{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T08:29:26Z","timestamp":1778142566995,"version":"3.51.4"},"reference-count":56,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,30]],"date-time":"2022-12-30T00:00:00Z","timestamp":1672358400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The unstable nonlinear Schr\u00f6dinger equations (UNLSEs) are universal equations of the class of nonlinear integrable systems, which reveal the temporal changing of disruption in slightly stable and unstable media. In current paper, an improved auxiliary equation technique is proposed to obtain the wave results of UNLSE and modified UNLSE. Numerous varieties of results are generated in the mode of some special Jacobi elliptic functions and trigonometric and hyperbolic functions, many of which are distinctive and have significant applications such as pulse propagation in optical fibers. The exact soliton solutions also give information on the soliton interaction in unstable media. Furthermore, with the assistance of the suitable parameter values, various kinds of structures such as bright-dark, multi-wave structures, breather and kink-type solitons, and several periodic solitary waves are depicted that aid in the understanding of the physical interpretation of unstable nonlinear models. The various constructed solutions demonstrate the effectiveness of the suggested approach, which proves that the current technique may be applied to other nonlinear physical problems encountered in mathematical physics.<\/jats:p>","DOI":"10.3390\/sym15010099","type":"journal-article","created":{"date-parts":[[2022,12,30]],"date-time":"2022-12-30T05:15:53Z","timestamp":1672377353000},"page":"99","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Construction of Novel Bright-Dark Solitons and Breather Waves of Unstable Nonlinear Schr\u00f6dinger Equations with Applications"],"prefix":"10.3390","volume":"15","author":[{"given":"Ambreen","family":"Sarwar","sequence":"first","affiliation":[{"name":"School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9100-6082","authenticated-orcid":false,"given":"Muhammad","family":"Arshad","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Sub-Campus Depalpur, University of Agriculture, Faisalabad 38000, Pakistan"},{"name":"Faculty of Science, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7616-0500","authenticated-orcid":false,"given":"Muhammad","family":"Farman","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Khawaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan"},{"name":"Department of Mathematics Near East Boulevard, Mathematics Research Center, Near East University, Mersin 99138, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9832-1424","authenticated-orcid":false,"given":"Ali","family":"Akg\u00fcl","sequence":"additional","affiliation":[{"name":"Department of Mathematics Near East Boulevard, Mathematics Research Center, Near East University, Mersin 99138, Turkey"},{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut 1102 2801, Lebanon"},{"name":"Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Iftikhar","family":"Ahmed","sequence":"additional","affiliation":[{"name":"Faculty of Science, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mustafa","family":"Bayram","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, Biruni University, Topkap\u0131, Istanbul 34010, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3463-2607","authenticated-orcid":false,"given":"Shahram","family":"Rezapour","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran"},{"name":"Department of Medical Research, China Medical University Hospital China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, 48940 Leioa, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mitschke, F., Mahnke, C., and Hause, A. 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