{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,9]],"date-time":"2026-09-09T08:22:49Z","timestamp":1788942169800,"version":"build-2803163510"},"reference-count":39,"publisher":"AIP Publishing","issue":"3","funder":[{"DOI":"10.13039\/501100004070","name":"ITB Research Program 2026 under the ITB International Research Scheme through the Directorate of Research and Innovation, Institut Teknologi Bandung","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004070","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004070","name":"Research & Innovation","doi-asserted-by":"publisher","award":["KU-INT RIG-2024-8474000789"],"award-info":[{"award-number":["KU-INT RIG-2024-8474000789"]}],"id":[{"id":"10.13039\/501100004070","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,3,1]]},"abstract":"<jats:p>We investigate the spectral and dynamical properties of the fractional nonlinear Schr\u00f6dinger equation with harmonic confinement. In this setting, the classical Laplacian is replaced by its fractional power (\u2212\u2202x2)\u03b1\/2 with \u03b1\u2208(1,2], introducing nonlocal, L\u00e9vy-type dispersion. This modification fundamentally alters the balance between nonlinearity, dispersion, and trapping, reshaping both the structure and stability of stationary states. Using a Fourier pseudo-spectral discretization, we compute stationary branches as functions of the temporal frequency \u03a9 in focusing (\u03c3=+1) and defocusing (\u03c3=\u22121) regimes, and assess spectral stability via the linearized eigenvalue problem. Direct simulations, performed with split-step and exponential time-differencing integrators, confirm these predictions and reveal \u03b1-dependent transitions between coherent oscillations, bounded breathing dynamics, and decoherence or fragmentation. Our results show that decreasing \u03b1 systematically shifts bifurcation curves, fragments stability windows for excited states, and amplifies instability in the focusing regime, while supporting robust coherence in the defocusing case. Beyond clarifying how harmonic confinement mediates the interplay between nonlinearity and fractional dispersion, the study also provides benchmarks for numerical treatments of fractional operators and points toward potential applications in nonlinear optics, Bose\u2013Einstein condensates, and anomalous transport phenomena.<\/jats:p>","DOI":"10.1063\/5.0307515","type":"journal-article","created":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T13:52:31Z","timestamp":1772545951000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Spectral and dynamical properties of the fractional nonlinear Schr\u00f6dinger equation under harmonic confinement"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8455-1963","authenticated-orcid":false,"given":"R.","family":"Kusdiantara","sequence":"first","affiliation":[{"name":"Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung 1 , Bandung 40132,","place":["Indonesia"]},{"name":"Center of Excellence in Predictive Risk and Simulation Modeling, Institut Teknologi Bandung 2 , Bandung 40132,","place":["Indonesia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-8030-9793","authenticated-orcid":false,"given":"M. F.","family":"Adhari","sequence":"additional","affiliation":[{"name":"Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung 1 , Bandung 40132,","place":["Indonesia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-8593-8670","authenticated-orcid":false,"given":"H. A.","family":"Mardi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Universitas Syiah Kuala 3 , Banda Aceh 23111,","place":["Indonesia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2116-498X","authenticated-orcid":false,"given":"I. W.","family":"Sudiarta","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Mataram 4 , Mataram 83125,","place":["Indonesia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0425-107X","authenticated-orcid":false,"given":"H.","family":"Susanto","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Khalifa University 5 , PO Box, 127788 Abu Dhabi,","place":["United Arab Emirates"]},{"name":"Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia 6 , Gedung D Lt. 2 FMIPA Kampus UI, Depok 16424,","place":["Indonesia"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,3,3]]},"reference":[{"key":"2026030308522776100_c1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walk\u2019s guide to anomalous diffusion: A fractional dynamics approach","volume":"339","year":"2000","journal-title":"Phys. Rep."},{"key":"2026030308522776100_c2","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/0370-1573(90)90099-N","article-title":"Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications","volume":"195","year":"1990","journal-title":"Phys. Rep."