{"id":"https://openalex.org/W2049937590","doi":"https://doi.org/10.1137/070707622","title":"Localized Hexagon Patterns of the Planar Swift\u2013Hohenberg Equation","display_name":"Localized Hexagon Patterns of the Planar Swift\u2013Hohenberg Equation","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2049937590","doi":"https://doi.org/10.1137/070707622","mag":"2049937590"},"language":"en","primary_location":{"id":"doi:10.1137/070707622","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070707622","pdf_url":null,"source":{"id":"https://openalex.org/S161160597","display_name":"SIAM Journal on Applied Dynamical Systems","issn_l":"1536-0040","issn":["1536-0040"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Dynamical Systems","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/A5056096547","display_name":"David J. B. Lloyd","orcid":"https://orcid.org/0000-0002-1902-0007"},"institutions":[{"id":"https://openalex.org/I4210160018","display_name":"Merchant Marine Academy","ror":"https://ror.org/04cdzwd37","country_code":"GR","type":"education","lineage":["https://openalex.org/I4210160018"]}],"countries":["GR"],"is_corresponding":true,"raw_author_name":"David J. B. Lloyd","raw_affiliation_strings":["Merchant Venturers' School of Engineering"],"affiliations":[{"raw_affiliation_string":"Merchant Venturers' School of Engineering","institution_ids":["https://openalex.org/I4210160018"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024502637","display_name":"Bj\u00f6rn Sandstede","orcid":"https://orcid.org/0000-0002-5432-1235"},"institutions":[{"id":"https://openalex.org/I28290843","display_name":"University of Surrey","ror":"https://ror.org/00ks66431","country_code":"GB","type":"education","lineage":["https://openalex.org/I28290843"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Bj\u00f6rn Sandstede","raw_affiliation_strings":["Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK","institution_ids":["https://openalex.org/I28290843"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5046360782","display_name":"Daniele Avitabile","orcid":"https://orcid.org/0000-0003-3714-7973"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Daniele Avitabile","raw_affiliation_strings":["Department of Engineering Mathematics"],"affiliations":[{"raw_affiliation_string":"Department of Engineering Mathematics","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5088563106","display_name":"Alan Champneys","orcid":"https://orcid.org/0000-0001-7772-3686"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Alan R. Champneys","raw_affiliation_strings":["Department of Engineering Mathematics"],"affiliations":[{"raw_affiliation_string":"Department of Engineering Mathematics","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5056096547"],"corresponding_institution_ids":["https://openalex.org/I4210160018"],"apc_list":null,"apc_paid":null,"fwci":17.1544,"has_fulltext":false,"cited_by_count":195,"citation_normalized_percentile":{"value":0.99279128,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"7","issue":"3","first_page":"1049","last_page":"1100"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9873999953269958,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9732999801635742,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.7881021499633789},{"id":"https://openalex.org/keywords/parameter-space","display_name":"Parameter space","score":0.6257084012031555},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5502880811691284},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5192880034446716},{"id":"https://openalex.org/keywords/numerical-continuation","display_name":"Numerical continuation","score":0.4712553322315216},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.46894264221191406},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4386546313762665},{"id":"https://openalex.org/keywords/continuation","display_name":"Continuation","score":0.43296974897384644},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.39441370964050293},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3172754645347595},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.182763010263443},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.12738648056983948},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10973167419433594}],"concepts":[{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.7881021499633789},{"id":"https://openalex.org/C73586568","wikidata":"https://www.wikidata.org/wiki/Q2600211","display_name":"Parameter