{"id":"https://openalex.org/W7161955008","doi":"https://doi.org/10.1007/s00332-026-10280-w","title":"Qualitative Aspects of Periodic Traveling Waves for the Sinh-Gordon equation","display_name":"Qualitative Aspects of Periodic Traveling Waves for the Sinh-Gordon equation","publication_year":2026,"publication_date":"2026-05-21","ids":{"openalex":"https://openalex.org/W7161955008","doi":"https://doi.org/10.1007/s00332-026-10280-w"},"language":"en","primary_location":{"id":"doi:10.1007/s00332-026-10280-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-026-10280-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-026-10280-w.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00332-026-10280-w.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038115142","display_name":"Beatriz Signori Lonardoni","orcid":"https://orcid.org/0009-0007-9707-3204"},"institutions":[{"id":"https://openalex.org/I123443094","display_name":"Universidade Estadual de Maring\u00e1","ror":"https://ror.org/04bqqa360","country_code":"BR","type":"education","lineage":["https://openalex.org/I123443094"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Beatriz Signori Lonardoni","raw_affiliation_strings":["Departamento de Matem\u00e1tica, Universidade Estadual de Maring\u00e1, Avenida Colombo, 5790, Maring\u00e1, PR, CEP 87020-900, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1tica, Universidade Estadual de Maring\u00e1, Avenida Colombo, 5790, Maring\u00e1, PR, CEP 87020-900, Brazil","institution_ids":["https://openalex.org/I123443094"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5136683331","display_name":"F\u00e1bio Natali","orcid":null},"institutions":[{"id":"https://openalex.org/I123443094","display_name":"Universidade Estadual de Maring\u00e1","ror":"https://ror.org/04bqqa360","country_code":"BR","type":"education","lineage":["https://openalex.org/I123443094"]}],"countries":["BR"],"is_corresponding":true,"raw_author_name":"F\u00e1bio Natali","raw_affiliation_strings":["Departamento de Matem\u00e1tica, Universidade Estadual de Maring\u00e1, Avenida Colombo, 5790, Maring\u00e1, PR, CEP 87020-900, Brazil"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1tica, Universidade Estadual de Maring\u00e1, Avenida Colombo, 5790, Maring\u00e1, PR, CEP 87020-900, Brazil","institution_ids":["https://openalex.org/I123443094"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5136683331"],"corresponding_institution_ids":["https://openalex.org/I123443094"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.93903743,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":null,"biblio":{"volume":"36","issue":"3","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.2587999999523163,"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"}},"topics":[{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.2587999999523163,"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"}},{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":0.17829999327659607,"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"}},{"id":"https://openalex.org/T11575","display_name":"Nonlinear Photonic Systems","score":0.1738000065088272,"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/linearization","display_name":"Linearization","score":0.652999997138977},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.5788999795913696},{"id":"https://openalex.org/keywords/instability","display_name":"Instability","score":0.521399974822998},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.49709999561309814},{"id":"https://openalex.org/keywords/monotonic-function","display_name":"Monotonic function","score":0.49070000648498535},{"id":"https://openalex.org/keywords/spectrum","display_name":"Spectrum (functional analysis)","score":0.4596000015735626},{"id":"https://openalex.org/keywords/fixed-point","display_name":"Fixed point","score":0.43160000443458557},{"id":"https://openalex.org/keywords/periodic-function","display_name":"Periodic function","score":0.3977999985218048},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.38499999046325684}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7598000168800354},{"id":"https://openalex.org/C11210021","wikidata":"https://www.wikidata.org/wiki/Q1520713","display_name":"Linearization","level":3,"score":0.652999997138977},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6523000001907349},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.5788999795913696},{"id":"https://openalex.org/C207821765","wikidata":"https://www.wikidata.org/wiki/Q405372","display_name":"Instability","level":2,"score":0.521399974822998},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.49709999561309814},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.49070000648498535},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.4596000015735626},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.43160000443458557},{"id":"https://openalex.org/C98644592","wikidata":"https://www.wikidata.org/wiki/Q184743","display_name":"Periodic function","level":2,"score":0.3977999985218048},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.38499999046325684},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.365200012922287},{"id":"https://openalex.org/C123958593","wikidata":"https://www.wikidata.org/wiki/Q938102","display_name":"Elliptic function","level":2,"score":0.35249999165534973},{"id":"https://openalex.org/C140031139","wikidata":"https://www.wikidata.org/wiki/Q79897","display_name":"Morse