{"id":"https://openalex.org/W7224841925","doi":"https://doi.org/10.48550/arxiv.2610.39611","title":"Existence and spectral instability of steady rolls in the Nikolaevskiy equation: a computer-assisted study","display_name":"Existence and spectral instability of steady rolls in the Nikolaevskiy equation: a computer-assisted study","publication_year":2026,"publication_date":"2026-10-29","ids":{"openalex":"https://openalex.org/W7224841925","doi":"https://doi.org/10.48550/arxiv.2610.39611"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.39611","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.39611","pdf_url":"https://arxiv.org/pdf/2610.39611","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.39611","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5160166363","display_name":"Nikita V. Konev","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I173089394","display_name":"ITMO University","ror":"https://ror.org/04txgxn49","country_code":"RU","type":"education","lineage":["https://openalex.org/I173089394"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Konev, Nikita V.","raw_affiliation_strings":["Faculty of Physics , ITMO University , Saint Petersburg 197101 , Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Physics , ITMO University , Saint Petersburg 197101 , Russia","institution_ids":["https://openalex.org/I173089394"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5160195519","display_name":"Fedor A. Vyazanitsyn","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I173089394","display_name":"ITMO University","ror":"https://ror.org/04txgxn49","country_code":"RU","type":"education","lineage":["https://openalex.org/I173089394"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Vyazanitsyn, Fedor A.","raw_affiliation_strings":["Computer Engineering Laboratory , Faculty of Software Engineering and Computer Systems , ITMO University , Saint Petersburg 197101 , Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Computer Engineering Laboratory , Faculty of Software Engineering and Computer Systems , ITMO University , Saint Petersburg 197101 , Russia","institution_ids":["https://openalex.org/I173089394"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5160208498","display_name":"Igor A. Knyazev","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I173089394","display_name":"ITMO University","ror":"https://ror.org/04txgxn49","country_code":"RU","type":"education","lineage":["https://openalex.org/I173089394"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Knyazev, Igor A.","raw_affiliation_strings":["Faculty of Physics , ITMO University , Saint Petersburg 197101 , Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Physics , ITMO University , Saint Petersburg 197101 , Russia","institution_ids":["https://openalex.org/I173089394"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I173089394"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.36820000410079956,"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":0.36820000410079956,"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.20350000262260437,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.1306000053882599,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/spectral-instability","display_name":"spectral instability","score":0.878000020980835},{"id":"https://openalex.org/keywords/linear-stability","display_name":"linear stability","score":0.4749999940395355},{"id":"https://openalex.org/keywords/newton-kantorovich-theorem","display_name":"Newton-Kantorovich theorem","score":0.7919999957084656},{"id":"https://openalex.org/keywords/desingularization","display_name":"desingularization","score":0.7609999775886536},{"id":"https://openalex.org/keywords/interval-arithmetic","display_name":"interval arithmetic","score":0.8109999895095825},{"id":"https://openalex.org/keywords/computer-assisted-proof","display_name":"computer-assisted proof","score":0.8619999885559082}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7746000289916992},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.7502999901771545},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6528000235557556},{"id":"https://openalex.org/C207821765","wikidata":"https://www.wikidata.org/wiki/Q405372","display_name":"Instability","level":2,"score":0.5710999965667725},{"id":"https://openalex.org/C8171440","wikidata":"https://www.wikidata.org/wiki/Q903414","display_name":"Steady state (chemistry)","level":2,"score":0.5331000089645386},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5145999789237976},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.5134999752044678},{"id":"https://openalex.org/C180205008","wikidata":"https://www.wikidata.org/wiki/Q159190","display_name":"Amplitude","level":2,"score":0.4645000100135803},{"id":"https://openalex.org/C11577676","wikidata":"https://www.wikidata.org/wiki/Q134237","display_name":"Square root","level":2,"score":0.439300000667572},{"id":"https://openalex.org/C121130766","wikidata":"https://www.wikidata.org/wiki/Q192510","display_name":"Wavenumber","level":2,"score":0.4228000044822693},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.39259999990463257},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.36820000410079956},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.3537999987602234},{"id":"https://openalex.org/C80845027","wikidata":"https://www.wikidata.org/wiki/Q15144062","display_name":"Continuous