{"id":"https://openalex.org/W2062713613","doi":"https://doi.org/10.1137/070710949","title":"Stationary Solutions of Driven Fourth- and Sixth-Order Cahn\u2013Hilliard-Type Equations","display_name":"Stationary Solutions of Driven Fourth- and Sixth-Order Cahn\u2013Hilliard-Type Equations","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2062713613","doi":"https://doi.org/10.1137/070710949","mag":"2062713613"},"language":"en","primary_location":{"id":"doi:10.1137/070710949","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070710949","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/0712.2482","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5017707470","display_name":"M. D. Korzec","orcid":null},"institutions":[{"id":"https://openalex.org/I4210092021","display_name":"Weierstrass Institute for Applied Analysis and Stochastics","ror":"https://ror.org/00h1x4t21","country_code":"DE","type":"facility","lineage":["https://openalex.org/I315704651","https://openalex.org/I4210092021"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"M. D. Korzec","raw_affiliation_strings":["Weierstrass Institute for Applied Analysis and Stochastics (WIAS), D-10117 Berlin, Germany"],"affiliations":[{"raw_affiliation_string":"Weierstrass Institute for Applied Analysis and Stochastics (WIAS), D-10117 Berlin, Germany","institution_ids":["https://openalex.org/I4210092021"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017746334","display_name":"Paul Evans","orcid":"https://orcid.org/0000-0002-1693-417X"},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"P. L. Evans","raw_affiliation_strings":["Institute for Mathematics, Humboldt University of Berlin, D-10099 Berlin, Germany"],"affiliations":[{"raw_affiliation_string":"Institute for Mathematics, Humboldt University of Berlin, D-10099 Berlin, Germany","institution_ids":["https://openalex.org/I39343248"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102978147","display_name":"Andreas M\u00fcnch","orcid":"https://orcid.org/0000-0002-8325-3809"},"institutions":[{"id":"https://openalex.org/I142263535","display_name":"University of Nottingham","ror":"https://ror.org/01ee9ar58","country_code":"GB","type":"education","lineage":["https://openalex.org/I142263535"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"A. M\u00fcnch","raw_affiliation_strings":["School of Mathematical Sciences, University of Nottingham, UK"],"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, University of Nottingham, UK","institution_ids":["https://openalex.org/I142263535"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5083611204","display_name":"Barbara Wagner","orcid":"https://orcid.org/0000-0001-8306-3645"},"institutions":[{"id":"https://openalex.org/I4210092021","display_name":"Weierstrass Institute for Applied Analysis and Stochastics","ror":"https://ror.org/00h1x4t21","country_code":"DE","type":"facility","lineage":["https://openalex.org/I315704651","https://openalex.org/I4210092021"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"B. Wagner","raw_affiliation_strings":["Weierstrass Institute for Applied Analysis and Stochastics (WIAS), D-10117 Berlin, Ger-many"],"affiliations":[{"raw_affiliation_string":"Weierstrass Institute for Applied Analysis and Stochastics (WIAS), D-10117 Berlin, Ger-many","institution_ids":["https://openalex.org/I4210092021"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5017707470"],"corresponding_institution_ids":["https://openalex.org/I4210092021"],"apc_list":null,"apc_paid":null,"fwci":0.965,"has_fulltext":false,"cited_by_count":36,"citation_normalized_percentile":{"value":0.77341871,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"69","issue":"2","first_page":"348","last_page":"374"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12202","display_name":"nanoparticles nucleation surface interactions","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/1902","display_name":"Atmospheric Science"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12202","display_name":"nanoparticles nucleation surface interactions","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/1902","display_name":"Atmospheric Science"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12141","display_name":"Fluid Dynamics and Thin Films","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9958000183105469,"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/cahn\u2013hilliard-equation","display_name":"Cahn\u2013Hilliard equation","score":0.8761720061302185},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.54425048828125},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5350265502929688},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.5317453145980835},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.507532000541687},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.494370698928833},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.42659685015678406},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.41326117515563965},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.4118826985359192},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3300187587738037},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.10045847296714783},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08420562744140625},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.08128497004508972}],"concepts":[{"id":"https://openalex.org/C24822716","wikidata":"https://www.wikidata.org/wiki/Q5017440","display_name":"Cahn\u2013Hilliard