{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T18:56:08Z","timestamp":1780599368322,"version":"3.54.1"},"reference-count":55,"publisher":"American Mathematical Society (AMS)","issue":"348","license":[{"start":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T00:00:00Z","timestamp":1732665600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000780","name":"European Commission","doi-asserted-by":"publisher","award":["101064261"],"award-info":[{"award-number":["101064261"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["101064261"],"award-info":[{"award-number":["101064261"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schr\u00f6dinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense of the underlying partial differential equation. The main idea thereby lies in treating the zeroth mode exactly within the discretization. For long-time error estimates, we rigorously establish the error bounds of different low-regularity integrators for the NLSE with small initial data characterized by a dimensionless parameter\n \n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\varepsilon \\in (0, 1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We begin with the low-regularity integrator for the quadratic NLSE in which the integral is computed exactly and the improved uniform first-order convergence in\n \n \n \n \n H<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msup>\n H^r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is proven at\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \u03b5\n \n <\/mml:mi>\n \n \u03c4\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n O(\\varepsilon \\tau )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for solutions in\n \n \n \n \n H<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msup>\n H^r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n r<\/mml:mi>\n ><\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n r > 1\/2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n up to the time\n \n \n \n \n \n T<\/mml:mi>\n \n \n \u03b5\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n T<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \u03b5\n \n <\/mml:mi>\n <\/mml:mrow>\n T_{\\varepsilon } = T\/\\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with fixed\n \n \n \n \n T<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n T > 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Then, the improved uniform long-time error bound is extended to a symmetric second-order low-regularity integrator in the long-time regime. For the cubic NLSE, we design new non-resonant first-order and symmetric second-order low-regularity integrators which treat the zeroth mode exactly and rigorously carry out the error analysis up to the time\n \n \n \n \n \n T<\/mml:mi>\n \n \n \u03b5\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n T<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \n \u03b5\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n T_{\\varepsilon } = T\/\\varepsilon ^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . With the help of the regularity compensation oscillation technique, the improved uniform error bounds are established for the new non-resonant low-regularity schemes, which further reduces the long-time error by a factor of\n \n \n \n \n \n \u03b5\n \n <\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \\varepsilon ^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n compared with classical low-regularity integrators for the cubic NLSE. Numerical examples are presented to validate the error estimates and compare with the classical time-splitting methods in the long-time simulations.\n <\/p>","DOI":"10.1090\/mcom\/3922","type":"journal-article","created":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T09:45:49Z","timestamp":1698227149000},"page":"1569-1598","source":"Crossref","is-referenced-by-count":11,"title":["Long-time error bounds of low-regularity integrators for nonlinear Schr\u00f6dinger equations"],"prefix":"10.1090","volume":"93","author":[{"given":"Yue","family":"Feng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Georg","family":"Maierhofer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Katharina","family":"Schratz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2023,11,27]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"F. 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