{"id":"https://openalex.org/W4286971439","doi":"https://doi.org/10.1090/mcom/3801","title":"Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schr\u00f6dinger equation","display_name":"Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schr\u00f6dinger equation","publication_year":2022,"publication_date":"2022-11-29","ids":{"openalex":"https://openalex.org/W4286971439","doi":"https://doi.org/10.1090/mcom/3801"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/3801","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3801","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100759028","display_name":"Weizhu Bao","orcid":"https://orcid.org/0000-0003-3418-9625"},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":true,"raw_author_name":"Weizhu Bao","raw_affiliation_strings":["Department of Mathematics, National University of Singapore, Singapore 119076"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, National University of Singapore, Singapore 119076","institution_ids":["https://openalex.org/I165932596"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013651928","display_name":"Yongyong Cai","orcid":"https://orcid.org/0000-0002-3890-0878"},"institutions":[{"id":"https://openalex.org/I25254941","display_name":"Beijing Normal University","ror":"https://ror.org/022k4wk35","country_code":"CN","type":"education","lineage":["https://openalex.org/I25254941"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yongyong Cai","raw_affiliation_strings":["Laboratory of Mathematics and Complex Systems and School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People\u00e2\u0080\u0099s Republic of China","Laboratory of Mathematics and Complex Systems and School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratory of Mathematics and Complex Systems and School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People\u00e2\u0080\u0099s Republic of China","institution_ids":["https://openalex.org/I25254941"]},{"raw_affiliation_string":"Laboratory of Mathematics and Complex Systems and School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of China","institution_ids":["https://openalex.org/I25254941"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039748802","display_name":"Yue Feng","orcid":"https://orcid.org/0000-0002-3895-3283"},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Yue Feng","raw_affiliation_strings":["Department of Mathematics, National University of Singapore, Singapore 119076"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, National University of Singapore, Singapore 119076","institution_ids":["https://openalex.org/I165932596"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5100759028"],"corresponding_institution_ids":["https://openalex.org/I165932596"],"apc_list":null,"apc_paid":null,"fwci":9.3817,"has_fulltext":false,"cited_by_count":27,"citation_normalized_percentile":{"value":0.99355783,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":100},"biblio":{"volume":"92","issue":"341","first_page":"1109","last_page":"1139"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9980999827384949,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9919999837875366,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8529237508773804},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5904441475868225},{"id":"https://openalex.org/keywords/nonlinear-schr\u00f6dinger-equation","display_name":"Nonlinear Schr\u00f6dinger equation","score":0.5308298468589783},{"id":"https://openalex.org/keywords/schr\u00f6dinger-equation","display_name":"Schr\u00f6dinger equation","score":0.5023367404937744},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4941711723804474},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.36088094115257263},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.1094784140586853},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08146274089813232}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8529237508773804},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5904441475868225},{"id":"https://openalex.org/C83774755","wikidata":"https://www.wikidata.org/wiki/Q2579914","display_name":"Nonlinear Schr\u00f6dinger equation","level":3,"score":0.5308298468589783},{"id":"https://openalex.org/C63036615","wikidata":"https://www.wikidata.org/wiki/Q165498","display_name":"Schr\u00f6dinger equation","level":2,"score":0.5023367404937744},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4941711723804474},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36088094115257263},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.1094784140586853},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08146274089813232}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/mcom/3801","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3801","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":57,"referenced_works":["https://openalex.org/W636566490","https://openalex.org/W1547431580","https://openalex.org/W1592378316","https://openalex.org/W1830089776","https://openalex.org/W1972140872","https://openalex.org/W1973279410","https://openalex.org/W1979983774","https://openalex.org/W1988469123","https://openalex.org/W1990790554","https://openalex.org/W1997190030","https://openalex.org/W1998794711","https://openalex.org/W2012220122","https://openalex.org/W2013529907","https://openalex.org/W2023195358","https://openalex.org/W2023290656","https://openalex.org/W2026509625","https://openalex.org/W2031373684","https://openalex.org/W2036516029","https://openalex.org/W2037065783","https://openalex.org/W2038167689","https://openalex.org/W2039264003","https://openalex.org/W2041905900","https://openalex.org/W2048216020","https://openalex.org/W2049161806