equation","level":3,"score":0.8761720061302185},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.54425048828125},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5350265502929688},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5317453145980835},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.507532000541687},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.494370698928833},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.42659685015678406},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.41326117515563965},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.4118826985359192},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3300187587738037},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.10045847296714783},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08420562744140625},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.08128497004508972},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/070710949","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070710949","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:0712.2482","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0712.2482","pdf_url":"https://arxiv.org/pdf/0712.2482","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:doi:10.20347/wias.preprint.1279","is_oa":false,"landing_page_url":"http://localhost:8291/receive/wias_mods_00001526","pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"preprint"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.190.9044","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.190.9044","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.wias-berlin.de/people/korzec/papers/KorzecEvansMuenchWagner07_preprint.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.91.8926","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.91.8926","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.matheon.de/preprints/4715_KEMW.pdf","raw_type":"text"},{"id":"pmh:oai:bibcyt.ucla.edu.ve:T070300015774/3","is_oa":false,"landing_page_url":"http://bibcyt.ucla.edu.ve/cgi-win/be_alex.exe?Acceso=T070300015774/3&Nombrebd=Bciucla","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:0712.2482","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0712.2482","pdf_url":"https://arxiv.org/pdf/0712.2482","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W85861326","https://openalex.org/W1554107248","https://openalex.org/W1646886396","https://openalex.org/W1971808848","https://openalex.org/W1989453633","https://openalex.org/W1994854223","https://openalex.org/W2003205185","https://openalex.org/W2010142289","https://openalex.org/W2028296976","https://openalex.org/W2037933894","https://openalex.org/W2041832550","https://openalex.org/W2043466506","https://openalex.org/W2054724344","https://openalex.org/W2072655083","https://openalex.org/W2073686912","https://openalex.org/W2074633705","https://openalex.org/W2079944289","https://openalex.org/W2080252049","https://openalex.org/W2082261293","https://openalex.org/W2082710148","https://openalex.org/W2169920821","https://openalex.org/W2271012468","https://openalex.org/W2474406097","https://openalex.org/W2515590532","https://openalex.org/W2605113454","https://openalex.org/W3127727104"],"related_works":["https://openalex.org/W2967558384","https://openalex.org/W2048454885","https://openalex.org/W2077028260","https://openalex.org/W57680709","https://openalex.org/W2059693824","https://openalex.org/W1969464363","https://openalex.org/W2968867992","https://openalex.org/W2183190086","https://openalex.org/W4298136942","https://openalex.org/W2000402520"],"abstract_inverted_index":{"New":[0],"types":[1],"of":[2,5,31,34,56,59,67,79,87,101,128],"stationary":[3],"solutions":[4,63,83,100,135],"a":[6,16,118],"one-dimensional":[7],"driven":[8,152],"sixthorder":[9],"Cahn-Hilliard":[10,154,161],"type":[11],"equation":[12],"that":[13,38,121],"arises":[14],"as":[15,23,51,53,138,158],"model":[17],"for":[18,48,110,147],"epitaxially":[19],"growing":[20],"nano-structures":[21],"such":[22],"quantum":[24,166],"dots,":[25,167],"are":[26,136],"derived":[27],"by":[28],"an":[29],"extension":[30],"the":[32,54,57,65,74,85,88,102,111,123,129,133,148,159],"method":[33,44],"matched":[35],"asymptotic":[36,134],"expansions":[37],"retains":[39],"exponentially":[40],"small":[41,68],"terms.":[42],"This":[43],"yields":[45],"analytical":[46],"expressions":[47],"far-field":[49],"behavior":[50],"well":[52],"widths":[55],"humps":[58],"these":[60],"spatially":[61],"non-monotone":[62],"in":[64,77],"limit":[66],"driving":[69],"force":[70],"strength":[71],"which":[72],"is":[73,97,144],"deposition":[75],"rate":[76],"case":[78],"epitaxial":[80],"growth.":[81],"These":[82],"extend":[84],"family":[86],"monotone":[89],"kink":[90],"and":[91,125],"antikink":[92],"solutions.":[93],"The":[94],"hump":[95],"spacing":[96],"related":[98,149],"to":[99],"Lambert":[103],"W":[104],"function.":[105],"Using":[106],"phase":[107],"space":[108],"analysis":[109],"corresponding":[112],"fifth-order":[113],"dynamical":[114,171],"system,":[115],"we":[116],"use":[117],"numerical":[119],"technique":[120],"enables":[122],"efficient":[124],"accurate":[126],"tracking":[127],"solution":[130],"branches,":[131],"where":[132],"used":[137],"initial":[139],"input.":[140],"Additionally,":[141],"our":[142],"approach":[143],"first":[145],"demonstrated":[146],"but":[150],"simpler":[151],"fourth-order":[153],"equation,":[155],"also":[156],"known":[157],"convective":[160,164],"equation.":[162],"Keywords:":[163],"Cahn-Hilliard,":[165],"exponential":[168],"asymptotics,":[169],"matching,":[170],"systems":[172],"1":[173]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":6}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}
