{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:08:05Z","timestamp":1776838085564,"version":"3.51.2"},"reference-count":46,"publisher":"American Mathematical Society (AMS)","issue":"348","license":[{"start":{"date-parts":[[2024,10,4]],"date-time":"2024-10-04T00:00:00Z","timestamp":1728000000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schr\u00f6dinger equation (NLSE) with semi-smooth nonlinearity\n \n \n \n \n f<\/mml:mi>\n (<\/mml:mo>\n \n \u03c1\n \n <\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n \u03c1\n \n <\/mml:mi>\n \n \u03c3\n \n <\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n f(\\rho ) = \\rho ^\\sigma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n \n \u03c1\n \n <\/mml:mi>\n =<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u03c8\n \n <\/mml:mi>\n \n \n |<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n \\rho =|\\psi |^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the density with\n \n \n \n \n \u03c8\n \n <\/mml:mi>\n \\psi<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the wave function and\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\sigma >0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the exponent of the semi-smooth nonlinearity. Under the assumption of\n \n \n \n \n H<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n H^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -solution of the NLSE, we prove error bounds at\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \n \u03c4\n \n <\/mml:mi>\n \n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n +<\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n h<\/mml:mi>\n \n 1<\/mml:mn>\n +<\/mml:mo>\n 2<\/mml:mn>\n \n \u03c3\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(\\tau ^{\\frac {1}{2}+\\sigma } + h^{1+2\\sigma })<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \u03c4\n \n <\/mml:mi>\n +<\/mml:mo>\n \n h<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(\\tau + h^{2})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n L^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -norm for\n \n \n \n \n 0<\/mml:mn>\n ><\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n 0>\\sigma \\leq \\frac {1}{2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n \\sigma \\geq \\frac {1}{2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , respectively, and an error bound at\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \n \u03c4\n \n <\/mml:mi>\n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:msup>\n +<\/mml:mo>\n h<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n O(\\tau ^\\frac {1}{2} + h)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n H^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -norm for\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n \\sigma \\geq \\frac {1}{2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \u03c4\n \n <\/mml:mi>\n \\tau<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are the mesh size and time step size, respectively. In addition, when\n \n \n \n \n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n ><\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n ><\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n \\frac {1}{2}>\\sigma >1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and under the assumption of\n \n \n \n \n H<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n H^3<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -solution of the NLSE, we show an error bound at\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \n \u03c4\n \n <\/mml:mi>\n \n \n \u03c3\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n h<\/mml:mi>\n \n 2<\/mml:mn>\n \n \u03c3\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(\\tau ^{\\sigma } + h^{2\\sigma })<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n H^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional\n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n L^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of\n \n \n \n \n 0<\/mml:mn>\n ><\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n 0 > \\sigma \\leq \\frac {1}{2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and to establish an\n \n \n \n \n l<\/mml:mi>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:msup>\n l^\\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -conditional\n \n \n \n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n H^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -stability to obtain the\n \n \n \n \n l<\/mml:mi>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:msup>\n l^\\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -bound of the numerical solution by using the mathematical induction and the error estimates for the case of\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mrow>\n \\sigma \\ge \\frac {1}{2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ; and the other one is to introduce a regularization technique to avoid the singularity of the semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.\n <\/p>","DOI":"10.1090\/mcom\/3900","type":"journal-article","created":{"date-parts":[[2023,9,6]],"date-time":"2023-09-06T10:04:01Z","timestamp":1693994641000},"page":"1599-1631","source":"Crossref","is-referenced-by-count":9,"title":["Error estimates of the time-splitting methods for the nonlinear Schr\u00f6dinger equation with semi-smooth nonlinearity"],"prefix":"10.1090","volume":"93","author":[{"given":"Weizhu","family":"Bao","sequence":"first","affiliation":[]},{"given":"Chushan","family":"Wang","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,10,4]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","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, Georgios D.","year":"1993","journal-title":"IMA J. Numer. 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Lett. 88 (2002), 160401.","DOI":"10.1103\/PhysRevLett.88.160401"},{"issue":"11","key":"25","doi-asserted-by":"publisher","first-page":"2147","DOI":"10.1142\/S0218202517500415","article-title":"Crank-Nicolson Galerkin approximations to nonlinear Schr\u00f6dinger equations with rough potentials","volume":"27","author":"Henning, Patrick","year":"2017","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"key":"26","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1017\/S0962492910000048","article-title":"Exponential integrators","volume":"19","author":"Hochbruck, Marlis","year":"2010","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"key":"27","doi-asserted-by":"crossref","unstructured":"Y. Hu, Y. Fei, X.-L. Chen, and Y. Zhang, Collisional dynamics of symmetric two-dimensional quantum droplets, Front. 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