{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T14:41:59Z","timestamp":1771339319956,"version":"3.50.1"},"reference-count":29,"publisher":"World Scientific Pub Co Pte Lt","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2010,8]]},"abstract":"<jats:p>In this paper, through qualitative analysis and integration, we study the explicit periodic wave solutions and their bifurcations for the generalized Camassa\u2013Holm equation [Formula: see text] When the parameter k satisfies k &lt; 3\/8 and the constant wave speed c satisfies [Formula: see text], we obtain two types of explicit periodic wave solutions, elliptic smooth periodic wave solution and elliptic periodic blow-up solutions. These solutions include a bifurcation parameter \u03b1 which has four bifurcation values \u03b1<jats:sub>i<\/jats:sub>(i = 1, 2, 3, 4). When \u03b1 tends to the bifurcation values, the elliptic periodic wave solutions become three types of other solutions, the hyperbolic smooth solitary wave solution, the hyperbolic blow-up solution and the trigonometric periodic blow-up solution. Especially, a new bifurcation phenomenon is found, that is, the periodic blow-up solution can become a smooth solitary wave solution when \u03b1 varies. When k &gt; 3\/8, we guess that there is no other explicit solution except the explicit periodic blow-up solution.<\/jats:p>","DOI":"10.1142\/s0218127410027131","type":"journal-article","created":{"date-parts":[[2010,9,7]],"date-time":"2010-09-07T11:21:59Z","timestamp":1283858519000},"page":"2507-2519","source":"Crossref","is-referenced-by-count":20,"title":["EXPLICIT PERIODIC WAVE SOLUTIONS AND THEIR BIFURCATIONS FOR GENERALIZED CAMASSA\u2013HOLM EQUATION"],"prefix":"10.1142","volume":"20","author":[{"given":"ZHENGRONG","family":"LIU","sequence":"first","affiliation":[{"name":"Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, P. R. China"}]},{"given":"HAO","family":"TANG","sequence":"additional","affiliation":[{"name":"Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, P. R. 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