space","level":2,"score":0.6257084012031555},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5502880811691284},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5192880034446716},{"id":"https://openalex.org/C2775978667","wikidata":"https://www.wikidata.org/wiki/Q7069654","display_name":"Numerical continuation","level":4,"score":0.4712553322315216},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.46894264221191406},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4386546313762665},{"id":"https://openalex.org/C88626702","wikidata":"https://www.wikidata.org/wiki/Q1128903","display_name":"Continuation","level":2,"score":0.43296974897384644},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.39441370964050293},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3172754645347595},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.182763010263443},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.12738648056983948},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10973167419433594},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/070707622","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070707622","pdf_url":null,"source":{"id":"https://openalex.org/S161160597","display_name":"SIAM Journal on Applied Dynamical Systems","issn_l":"1536-0040","issn":["1536-0040"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Dynamical Systems","raw_type":"journal-article"},{"id":"pmh:oai:alma.44SUR_INST:11138535930002346","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4210197018","display_name":"View","issn_l":"2688-268X","issn":["2688-268X","2688-3988"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"","raw_type":""},{"id":"pmh:oai:research-information.bris.ac.uk:openaire_cris_publications/0d91ee70-e673-4512-b09f-7ab2a8d014e2","is_oa":false,"landing_page_url":"https://research-information.bris.ac.uk/en/publications/0d91ee70-e673-4512-b09f-7ab2a8d014e2","pdf_url":null,"source":{"id":"https://openalex.org/S7407055359","display_name":"Explore Bristol Research","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lloyd, D J B, Sandstede, B, Avitabile, D & Champneys, A R 2008, 'Localized Hexagon Patterns of the Planar Swift-Hohenberg Equation', SIAM Journal on Applied Dynamical Systems, vol. 7, no. 3, pp. 1049 - 1100. https://doi.org/10.1137/070707622","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.379.5629","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.379.5629","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.dam.brown.edu/lcds/publications/documents/Sandstede_2.pdf","raw_type":"text"},{"id":"pmh:oai:epubs.surrey.ac.uk:1449","is_oa":false,"landing_page_url":"http://epubs.surrey.ac.uk/1449/1/fulltext.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400680","display_name":"Surrey Research Insight Open Access (The University of Surrey)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28290843","host_organization_name":"University of Surrey","host_organization_lineage":["https://openalex.org/I28290843"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:epubs.surrey.ac.uk:294644","is_oa":false,"landing_page_url":"http://epubs.surrey.ac.uk/294644/2/SRI_deposit_agreement.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400680","display_name":"Surrey Research Insight Open Access (The University of Surrey)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28290843","host_organization_name":"University of Surrey","host_organization_lineage":["https://openalex.org/I28290843"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":87,"referenced_works":["https://openalex.org/W141185090","https://openalex.org/W596805671","https://openalex.org/W1490180844","https://openalex.org/W1524491377","https://openalex.org/W1563060612","https://openalex.org/W1866869231","https://openalex.org/W1964109380","https://openalex.org/W1965778005","https://openalex.org/W1967308282","https://openalex.org/W1969189816","https://openalex.org/W1970001584","https://openalex.org/W1970301131","https://openalex.org/W1970367503","https://openalex.org/W1970545034","https://openalex.org/W1974742255","https://openalex.org/W1976618166","https://openalex.org/W1982795007","https://openalex.org/W1985434612","https://openalex.org/W1988031493","https://openalex.org/W1993814570","https://openalex.org/W1994456667","https://openalex.org/W1994861315","https://openalex.org/W1996036666","https://openalex.org/W1997341646","https://openalex.org/W1999138544","https://openalex.org/W2000829282","https://openalex.org/W2000883601","https://openalex.org/W2001292665","https://openalex.org/W2003884541","https://openalex.org/W2004710798","https://openalex.org/W2004785198","https://openalex.org/W2005306494","https://openalex.org/W2008421829","https://openalex.org/W2014301409","https://openalex.org/W2016656927","https://openalex.org