},{"key":"2026030308522776100_c3","volume-title":"Physics of Fractal Operators","year":"2003"},{"key":"2026030308522776100_c4","volume-title":"L\u00e9vy Flights and Related Topics in Physics","year":"1995"},{"key":"2026030308522776100_c5","volume-title":"Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media","year":"2011"},{"key":"2026030308522776100_c6","doi-asserted-by":"publisher","first-page":"056108","DOI":"10.1103\/PhysRevE.66.056108","article-title":"Fractional Schr\u00f6dinger equation","volume":"66","year":"2002","journal-title":"Phys. Rev. E"},{"key":"2026030308522776100_c7","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.1364\/OL.40.001117","article-title":"Fractional Schr\u00f6dinger equation in optics","volume":"40","year":"2015","journal-title":"Opt. Lett."},{"key":"2026030308522776100_c8","doi-asserted-by":"publisher","first-page":"353","DOI":"10.3390\/photonics8090353","article-title":"Optical solitons and vortices in fractional media: A mini-review of recent results","volume":"8","year":"2021","journal-title":"Photonics"},{"key":"2026030308522776100_c9","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1016\/j.bulsci.2011.12.004","article-title":"Hitchhiker\u2019s guide to the fractional Sobolev spaces","volume":"136","year":"2012","journal-title":"Bull. Sci. Math."},{"key":"2026030308522776100_c10","doi-asserted-by":"publisher","first-page":"2140003","DOI":"10.1142\/S0217732321400034","article-title":"Fractional Schr\u00f6dinger equation in gravitational optics","volume":"36","year":"2021","journal-title":"Mod. Phys. Lett. A"},{"key":"2026030308522776100_c11","doi-asserted-by":"publisher","first-page":"5469","DOI":"10.1038\/s41467-025-60625-4","article-title":"Nonlinear wave propagation governed by a fractional derivative","volume":"16","year":"2025","journal-title":"Nat. Commun."},{"key":"2026030308522776100_c12","doi-asserted-by":"publisher","first-page":"eaea5872","DOI":"10.1126\/sciadv.aea5872","article-title":"Nyquist\u2013Hilbert\u2013nonlinear Schr\u00f6dinger solitons: A continuous family of fractional nonlinear waves","volume":"12","year":"2026","journal-title":"Sci. Adv."},{"key":"2026030308522776100_c13","doi-asserted-by":"publisher","first-page":"014222","DOI":"10.1103\/PhysRevE.109.014222","article-title":"Spin-orbit-coupled fractional oscillators and trapped Bose-Einstein condensates","volume":"109","year":"2024","journal-title":"Phys. Rev. E"},{"key":"2026030308522776100_c14","doi-asserted-by":"publisher","first-page":"012120","DOI":"10.1103\/PhysRevE.88.012120","article-title":"Potential condensed-matter realization of space-fractional quantum mechanics: The one-dimensional L\u00e9vy crystal","volume":"88","year":"2013","journal-title":"Phys. Rev. E: Stat. Nonlinear Soft Matter Phys."},{"key":"2026030308522776100_c15","unstructured":"B. P.\n            Epps\n           and B.Cushman-Roisin, \u201cTurbulence modeling via the fractional Laplacian,\u201d arXiv:1803.05286 (2018)."},{"key":"2026030308522776100_c16","doi-asserted-by":"publisher","first-page":"013131","DOI":"10.1063\/5.0242142","article-title":"The fractional nonlinear Schr\u00f6dinger equation: Soliton turbulence, modulation instability, and extreme rogue waves","volume":"35","year":"2025","journal-title":"Chaos"},{"key":"2026030308522776100_c17","doi-asserted-by":"publisher","first-page":"170403","DOI":"10.1103\/tx71-1cd9","article-title":"Measurement-induced L\u00e9vy flights of quantum information","volume":"135","year":"2025","journal-title":"Phys. Rev. Lett."},{"key":"2026030308522776100_c18","doi-asserted-by":"publisher","first-page":"222","DOI":"10.1038\/s41467-023-35892-8","article-title":"Experimental realisations of the fractional Schr\u00f6dinger equation in the temporal domain","volume":"14","year":"2023","journal-title":"Nat. Commun."},{"key":"2026030308522776100_c19","doi-asserted-by":"publisher","first-page":"2401714","DOI":"10.1002\/lpor.202401714","article-title":"Experimental emulator of pulse dynamics in fractional nonlinear Schr\u00f6dinger equation","volume":"19","year":"2025","journal-title":"Laser Photon. Rev."},{"key":"2026030308522776100_c20","doi-asserted-by":"publisher","first-page":"5943","DOI":"10.1364\/AO.40.005943","article-title":"Fractional derivatives\u2014Analysis and experimental implementation","volume":"40","year":"2001","journal-title":"Appl. Opt."},{"key":"2026030308522776100_c21","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1016\/S0375-9601(00)00774-X","article-title":"Nonlinear modes of a macroscopic quantum oscillator","volume":"278","year":"2001","journal-title":"Phys. Lett. A"},{"key":"2026030308522776100_c22","doi-asserted-by":"publisher","first-page":"013606","DOI":"10.1103\/PhysRevA.78.013606","article-title":"Stability of excited states of a Bose\u2013Einstein condensate in an anharmonic trap","volume":"78","year":"2008","journal-title":"Phys. Rev. A"},{"key":"2026030308522776100_c23","doi-asserted-by":"publisher","first-page":"1674","DOI":"10.1137\/S1064827503422956","article-title":"Computing the ground state solution of Bose\u2013Einstein condensates by a normalized gradient flow","volume":"25","year":"2004","journal-title":"SIAM J. Sci. Comput."