{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:13:48Z","timestamp":1760148828193,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T00:00:00Z","timestamp":1686268800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Fundamental Research Funds for the Central Universities","award":["2018ZY14","2019ZY20","2015ZCQ-LY-01","YETP0769","61571002","61179034","61370193"],"award-info":[{"award-number":["2018ZY14","2019ZY20","2015ZCQ-LY-01","YETP0769","61571002","61179034","61370193"]}]},{"name":"the Beijing Higher Education Young Elite Teacher Project","award":["2018ZY14","2019ZY20","2015ZCQ-LY-01","YETP0769","61571002","61179034","61370193"],"award-info":[{"award-number":["2018ZY14","2019ZY20","2015ZCQ-LY-01","YETP0769","61571002","61179034","61370193"]}]},{"name":"the National Natural Science Foundation of China","award":["2018ZY14","2019ZY20","2015ZCQ-LY-01","YETP0769","61571002","61179034","61370193"],"award-info":[{"award-number":["2018ZY14","2019ZY20","2015ZCQ-LY-01","YETP0769","61571002","61179034","61370193"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Schr\u00f6dinger equation is one of the most basic equations in quantum mechanics. In this paper, we study the convergence of symmetric discretization models for the nonlinear Schr\u00f6dinger equation in dark solitons\u2019 motion and verify the theoretical results through numerical experiments. Via the second-order symmetric difference, we can obtain two popular space-symmetric discretization models of the nonlinear Schr\u00f6dinger equation in dark solitons\u2019 motion: the direct-discrete model and the Ablowitz\u2013Ladik model. Furthermore, applying the midpoint scheme with symmetry to the space discretization models, we obtain two time\u2013space discretization models: the Crank\u2013Nicolson method and the new difference method. Secondly, we demonstrate that the solutions of the two space-symmetric discretization models converge to the solution of the nonlinear Schr\u00f6dinger equation. Additionally, we prove that the convergence order of the two time\u2013space discretization models is O(h2+\u03c42) in discrete L2-norm error estimates. Finally, we present some numerical experiments to verify the theoretical results and show that our numerical experiments agree well with the proven theoretical results.<\/jats:p>","DOI":"10.3390\/sym15061229","type":"journal-article","created":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T03:30:03Z","timestamp":1686281403000},"page":"1229","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Convergence of Symmetric Discretization Models for Nonlinear Schr\u00f6dinger Equation in Dark Solitons\u2019 Motion"],"prefix":"10.3390","volume":"15","author":[{"given":"Yazhuo","family":"Li","sequence":"first","affiliation":[{"name":"College of Science, Beijing Forestry University, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qian","family":"Luo","sequence":"additional","affiliation":[{"name":"College of Science, Beijing Forestry University, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Quandong","family":"Feng","sequence":"additional","affiliation":[{"name":"College of Science, Beijing Forestry University, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ablowitz, M.J., and Segur, H. (1981). Solitons and the Inverse Scattering Transform, SIAM.","DOI":"10.1137\/1.9781611970883"},{"key":"ref_2","unstructured":"Dodd, R.K., Eilbeck, J.C., Gibbon, J.D., and Morris, H.C. (1982). Solitons and Nonlinear Wave Equations, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hasegawa, A. (1989). Optical Solitons in Fibers, Springer.","DOI":"10.1007\/BFb0041283"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Konotop, V.V. (1994). Nonlinear Random Waves, World Scientific.","DOI":"10.1142\/2320"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1088\/0305-4470\/24\/4\/013","article-title":"Randomly modulated dark soliton","volume":"24","author":"Konotop","year":"1991","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_6","first-page":"823","article-title":"Interaction between solitons in a stable medium","volume":"37","author":"Zakharov","year":"1973","journal-title":"Sov. Phys. JETP"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1090\/S0025-5718-1984-0744922-X","article-title":"Methods for the numerical solution of the nonlinear Schr\u00f6dinger equation","volume":"43","year":"1984","journal-title":"Math. Comput."},{"key":"ref_8","first-page":"178","article-title":"A high accurate and conservative finite difference scheme for nonlinear Schr\u00f6dinger equation","volume":"28","author":"Zhang","year":"2005","journal-title":"Acta Math. Appl. Sin."},{"key":"ref_9","first-page":"165","article-title":"Numerical simulation of nonlinear Schr\u00f6dinger systems: A new conservative scheme","volume":"71","author":"Fei","year":"1995","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.jcp.2004.11.001","article-title":"Local discontinuous Galerkin methods for nonlinear Schr\u00f6dinger equations","volume":"205","author":"Xu","year":"2005","journal-title":"J. Comput. Phys."