{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:51:29Z","timestamp":1760230289700,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T00:00:00Z","timestamp":1658275200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Fujian Provincial","award":["No.2022J01303"],"award-info":[{"award-number":["No.2022J01303"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The traveling wave solutions of a generalized HD type equation are investigated in this study. The traveling wave system is a singular system of the first class with given parameter conditions. From the standpoint of dynamical systems, the bifurcations of traveling wave solutions in parameter space are examined. It is demonstrated that solitary wave solutions, periodic peakons, pseudo-peakons, and compacton solutions exist. All conceivable exact explicit parametric representations of various solutions are presented.<\/jats:p>","DOI":"10.3390\/sym14071480","type":"journal-article","created":{"date-parts":[[2022,7,21]],"date-time":"2022-07-21T03:34:40Z","timestamp":1658374480000},"page":"1480","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Exact Traveling Wave Solutions in a Generalized Harry Dym Type Equation"],"prefix":"10.3390","volume":"14","author":[{"given":"Rong","family":"Wu","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yan","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.1103\/PhysRevLett.71.1661","article-title":"An integrable shallow water equation with peak solutions","volume":"71","author":"Camassa","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0065-2156(08)70254-0","article-title":"A new integrable shallow water equation","volume":"31","author":"Camassa","year":"1994","journal-title":"Adv. Appl. Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0167-2789(98)00108-0","article-title":"Nonlinear waves and solitons in physical sysytems","volume":"123","author":"Camassa","year":"1998","journal-title":"Physics D"},{"key":"ref_4","unstructured":"Li, J. (2013). Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions, Science Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"123501","DOI":"10.1063\/1.4835395","article-title":"Peakon, pseudo-peakon and cuspon solutions for two generalized Cammasa-Holm equations","volume":"54","author":"Li","year":"2013","journal-title":"J. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4049","DOI":"10.1142\/S0218127407019858","article-title":"On a class of singular nonlinear traveling wave equations","volume":"17","author":"Li","year":"2007","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1650207","DOI":"10.1142\/S0218127416502072","article-title":"Understanding peakons, periodic peakons and compactons via a shallow water wave equation","volume":"26","author":"Li","year":"2016","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1450160","DOI":"10.1142\/S0218127414501600","article-title":"Variform exact one-peakon solutions for some singular nonlinear traveling wave equations of the first kind","volume":"24","author":"Li","year":"2014","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.aml.2015.02.006","article-title":"A new integrable equation with peakons and cuspons and its bi-Hamiltonian structure","volume":"46","author":"Geng","year":"2015","journal-title":"Appl. Math. Lett."},{"key":"ref_10","unstructured":"Oono, H. (1996). N soliton solution of Harry Dym equation by inverse scattering method. Nonlinear Physics: Theory and Experiment, World Scientific Publishing."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1088\/0305-4470\/25\/1\/025","article-title":"Explicit solutions for the Harry Dym equation","volume":"25","author":"Fuchssteiner","year":"1992","journal-title":"J. Phys. A"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Byrd, P.F., and Fridman, M.D. (1971). Handbook of Elliptic Integrals for Engineers and Scientists, Springer.","DOI":"10.1007\/978-3-642-65138-0"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1480\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:54:14Z","timestamp":1760140454000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1480"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,20]]},"references-count":12,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["sym14071480"],"URL":"https:\/\/doi.org\/10.3390\/sym14071480","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,7,20]]}}}