{"id":"https://openalex.org/W1966734519","doi":"https://doi.org/10.1137/090762865","title":"Localized States in a Model of Pattern Formation in a Vertically Vibrated Layer","display_name":"Localized States in a Model of Pattern Formation in a Vertically Vibrated Layer","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W1966734519","doi":"https://doi.org/10.1137/090762865","mag":"1966734519"},"language":"en","primary_location":{"id":"doi:10.1137/090762865","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090762865","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/A5064073614","display_name":"Jonathan Dawes","orcid":"https://orcid.org/0000-0002-4347-9985"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"J. H. P. Dawes","raw_affiliation_strings":["jhpd20@bath.ac.uk#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"jhpd20@bath.ac.uk#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5028950528","display_name":"S. Lilley","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"S. Lilley","raw_affiliation_strings":["sarah.l.lilley@googlemail.com#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"sarah.l.lilley@googlemail.com#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.0477,"has_fulltext":false,"cited_by_count":26,"citation_normalized_percentile":{"value":0.93425627,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"9","issue":"1","first_page":"238","last_page":"260"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9995999932289124,"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":0.9995999932289124,"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/T10474","display_name":"Material Dynamics and Properties","score":0.9955000281333923,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11575","display_name":"Nonlinear Photonic Systems","score":0.992900013923645,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear 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/limit","display_name":"Limit (mathematics)","score":0.7297009229660034},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6835747957229614},{"id":"https://openalex.org/keywords/viscoelasticity","display_name":"Viscoelasticity","score":0.6562635898590088},{"id":"https://openalex.org/keywords/asymptotic-analysis","display_name":"Asymptotic analysis","score":0.5780518651008606},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5729628801345825},{"id":"https://openalex.org/keywords/amplitude","display_name":"Amplitude","score":0.5523170828819275},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.5516234636306763},{"id":"https://openalex.org/keywords/layer","display_name":"Layer (electronics)","score":0.5513927936553955},{"id":"https://openalex.org/keywords/pattern-formation","display_name":"Pattern formation","score":0.5103806853294373},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.50014328956604},{"id":"https://openalex.org/keywords/granular-material","display_name":"Granular material","score":0.4851420819759369},{"id":"https://openalex.org/keywords/diffusion","display_name":"Diffusion","score":0.47763100266456604},{"id":"https://openalex.org/keywords/mechanics","display_name":"Mechanics","score":0.4383237957954407},{"id":"https://openalex.org/keywords/excitation","display_name":"Excitation","score":0.4310319125652313},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.42587313055992126},{"id":"https://openalex.org/keywords/boundary-layer","display_name":"Boundary layer","score":0.4111037254333496},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3699401617050171},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3254554271697998},{"id":"https://openalex.org/keywords/materials-science","display_name":"Materials science","score":0.14423465728759766},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.08775746822357178},{"id":"https://openalex.org/keywords/nanotechnology","display_name":"Nanotechnology","score":0.08369600772857666},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08248269557952881}],"concepts":[{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.7297009229660034},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6835747957229614},{"id":"https://openalex.org/C186541917","wikidata":"https://www.wikidata.org/wiki/Q910483","display_name":"Viscoelasticity","level":2,"score":0.6562635898590088},{"id":"https://openalex.org/C205147927","wikidata":"https://www.wikidata.org/wiki/Q752718","display_name":"Asymptotic analysis","level":2,"score":0.5780518651008606},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5729628801345825},{"id":"https://openalex.org/C180205008","wikidata":"https://www.wikidata.org/wiki/Q159190","display_name":"Amplitude","level":2,"score":0.5523170828819275},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5516234636306763},{"id":"https://openalex.org/C2779227376","wikidata":"https://www.wikidata.org/wiki/Q6505497","display_name":"Layer (electronics)","level":2,"score":0.5513927936553955},{"id":"https://openalex.org/C207016750","wikidata":"https://www.wikidata.org/wiki/Q1955627","display_name":"Pattern formation","level":2,"score":0.5103806853294373},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.50014328956604},{"id":"https://openalex.org/C32568104","wikidata":"https://www.wikidata.org/wiki/Q1155083","display_name":"Granular material","level":2,"score":0.4851420819759369},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.47763100266456604},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.4383237957954407},{"id":"https://openalex.org/C83581075","wikidata":"https://www.wikidata.org/wiki/Q1361503","display_name":"Excitation","level":2,"score":0.4310319125652313},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.42587313055992126},{"id":"https://openalex.org/C111603439","wikidata":"https://www.wikidata.org/wiki/Q752193","display_name":"Boundary layer","level":2,"score":0.4111037254333496},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3699401617050171},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3254554271697998},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.14423465728759766},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.08775746822357178},{"id":"https://openalex.org/C171250308","wikidata":"https://www.wikidata.org/wiki/Q11468","display_name":"Nanotechnology","level":1,"score":0.08369600772857666},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08248269557952881},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/090762865","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090762865","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":34,"referenced_works":["https://openalex.org/W604181985","https://openalex.org/W1563060612","https://openalex.org/W1833129238","https://openalex.org/W1969217054","https://openalex.org/W1970367503","https://openalex.org/W1974152558","https://openalex.org/W1976618166","https://openalex.org/W1982795007","https://openalex.org/W1985434612","https://openalex.org/W1987577018","https://openalex.org/W1992702817","https://openalex.org/W2007275880","https://openalex.org/W2014301409","https://openalex.org/W2016656927","https://openalex.org/W2021046419","https://openalex.org/W2024211775","https://openalex.org/W2030959643","https://openalex.org/W2033311861","https://openalex.org/W2051982903","https://openalex.org/W2054033101","https://openalex.org/W2056101661","https://openalex.org/W2061883026","https://openalex.org/W2069457875","https://openalex.org/W2071191918","https://openalex.org/W2073063853","https://openalex.org/W2077653620","https://openalex.org/W2077821020","https://openalex.org/W2084511990","https://openalex.org/W2086126894","https://openalex.org/W2086575495","https://openalex.org/W2092588199","https://openalex.org/W2095498348","https://openalex.org/W3104261536","https://openalex.org/W3106534923"],"related_works":["https://openalex.org/W4221057967","https://openalex.org/W4243019676","https://openalex.org/W2517352629","https://openalex.org/W2783562378","https://openalex.org/W4293192718","https://openalex.org/W2045876794","https://openalex.org/W2255703620","https://openalex.org/W807319463","https://openalex.org/W2040929039","https://openalex.org/W2784052753"],"abstract_inverted_index":{"We":[0],"consider":[1],"a":[2,18,59,67,104],"novel":[3,68],"asymptotic":[4,32],"limit":[5,33,65],"of":[6,14,22,46,80,108],"model":[7],"equations":[8],"proposed":[9],"to":[10,52,57,72,84],"describe":[11],"the":[12,31,64,77,81,91],"formation":[13],"localized":[15,40],"states":[16,41],"in":[17],"vertically":[19],"vibrated":[20],"layer":[21,83],"granular":[23,82],"material":[24,54],"or":[25],"viscoelastic":[26],"fluid.":[27],"In":[28],"physical":[29],"terms,":[30],"is":[34,94],"motivated":[35],"by":[36,86],"experimental":[37,110],"observations":[38],"that":[39],"(\u201coscillons\u201d)":[42],"arise":[43],"when":[44,90],"regions":[45],"weak":[47],"excitation":[48],"are":[49],"nevertheless":[50],"able":[51],"expel":[53],"rapidly":[55],"enough":[56],"reach":[58],"balance":[60],"with":[61],"diffusion.":[62],"Mathematically,":[63],"enables":[66],"weakly":[69,97],"nonlinear":[70,98],"analysis":[71,99],"be":[73],"performed":[74],"which":[75],"allows":[76],"local":[78],"depth":[79],"vary":[85],"$O(1)$":[87],"amounts":[88],"even":[89],"pattern":[92],"amplitude":[93],"small.":[95],"The":[96],"and":[100],"numerical":[101],"computations":[102],"provide":[103],"robust":[105],"possible":[106],"explanation":[107],"past":[109],"results.":[111]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":2}],"updated_date":"2026-07-30T17:31:21.811387","created_date":"2025-10-10T00:00:00"}
