/mml:mi><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>. Our procedure shows that the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:msup><mml:mrow><mml:mi>K<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>*<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn mathvariant=\"normal\">4,1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>equation either has peakon, cuspon, and smooth soliton solutions when sitting on a nonzero constant pedestal<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:msub><mml:mrow><mml:mtext>lim<\/mml:mtext><\/mml:mrow><mml:mrow><mml:mi>\u03be<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mo>\u00b1<\/mml:mo><mml:mi>\u221e<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mi>u<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo>\u2260<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:math>or possesses compacton solutions only when<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:msub><mml:mrow><mml:mtext>lim<\/mml:mtext><\/mml:mrow><mml:mrow><mml:mi>\u03be<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mo>\u00b1<\/mml:mo><mml:mi>\u221e<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mi>u<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:math>. We present a new smooth soliton solution in an explicit form. Mathematical analysis and numeric graphs are provided for those soliton solutions of the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:msup><mml:mrow><mml:mi>K<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>*<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn mathvariant=\"normal\">4,1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>equation.<\/jats:p>","DOI":"10.1155\/2013\/518415","type":"journal-article","created":{"date-parts":[[2013,12,14]],"date-time":"2013-12-14T16:00:44Z","timestamp":1387036844000},"page":"1-10","source":"Crossref","is-referenced-by-count":3,"title":["Peaked and Smooth Solitons for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:msup><mml:mrow><mml:mi>K<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>*<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>4,1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>Equation"],"prefix":"10.1155","volume":"2013","author":[{"given":"Yongan","family":"Xie","sequence":"first","affiliation":[{"name":"School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hualiang","family":"Fu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shengqiang","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"doi-asserted-by":"publisher","key":"1","DOI":"10.1103\/PhysRevE.48.4027"},{"doi-asserted-by":"publisher","key":"2","DOI":"10.1103\/PhysRevLett.70.564"},{"doi-asserted-by":"publisher","key":"3","DOI":"10.1016\/j.amc.2005.09.089"},{"doi-asserted-by":"publisher","key":"4","DOI":"10.1016\/j.amc.2009.09.019"},{"doi-asserted-by":"publisher","key":"5","DOI":"10.1142\/S0218127410025582"},{"doi-asserted-by":"publisher","key":"6","DOI":"10.1155\/2012\/486458"},{"doi-asserted-by":"publisher","key":"7","DOI":"10.1155\/2012\/575387"},{"doi-asserted-by":"publisher","key":"8","DOI":"10.1007\/s12648-013-0255-y"},{"doi-asserted-by":"publisher","key":"9","DOI":"10.1209\/epl\/i2005-10453-y"},{"doi-asserted-by":"publisher","key":"10","DOI":"10.1007\/s11040-007-9027-2"},{"doi-asserted-by":"publisher","key":"11","DOI":"10.1016\/j.jmaa.2010.04.018"},{"doi-asserted-by":"publisher","key":"12","DOI":"10.1016\/j.amc.2011.10.025"},{"year":"1954","key":"13"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/518415.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/518415.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/518415.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,30]],"date-time":"2016-07-30T09:24:39Z","timestamp":1469870679000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2013\/518415\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":13,"alternative-id":["518415","518415"],"URL":"https:\/\/doi.org\/10.1155\/2013\/518415","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2013]]}}}