{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T14:44:03Z","timestamp":1773413043485,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,16]],"date-time":"2025-05-16T00:00:00Z","timestamp":1747353600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The well-known nonlinear Schr\u00f6dinger equation (NLSE) plays a crucial role in describing the temporal evolution of disturbances in marginally stable or unstable media. However, when the media is a fractal form, it becomes ineffective. Thus, the fractal modification to the NLSE is presented based on the fractal derivative in this work for the first time. The semi-inverse method is employed to establish the fractal variational principle. The entire process of deriving the fractal variational principle is presented in detail. To our knowledge, the fractal variational principle mentioned in this article is the first exploration and report to date. The fractal variational principle established in this paper is expected to deepen our understanding of the essence of physical phenomena in the fractal space and offer new ideas for the application and exploration of the variational approaches.<\/jats:p>","DOI":"10.3390\/axioms14050376","type":"journal-article","created":{"date-parts":[[2025,5,16]],"date-time":"2025-05-16T08:32:29Z","timestamp":1747384349000},"page":"376","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Variational Principle of the Unstable Nonlinear Schr\u00f6dinger Equation with Fractal Derivatives"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3905-0844","authenticated-orcid":false,"given":"Kang-Jia","family":"Wang","sequence":"first","affiliation":[{"name":"School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, China"}]},{"given":"Ming","family":"Li","sequence":"additional","affiliation":[{"name":"School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s12043-020-02067-9","article-title":"Study of travelling, periodic, quasiperiodic and chaotic structures of perturbed Fokas\u2013Lenells model","volume":"95","author":"Jhangeer","year":"2021","journal-title":"Pramana"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2550034","DOI":"10.1142\/S0218348X25500343","article-title":"New perspective to the coupled fractional nonlinear Schr\u00f6dinger equations in dual-core optical fibers","volume":"33","author":"Wang","year":"2025","journal-title":"Fractals"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1007\/s11082-021-02823-0","article-title":"Conservation laws, optical molecules, modulation instability and Painlev\u00e9 analysis for the Chen\u2013Lee\u2013Liu model","volume":"53","author":"Seadawy","year":"2021","journal-title":"Opt. 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