},{"key":"ref_11","first-page":"190","article-title":"A discrete Adomian decomposition method for discrete nonlinear Schr\u00f6dinger equations","volume":"197","author":"Bratsos","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_12","first-page":"73","article-title":"Homotopy perturbation method: A new nonlinear analytical technique","volume":"135","author":"He","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1093\/imanum\/13.1.115","article-title":"Finite difference discretization of the cubic Schr\u00f6dinger equation","volume":"13","author":"Akrivis","year":"1993","journal-title":"IMA J. Numer. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2026","DOI":"10.1016\/j.optcom.2010.01.046","article-title":"Numerical study of nonlinear Schr\u00f6dinger and coupled Schr\u00f6dinger equations by differential transformation method","volume":"283","author":"Borhanifar","year":"2010","journal-title":"Opt. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1007\/s11075-019-00708-8","article-title":"Symplectic simulation of dark solitons motion for nonlinear Schr\u00f6dinger equation","volume":"81","author":"Zhu","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_16","first-page":"2150056","article-title":"Symplectic schemes and symmetric schemes for nonlinear Schr\u00f6dinger equation in the case of dark solitons motion","volume":"12","author":"Yao","year":"2021","journal-title":"Int. J. Model. Simul. Sci."},{"key":"ref_17","first-page":"277","article-title":"Implementing arbitrarily high-order symplectic methods via Krylov deferred correction technique","volume":"1","author":"Feng","year":"2010","journal-title":"Int. J. Model. Simul. Sci."},{"key":"ref_18","first-page":"116","article-title":"Implicit difference schemes for the generalized non-linear Schr\u00f6dinger system","volume":"1","author":"Zhu","year":"1983","journal-title":"J. Comput. Math."},{"key":"ref_19","first-page":"121","article-title":"The convergence of numerical method for nonlinear Schrodinger equation","volume":"4","author":"Guo","year":"1986","journal-title":"J. Comput. Math."},{"key":"ref_20","first-page":"603","article-title":"A conservative numerical scheme for a class of nonlinear Schr\u00f6dinger equation with wave operator","volume":"145","author":"Zhang","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1016\/j.cma.2008.11.011","article-title":"Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schr\u00f6dinger equation","volume":"198","author":"Xie","year":"2009","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1360\/012010-846","article-title":"Unconditional convergence of two conservative compact difference schemes for non-linear Schr\u00f6dinger equation in one dimension","volume":"41","author":"Wang","year":"2011","journal-title":"Sci. Sin. Math."},{"key":"ref_23","first-page":"3187","article-title":"A compact finite difference scheme for the nonlinear Schr\u00f6dinger equation with wave operator","volume":"219","author":"Li","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s12190-016-1000-4","article-title":"A new numerical scheme for the nonlinear Schr\u00f6dinger equation with wave operator","volume":"54","author":"Li","year":"2017","journal-title":"J. Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0898-1221(96)00136-8","article-title":"Symplectic methods for the nonlinear Schr\u00f6dinger equation","volume":"32","author":"Tang","year":"1996","journal-title":"Comput. Math. with Appl."},{"key":"ref_26","first-page":"17","article-title":"Symplectic methods for the Ablowitz-Ladik model","volume":"82","author":"Tang","year":"1997","journal-title":"Appl. Math. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Hairer, E., Lubich, C., and Wanner, G. (2002). Geometric Numerical Integration, Springer.","DOI":"10.1007\/978-3-662-05018-7"},{"key":"ref_28","first-page":"24","article-title":"Symplectic methods for the Ablowitz\u2013Ladik discrete nonlinear Schr\u00f6dinger equation","volume":"40","author":"Tang","year":"2007","journal-title":"J. Phys. Math. Theor."},{"key":"ref_29","first-page":"1","article-title":"Numerical solution of the sine-Gordon equation","volume":"18","author":"Guo","year":"1986","journal-title":"Appl. Math. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1229\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:51:27Z","timestamp":1760125887000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1229"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,9]]},"references-count":29,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["sym15061229"],"URL":"https:\/\/doi.org\/10.3390\/sym15061229","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,6,9]]}}}