/W2017288987","https://openalex.org/W2022894121","https://openalex.org/W2022916138","https://openalex.org/W2027605763","https://openalex.org/W2028885825","https://openalex.org/W2030170709","https://openalex.org/W2033311861","https://openalex.org/W2033566973","https://openalex.org/W2037104303","https://openalex.org/W2038477894","https://openalex.org/W2040122443","https://openalex.org/W2046408468","https://openalex.org/W2048678256","https://openalex.org/W2049165066","https://openalex.org/W2049546479","https://openalex.org/W2051982903","https://openalex.org/W2054371931","https://openalex.org/W2055541567","https://openalex.org/W2056658486","https://openalex.org/W2059849709","https://openalex.org/W2062561585","https://openalex.org/W2063907161","https://openalex.org/W2069638784","https://openalex.org/W2070451295","https://openalex.org/W2071191918","https://openalex.org/W2071348013","https://openalex.org/W2071466514","https://openalex.org/W2072819947","https://openalex.org/W2074576851","https://openalex.org/W2076313740","https://openalex.org/W2077653620","https://openalex.org/W2077821020","https://openalex.org/W2084202510","https://openalex.org/W2084511990","https://openalex.org/W2086126894","https://openalex.org/W2087278084","https://openalex.org/W2092588199","https://openalex.org/W2095498348","https://openalex.org/W2099280505","https://openalex.org/W2100386495","https://openalex.org/W2112400625","https://openalex.org/W2114892458","https://openalex.org/W2122329937","https://openalex.org/W2122638156","https://openalex.org/W2123514264","https://openalex.org/W2148507681","https://openalex.org/W2152647360","https://openalex.org/W2333882743","https://openalex.org/W2791548756","https://openalex.org/W2798320239","https://openalex.org/W3104261536","https://openalex.org/W3106534923"],"related_works":["https://openalex.org/W2466843756","https://openalex.org/W2299007595","https://openalex.org/W2116098329","https://openalex.org/W2157680900","https://openalex.org/W2018102019","https://openalex.org/W1966113063","https://openalex.org/W28777054","https://openalex.org/W2164384956","https://openalex.org/W4387614905","https://openalex.org/W1591565880"],"abstract_inverted_index":{"We":[0,62,182],"investigate":[1,123],"stationary":[2],"spatially":[3],"localized":[4,40,119],"hexagon":[5,23,41,110,120,136],"patterns":[6,24,66,81,153],"of":[7,38,44,49,80,104,108,116,130,145],"the":[8,14,18,35,59,74,102,105,117,127,146,158,166,171,175,179,185,190],"two-dimensional":[9],"(2D)":[10],"Swift\u2013Hohenberg":[11],"equation":[12],"in":[13,58,73,86,134,157],"parameter":[15,71,87],"region":[16],"where":[17],"trivial":[19,60,180],"state":[20],"and":[21,43,89,115,161],"regular":[22],"are":[25,84,132,155],"both":[26],"stable.":[27],"Using":[28],"numerical":[29],"continuation":[30],"techniques,":[31],"we":[32,140],"trace":[33],"out":[34],"existence":[36],"regions":[37,107],"fully":[39,118],"patches":[42],"planar":[45,109,152],"pulses":[46,111],"which":[47,124,154,170],"consist":[48],"a":[50,99,135,142],"strip":[51],"filled":[52],"with":[53,112],"hexagons":[54,125,131,173],"that":[55,64,83,150,184],"is":[56],"embedded":[57],"state.":[61,181],"find":[63,183],"these":[65],"exhibit":[67],"snaking:":[68],"for":[69],"each":[70],"value":[72],"snaking":[75,106,191],"region,":[76,192],"an":[77],"infinite":[78],"number":[79],"exist":[82],"connected":[85],"space":[88],"whose":[90],"width":[91],"increases":[92],"without":[93],"bound.":[94],"Our":[95],"computations":[96],"also":[97],"indicate":[98],"relation":[100],"between":[101],"limits":[103],"different":[113],"orientations":[114],"patches.":[121],"To":[122],"among":[126],"one-parameter":[128],"family":[129],"selected":[133,172],"pulse":[137],"or":[138],"front,":[139],"derive":[141],"conserved":[143],"quantity":[144],"spatial":[147],"dynamical":[148],"system":[149],"describes":[151],"periodic":[156],"transverse":[159],"direction":[160],"use":[162],"it":[163],"to":[164],"calculate":[165],"Maxwell":[167,186],"curves":[168],"along":[169],"have":[174],"same":[176],"energy":[177],"as":[178,193],"curve":[187],"lies":[188],"within":[189],"expected":[194],"from":[195],"heuristic":[196],"arguments.":[197]},"counts_by_year":[{"year":2025,"cited_by_count":11},{"year":2024,"cited_by_count":8},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":17},{"year":2020,"cited_by_count":11},{"year":2019,"cited_by_count":15},{"year":2018,"cited_by_count":13},{"year":2017,"cited_by_count":8},{"year":2016,"cited_by_count":8},{"year":2015,"cited_by_count":16},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":17},{"year":2012,"cited_by_count":8}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}