code","level":2,"score":0.3382999897003174},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.33169999718666077},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.31709998846054077},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.31369999051094055},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.31290000677108765},{"id":"https://openalex.org/C49344536","wikidata":"https://www.wikidata.org/wiki/Q726441","display_name":"Cauchy distribution","level":2,"score":0.2953999936580658},{"id":"https://openalex.org/C80845027","wikidata":"https://www.wikidata.org/wiki/Q15144062","display_name":"Continuous spectrum","level":2,"score":0.2904999852180481},{"id":"https://openalex.org/C70610323","wikidata":"https://www.wikidata.org/wiki/Q427625","display_name":"Elliptic operator","level":2,"score":0.28679999709129333},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.2847999930381775},{"id":"https://openalex.org/C98306654","wikidata":"https://www.wikidata.org/wiki/Q2409122","display_name":"Spectral theory","level":3,"score":0.27480000257492065},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.273499995470047},{"id":"https://openalex.org/C23463724","wikidata":"https://www.wikidata.org/wiki/Q2308831","display_name":"Spectral method","level":2,"score":0.2644999921321869},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.26350000500679016},{"id":"https://openalex.org/C2983668108","wikidata":"https://www.wikidata.org/wiki/Q280453","display_name":"Spectral analysis","level":3,"score":0.26010000705718994}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00332-026-10280-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-026-10280-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-026-10280-w.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00332-026-10280-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-026-10280-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-026-10280-w.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8028430551","display_name":null,"funder_award_id":"303907/2021-5","funder_id":"https://openalex.org/F4320322025","funder_display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico"}],"funders":[{"id":"https://openalex.org/F4320321091","display_name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","ror":"https://ror.org/00x0ma614"},{"id":"https://openalex.org/F4320322025","display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","ror":"https://ror.org/03swz6y49"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7161955008.pdf","grobid_xml":"https://content.openalex.org/works/W7161955008.grobid-xml"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W360456325","https://openalex.org/W1427593441","https://openalex.org/W1527296550","https://openalex.org/W1541040343","https://openalex.org/W1575147392","https://openalex.org/W1596163213","https://openalex.org/W1966841514","https://openalex.org/W1981288703","https://openalex.org/W2023443043","https://openalex.org/W2038664558","https://openalex.org/W2051195713","https://openalex.org/W2071895951","https://openalex.org/W2472647891","https://openalex.org/W2503658163","https://openalex.org/W2963058109","https://openalex.org/W2963446494","https://openalex.org/W2983739449","https://openalex.org/W3048005729","https://openalex.org/W3158413257","https://openalex.org/W3165791484","https://openalex.org/W4213132260","https://openalex.org/W4242981930","https://openalex.org/W4289981527","https://openalex.org/W4395701837","https://openalex.org/W4402486485"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"This":[1],"paper":[2],"presents":[3],"a":[4,15,34,49,98,193],"comprehensive":[5],"analysis":[6,101,118],"of":[7,10,30,57,63,85,89,102,122,141,148,166,175,196,206],"several":[8],"aspects":[9,174],"the":[11,28,43,71,78,103,107,120,123,138,142,146,161,176,181,197,200,203,207],"sinh-Gordon":[12,182],"equation":[13],"in":[14,21,61,116,145],"periodic":[16,31,58,151],"setting.":[17],"Our":[18],"investigation":[19],"proceeds":[20],"three":[22],"main":[23],"stages.":[24],"First,":[25],"we":[26,52,159],"establish":[27],"existence":[29],"solutions":[32,91],"for":[33,48],"fixed":[35,50,111],"wave":[36,72,109],"speed":[37],"and":[38,164,187],"varying":[39],"periods":[40],"by":[41,154],"applying":[42],"mountain":[44],"pass":[45],"theorem.":[46,81],"Subsequently,":[47],"period,":[51],"construct":[53],"an":[54],"explicit":[55,90,108,168],"family":[56],"solutions,":[59],"expressed":[60],"terms":[62],"Jacobi":[64],"elliptic":[65],"functions,":[66],"which":[67],"depend":[68],"smoothly":[69],"on":[70],"speed;":[73],"this":[74,86,117],"is":[75,92,119],"achieved":[76],"via":[77],"inverse":[79],"function":[80],"The":[82,190],"spectral":[83,100,162,204],"stability":[84,163],"smooth":[87],"curve":[88],"then":[93],"rigorously":[94],"addressed.":[95],"We":[96,170],"perform":[97],"detailed":[99],"linearized":[104],"operator":[105,144],"around":[106],"with":[110,129,180],"period.":[112],"A":[113],"key":[114],"element":[115],"monotonicity":[121],"period":[124],"map,":[125],"which,":[126],"when":[127],"combined":[128],"Morse":[130],"index":[131,157],"theory,":[132,158],"enables":[133],"us":[134],"to":[135],"fully":[136],"characterize":[137],"nonpositive":[139],"spectrum":[140],"projected":[143],"space":[147],"zero":[149],"mean":[150],"functions.":[152],"Finally,":[153],"employing":[155],"Hamiltonian\u2013Krein":[156],"determine":[160],"instability":[165,205],"these":[167],"waves.":[169],"also":[171],"discuss":[172],"qualitative":[173],"Cauchy":[177],"problem":[178],"associated":[179],"equation,":[183],"including":[184],"local":[185],"well-posedness":[186],"blow-up":[188],"phenomena.":[189],"former":[191],"supports":[192],"new":[194],"linearization":[195],"problem,":[198],"while":[199],"latter":[201],"predicts":[202],"wave.":[208]},"counts_by_year":[],"updated_date":"2026-05-22T06:22:51.596421","created_date":"2026-05-22T00:00:00"}