},{"key":"2026030308522776100_c24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3934\/krm.2013.6.1","article-title":"Mathematical theory and numerical methods for Bose\u2013Einstein condensation","volume":"6","year":"2013","journal-title":"Kinet. Relat. Model."},{"key":"2026030308522776100_c25","doi-asserted-by":"publisher","first-page":"095003","DOI":"10.1088\/0953-4075\/44\/9\/095003","article-title":"Josephson tunnelling of dark solitons in a double-well potential","volume":"44","year":"2011","journal-title":"J. Phys. B: At. Mol. Opt. Phys."},{"key":"2026030308522776100_c26"},{"key":"2026030308522776100_c27","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.physd.2016.05.015","article-title":"Fractional Schr\u00f6dinger dynamics and decoherence","volume":"332","year":"2016","journal-title":"Physica D"},{"key":"2026030308522776100_c28","doi-asserted-by":"publisher","first-page":"110222","DOI":"10.1016\/j.chaos.2020.110222","article-title":"Stabilization of single- and multi-peak solitons in the fractional nonlinear Schr\u00f6dinger equation with a trapping potential","volume":"140","year":"2020","journal-title":"Chaos, Solitons Fractals"},{"key":"2026030308522776100_c29","doi-asserted-by":"publisher","first-page":"180403","DOI":"10.1103\/PhysRevLett.115.180403","article-title":"Propagation dynamics of a light beam in a fractional Schr\u00f6dinger equation","volume":"115","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"2026030308522776100_c30","volume-title":"The Nonlinear Schr\u00f6dinger Equation: Self-Focusing and Wave Collapse","year":"2007"},{"key":"2026030308522776100_c31","doi-asserted-by":"publisher","first-page":"1394","DOI":"10.3390\/sym13081394","article-title":"Dynamics of plane waves in the fractional nonlinear Schr\u00f6dinger equation with long-range dispersion","volume":"13","year":"2021","journal-title":"Symmetry"},{"key":"2026030308522776100_c32","doi-asserted-by":"publisher","first-page":"20140364","DOI":"10.1098\/rspa.2014.0364","article-title":"Numerical study of fractional nonlinear Schr\u00f6dinger equations","volume":"470","year":"2014","journal-title":"Proc. R. Soc. A"},{"key":"2026030308522776100_c33","doi-asserted-by":"publisher","first-page":"022211","DOI":"10.1103\/PhysRevE.98.022211","article-title":"Optical solitons, self-focusing, and wave collapse in a space-fractional Schr\u00f6dinger equation with a Kerr-type nonlinearity","volume":"98","year":"2018","journal-title":"Phys. Rev. E"},{"key":"2026030308522776100_c34","doi-asserted-by":"publisher","first-page":"2569","DOI":"10.1016\/j.jfa.2016.08.011","article-title":"Blowup for fractional NLS","volume":"271","year":"2016","journal-title":"J. Funct. Anal."},{"key":"2026030308522776100_c35","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00332-025-10145-8","article-title":"Numerical evidence for singularity formation in defocusing fractional NLS in one space dimension","volume":"35","year":"2025","journal-title":"J. Nonlinear Sci."},{"key":"2026030308522776100_c36","volume-title":"Spectral and High-Order Methods with Applications","year":"2006"},{"key":"2026030308522776100_c37","doi-asserted-by":"publisher","first-page":"043840","DOI":"10.1103\/PhysRevA.85.043840","article-title":"Nonlinear modes in the harmonic PT-symmetric potential","volume":"85","year":"2012","journal-title":"Phys. Rev. A: At. Mol. Opt. Phys."},{"key":"2026030308522776100_c38","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1016\/j.cnsns.2018.06.019","article-title":"Localized modes in the Gross-Pitaevskii equation with a parabolic trapping potential and a nonlinear lattice pseudopotential","volume":"66","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"2026030308522776100_c39","doi-asserted-by":"publisher","first-page":"115254","DOI":"10.1016\/j.chaos.2024.115254","article-title":"Three-dimensional solitons in fractional nonlinear Schr\u00f6dinger equation with exponential saturating nonlinearity","volume":"186","year":"2024","journal-title":"Chaos, Solitons Fractals"}],"container-title":["Chaos: An Interdisciplinary Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0307515\/20925498\/033109_1_5.0307515.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0307515\/20925498\/033109_1_5.0307515.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T13:52:37Z","timestamp":1772545957000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/cha\/article\/36\/3\/033109\/3381975\/Spectral-and-dynamical-properties-of-the"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,1]]},"references-count":39,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0307515","relation":{},"ISSN":["1054-1500","1089-7682"],"issn-type":[{"value":"1054-1500","type":"print"},{"value":"1089-7682","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,3]]},"published":{"date-parts":[[2026,3,1]]},"article-number":"033109"}}