spectrum","level":2,"score":0.32109999656677246},{"id":"https://openalex.org/C161677786","wikidata":"https://www.wikidata.org/wiki/Q2478475","display_name":"Neighbourhood (mathematics)","level":2,"score":0.32030001282691956},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.31520000100135803},{"id":"https://openalex.org/C2779844894","wikidata":"https://www.wikidata.org/wiki/Q5421516","display_name":"Exponential dichotomy","level":3,"score":0.3118000030517578},{"id":"https://openalex.org/C174084160","wikidata":"https://www.wikidata.org/wiki/Q908049","display_name":"Bound state","level":2,"score":0.2946999967098236},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.29409998655319214},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.2872999906539917},{"id":"https://openalex.org/C173613240","wikidata":"https://www.wikidata.org/wiki/Q1150675","display_name":"Computer-assisted proof","level":3,"score":0.2856999933719635},{"id":"https://openalex.org/C2988173416","wikidata":"https://www.wikidata.org/wiki/Q37172","display_name":"Traveling wave","level":2,"score":0.2702000141143799},{"id":"https://openalex.org/C53744967","wikidata":"https://www.wikidata.org/wiki/Q1062971","display_name":"Characteristic equation","level":3,"score":0.27000001072883606},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.26499998569488525},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.2623000144958496},{"id":"https://openalex.org/C37689646","wikidata":"https://www.wikidata.org/wiki/Q4454926","display_name":"Bloch wave","level":2,"score":0.25440001487731934},{"id":"https://openalex.org/C2780297109","wikidata":"https://www.wikidata.org/wiki/Q917390","display_name":"Essential spectrum","level":3,"score":0.2500999867916107}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.39611","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.39611","pdf_url":"https://arxiv.org/pdf/2610.39611","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"doi:10.48550/arxiv.2610.39611","doi":"https://doi.org/10.48550/arxiv.2610.39611","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.39611","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.39611","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.39611","pdf_url":"https://arxiv.org/pdf/2610.39611","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7224841925.pdf","grobid_xml":"https://content.openalex.org/works/W7224841925.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4414014291","https://openalex.org/W1997268683","https://openalex.org/W1525648481","https://openalex.org/W7164836406","https://openalex.org/W4402095705","https://openalex.org/W2025706422","https://openalex.org/W2792818998","https://openalex.org/W2062060473","https://openalex.org/W4402814711","https://openalex.org/W2084806335"],"abstract_inverted_index":{"The":[0,189],"Nikolaevskiy":[1],"equation":[2,71],"is":[3,18,121,172],"a":[4,12,38,73,82,131,156,192,196,206],"model":[5],"of":[6,22,41,47,105,113],"short-wave":[7],"pattern":[8],"formation":[9],"coupled":[10],"to":[11,37,101,108],"neutral":[13],"long-wave":[14,166],"mode,":[15],"and":[16,58,68,140,177,205],"it":[17],"believed":[19],"that":[20,87],"all":[21,187,213],"its":[23,128,169],"spatially":[24],"periodic":[25],"steady":[26,77],"states":[27],"are":[28,32,215],"unstable.":[29],"Existing":[30],"results":[31],"formal,":[33],"numerical,":[34],"or":[35],"restricted":[36],"small":[39,157],"neighbourhood":[40],"onset.":[42,109],"We":[43],"give":[44],"computer-assisted":[45],"proofs":[46,190],"three":[48],"results.":[49],"First,":[50],"for":[51,144,209],"every":[52,59,111],"wavenumber":[53],"q":[54],"in":[55,118,217],"[0.9,":[56],"1.1]":[57],"control":[60],"parameter":[61,203],"r":[62,117,145],"between":[63],"the":[64,70,90,102,106,125,160,165,210],"onset":[65],"value":[66],"(1-q^2)^2":[67],"4,":[69],"has":[72,130],"nontrivial,":[74],"real-analytic,":[75],"odd":[76],"roll;":[78],"these":[79],"rolls":[80],"form":[81],"single":[83],"continuous":[84],"two-parameter":[85],"family":[86,115],"bifurcates":[88],"from":[89],"trivial":[91],"state":[92],"at":[93,137,141,149,155,174,179,186],"onset,":[94,199],"with":[95,116,134,164],"an":[96],"explicit":[97],"amplitude":[98],"bound":[99],"proportional":[100],"square":[103],"root":[104],"distance":[107],"Second,":[110],"roll":[112],"this":[114],"[0.25,":[119],"4]":[120],"spectrally":[122],"unstable":[123,183],"on":[124,202],"real":[126,135,170],"line:":[127],"linearisation":[129],"Bloch":[132,158,211],"eigenvalue":[133,161],"part":[136,171],"least":[138,142],"6.7e-4,":[139],"0.13":[143],">=":[146],"0.575.":[147],"Third,":[148],"nine":[150],"sample":[151],"points":[152,176],"we":[153],"enclose,":[154],"number,":[159],"numerically":[162],"identified":[163],"(Matthews-Cox)":[167],"branch;":[168],"positive":[173],"seven":[175],"negative":[178],"two,":[180],"while":[181],"strongly":[182],"eigenvalues":[184],"exist":[185],"nine.":[188],"combine":[191],"parametrised":[193],"Newton-Kantorovich":[194],"theorem,":[195],"desingularisation":[197],"near":[198],"Taylor-model":[200],"bounds":[201],"boxes,":[204],"Weyl-sequence":[207],"argument":[208],"spectrum;":[212],"estimates":[214],"evaluated":[216],"rigorous":[218],"interval":[219],"arithmetic.":[220]},"counts_by_year":[],"updated_date":"2026-11-02T09:45:03.842371","created_date":"2026-10-31T00:00:00"}