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Math."],"published-print":{"date-parts":[[2026,2,28]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this paper, we present new results regarding the orbital stability of solitary standing waves for the general fourth-order Schr\u00f6dinger equation with mixed dispersion. The existence of solitary waves can be determined both explicitly and by using a numerical approach. These explicit solutions cannot be seen as a smooth curve of solitary waves, and this fact prevents their determination of stability using classical approaches in the current literature. To overcome this difficulty, we employ a numerical approach to construct a smooth curve of solitary waves. The existence of a smooth curve is useful for showing the existence of a threshold power [Formula: see text] of the nonlinear term such that if [Formula: see text] the explicit solitary wave is stable, and if [Formula: see text], the wave is unstable. An important feature of our work, caused by the presence of the mixed dispersion term, concerns the fact that the threshold value [Formula: see text] is not the same as that established for proving the existence of global solutions in the energy space, as is well known for the classical nonlinear Schr\u00f6dinger equation.<\/jats:p>","DOI":"10.1137\/24m1665878","type":"journal-article","created":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T08:08:42Z","timestamp":1768291722000},"page":"187-204","source":"Crossref","is-referenced-by-count":2,"title":["On the Orbital Stability of Solitary Waves for the Fourth Order Nonlinear Schr\u00f6dinger Equation"],"prefix":"10.1137","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0491-9562","authenticated-orcid":true,"given":"Handan","family":"Borluk","sequence":"first","affiliation":[{"name":"Department of Mathematical Engineering, Ozyegin University, Cekmekoy, Istanbul, T\u00fcrkiye."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2268-3992","authenticated-orcid":true,"given":"Gulcin M.","family":"Muslu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Istanbul Technical University, Maslak, 34469 Istanbul, T\u00fcrkiye."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6458-2730","authenticated-orcid":true,"given":"F\u00e1bio","family":"Natali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, State University of Maring\u00e1, Maring\u00e1, PR, Brazil."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,1,13]]},"reference":[{"key":"ref1","unstructured":"M. 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