{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T13:07:24Z","timestamp":1774530444332,"version":"3.50.1"},"reference-count":52,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,8,30]],"date-time":"2021-08-30T00:00:00Z","timestamp":1630281600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Liaoning BaiQianWan Talents Program of China","award":["2019"],"award-info":[{"award-number":["2019"]}]},{"name":"Natural Science Foundation of Education Department of Liaoning Province of China","award":["LJ2020002"],"award-info":[{"award-number":["LJ2020002"]}]},{"name":"Natural Science Foundation of Xinjiang Autonomous Region of China","award":["2020D01B01"],"award-info":[{"award-number":["2020D01B01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the N-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schr\u00f6dinger (NLS)-type equation, are obtained by extending the Riemann\u2013Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales.<\/jats:p>","DOI":"10.3390\/sym13091593","type":"journal-article","created":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T22:58:15Z","timestamp":1630450695000},"page":"1593","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Riemann\u2013Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schr\u00f6dinger Type Equation"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7956-4882","authenticated-orcid":false,"given":"Bo","family":"Xu","sequence":"first","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"School of Educational Sciences, Bohai University, Jinzhou 121013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sheng","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Bohai University, Jinzhou 121013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.rinp.2018.06.011","article-title":"Fractal calculus and its geometrical explanation","volume":"10","author":"He","year":"2018","journal-title":"Results Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"S145","DOI":"10.2298\/TSCI11S1145H","article-title":"A new fractal derivation","volume":"15","author":"He","year":"2011","journal-title":"Therm. 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