","https://openalex.org/W2050338742","https://openalex.org/W2051494558","https://openalex.org/W2060005168","https://openalex.org/W2070781491","https://openalex.org/W2073800838","https://openalex.org/W2075696551","https://openalex.org/W2080454515","https://openalex.org/W2083561837","https://openalex.org/W2088291682","https://openalex.org/W2089752155","https://openalex.org/W2092400970","https://openalex.org/W2098595725","https://openalex.org/W2117988589","https://openalex.org/W2119372460","https://openalex.org/W2136491192","https://openalex.org/W2182274314","https://openalex.org/W2283547420","https://openalex.org/W2565219097","https://openalex.org/W2900117811","https://openalex.org/W2962953794","https://openalex.org/W3008781546","https://openalex.org/W3101047577","https://openalex.org/W3101835573","https://openalex.org/W3105700720","https://openalex.org/W3127420347","https://openalex.org/W3155908030","https://openalex.org/W3203396613","https://openalex.org/W4200630032","https://openalex.org/W4211077197","https://openalex.org/W4226033512","https://openalex.org/W4245463005","https://openalex.org/W4247381331","https://openalex.org/W4253937035"],"related_works":["https://openalex.org/W2154204858","https://openalex.org/W3209622820","https://openalex.org/W2382250648","https://openalex.org/W2245214771","https://openalex.org/W2333053900","https://openalex.org/W2951961089","https://openalex.org/W3101208931","https://openalex.org/W2181923191","https://openalex.org/W3213209941","https://openalex.org/W2074095400"],"abstract_inverted_index":{"We":[0],"establish":[1],"improved":[2],"uniform":[3,114],"error":[4,115],"bounds":[5],"for":[6,10,286,309],"the":[7,11,15,22,31,80,85,359],"time-splitting":[8,87],"methods":[9],"long-time":[12],"dynamics":[13],"of":[14,84],"Schr\u00f6dinger":[16,24,32],"equation":[17,25,33],"with":[18,27,34],"small":[19,35],"potential":[20,36],"and":[21,307],"nonlinear":[23],"(NLSE)":[26],"weak":[28],"nonlinearity.":[29],"For":[30],"characterized":[37],"by":[38],"a":[39,113],"dimensionless":[40],"parameter":[41],"<inline-formula":[42,93,119,156,231,288,310,346,362],"content-type=\"math/mathml\">":[43,94,120,157,232,289,311,347,363],"<mml:math":[44,95,121,158,233,290,312,348,364],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[45,96,122,159,234,291,313,349,365],"alttext=\"epsilon":[46,314],"element-of":[47,315],"left-parenthesis":[48,162,170,174,316],"0":[49,317],"comma":[50,318],"1":[51,319],"right-bracket\">":[52,320],"<mml:semantics>":[53,100,131,183,250,296,321,351,367],"<mml:mrow>":[54,132,184,251,297,322],"<mml:mi>":[55,135,145,213,254,263,273,323,368],"\u03b5":[56,136,146,255,264,274,324],"</mml:mi>":[57,137,147,215,256,265,275,325,370],"<mml:mo>":[58,195,258,326],"\u2208":[59,327],"</mml:mo>":[60,197,260,328],"<mml:mo":[61,66,186,189,200,203,205,218,329,334],"stretchy=\"false\">(</mml:mo>":[62,187,201,206,330],"<mml:mn>0</mml:mn>":[63,300,331],"<mml:mo>,</mml:mo>":[64,332],"<mml:mn>1</mml:mn>":[65,333],"stretchy=\"false\">]</mml:mo>":[67,335],"</mml:mrow>":[68,144,148,199,220,272,276,301,336],"<mml:annotation":[69,105,149,221,277,302,337,353,371],"encoding=\"application/x-tex\">\\varepsilon":[70,338],"\\in":[71,339],"(0,":[72],"1]</mml:annotation>":[73],"</mml:semantics>":[74,107,152,226,283,304,341,355],"</mml:math>":[75,108,153,227,284,305,342,356],"</inline-formula>":[76,109,154,228,285,306,343,357],",":[77,344],"we":[78],"employ":[79],"unitary":[81],"flow":[82],"property":[83],"(second-order)":[86],"Fourier":[88],"pseudospectral":[89],"(TSFP)":[90],"method":[91],"in":[92],"alttext=\"upper":[97,160,292],"L":[98],"squared\">":[99],"<mml:msup>":[101,207,212],"<mml:mi>L</mml:mi>":[102],"<mml:mn>2</mml:mn>":[103,216],"</mml:msup>":[104,210,217],"encoding=\"application/x-tex\">L^2</mml:annotation>":[106],"-norm":[110],"to":[111,230],"prove":[112],"bound":[116],"at":[117],"time":[118],"alttext=\"t":[123,235],"Subscript":[124,236,242],"epsilon":[125,237,243],"Baseline":[126,178,238,244],"equals":[127,245],"t":[128,163],"slash":[129,248],"epsilon\">":[130,249],"<mml:msub>":[133,252,261],"<mml:mi>t</mml:mi>":[134,140,188,253],"</mml:msub>":[138,257,266],"<mml:mo>=</mml:mo>":[139,267],"<mml:mrow":[141,191,269],"class=\"MJX-TeXAtom-ORD\">":[142,192,270],"<mml:mo>/</mml:mo>":[143,271],"encoding=\"application/x-tex\">t_\\varepsilon":[150,278],"=t/\\varepsilon</mml:annotation>":[151],"as":[155],"C":[161,167],"right-parenthesis":[164,173],"ModifyingAbove":[165],"upper":[166,171,240,246],"With":[168],"tilde":[169],"T":[172,241,247,293],"h":[175],"Superscript":[176],"m":[177],"plus":[179],"tau":[180],"squared":[181],"right-parenthesis\">":[182],"<mml:mi>C</mml:mi>":[185,194],"stretchy=\"false\">)</mml:mo>":[190,204,219],"<mml:mover>":[193],"~":[196],"</mml:mover>":[198],"<mml:mi>T</mml:mi>":[202,262,268,298],"<mml:mi>h</mml:mi>":[208,352],"<mml:mi>m</mml:mi>":[209],"<mml:mo>+</mml:mo>":[211],"\u03c4":[214,369],"encoding=\"application/x-tex\">C(t)\\widetilde":[222],"{C}(T)(h^m":[223],"+\\tau":[224],"^2)</mml:annotation>":[225],"up":[229],"less-than-or-equal-to":[239],"\u2264":[259],"\\leq":[279],"T_\\varepsilon":[280],"=":[281],"T/\\varepsilon</mml:annotation>":[282],"any":[287],"greater-than":[294],"0\">":[295],"<mml:mo>&gt;</mml:mo>":[299],"encoding=\"application/x-tex\">T&gt;0</mml:annotation>":[303],"uniformly":[308],"(0,1]</mml:annotation>":[340],"while":[345],"alttext=\"h\">":[350],"encoding=\"application/x-tex\">h</mml:annotation>":[354],"is":[358],"mesh":[360],"size,":[361],"alttext=\"tau\">":[366],"encoding=\"a":[372]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":15},{"year":2023,"cited_by_count":5}],"updated_date":"2026-04-23T06:14:38.165362","created_date":"2025-10-10T00:00:00"}
