{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T10:57:20Z","timestamp":1775818640773,"version":"3.50.1"},"reference-count":9,"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":[[2016,7]]},"abstract":"<jats:p> Propagating modes in a class of nonic derivative nonlinear Schr\u00f6dinger equations incorporating ninth order nonlinearity are investigated by the method of dynamical systems. Because the functions [Formula: see text] and [Formula: see text] in the solutions [Formula: see text], [Formula: see text] satisfy a four-dimensional integral system having two first integrals (i.e. the invariants of motion), a planar dynamical system for the squared wave amplitude [Formula: see text] can be derived in the invariant manifold of the four-dimensional integrable system. By using the bifurcation theory of dynamical systems, under different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions for this planar dynamical system can be given. Therefore, under some parameter conditions, solutions [Formula: see text] and [Formula: see text] can be exactly obtained. Thirty six exact explicit solutions of equation are derived. <\/jats:p>","DOI":"10.1142\/s0218127416501364","type":"journal-article","created":{"date-parts":[[2016,8,5]],"date-time":"2016-08-05T09:03:24Z","timestamp":1470387804000},"page":"1650136","source":"Crossref","is-referenced-by-count":6,"title":["Exact Solutions and Bifurcations in Invariant Manifolds for a Nonic Derivative Nonlinear Schr\u00f6dinger Equation"],"prefix":"10.1142","volume":"26","author":[{"given":"Jibin","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China"},{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2016,8,5]]},"reference":[{"key":"S0218127416501364BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65138-0"},{"key":"S0218127416501364BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2013.07.041"},{"key":"S0218127416501364BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"S0218127416501364BIB004","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127405014416"},{"key":"S0218127416501364BIB005","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127407019858"},{"key":"S0218127416501364BIB006","volume-title":"Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions","author":"Li J.","year":"2013"},{"key":"S0218127416501364BIB007","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.86.037601"},{"key":"S0218127416501364BIB008","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.81.094005"},{"key":"S0218127416501364BIB009","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/45\/15\/155205"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127416501364","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T18:24:36Z","timestamp":1565115876000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127416501364"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,7]]},"references-count":9,"journal-issue":{"issue":"08","published-online":{"date-parts":[[2016,8,5]]},"published-print":{"date-parts":[[2016,7]]}},"alternative-id":["10.1142\/S0218127416501364"],"URL":"https:\/\/doi.org\/10.1142\/s0218127416501364","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,7]]}}}