{"id":"https://openalex.org/W4410439834","doi":"https://doi.org/10.3390/axioms14050376","title":"Variational Principle of the Unstable Nonlinear Schr\u00f6dinger Equation with Fractal Derivatives","display_name":"Variational Principle of the Unstable Nonlinear Schr\u00f6dinger Equation with Fractal Derivatives","publication_year":2025,"publication_date":"2025-05-16","ids":{"openalex":"https://openalex.org/W4410439834","doi":"https://doi.org/10.3390/axioms14050376"},"language":"en","primary_location":{"id":"doi:10.3390/axioms14050376","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14050376","pdf_url":"https://www.mdpi.com/2075-1680/14/5/376/pdf?version=1747400500","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/14/5/376/pdf?version=1747400500","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064814377","display_name":"Kang\u2010Jia Wang","orcid":"https://orcid.org/0000-0002-3905-0844"},"institutions":[{"id":"https://openalex.org/I4210166499","display_name":"Henan Polytechnic University","ror":"https://ror.org/05vr1c885","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210166499"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Kang-Jia Wang","raw_affiliation_strings":["School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, China"],"affiliations":[{"raw_affiliation_string":"School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, China","institution_ids":["https://openalex.org/I4210166499"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100351409","display_name":"Ming Li","orcid":"https://orcid.org/0000-0002-2725-353X"},"institutions":[{"id":"https://openalex.org/I4210166499","display_name":"Henan Polytechnic University","ror":"https://ror.org/05vr1c885","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210166499"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Ming Li","raw_affiliation_strings":["School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, China"],"affiliations":[{"raw_affiliation_string":"School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, China","institution_ids":["https://openalex.org/I4210166499"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5100351409"],"corresponding_institution_ids":["https://openalex.org/I4210166499"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":21.4492,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.99566275,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"14","issue":"5","first_page":"376","last_page":"376"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9937000274658203,"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/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9937000274658203,"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/T13234","display_name":"advanced mathematical theories","score":0.989300012588501,"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"}},{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9779000282287598,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/fractal","display_name":"Fractal","score":0.6926348209381104},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6326242089271545},{"id":"https://openalex.org/keywords/variational-principle","display_name":"Variational principle","score":0.5546647906303406},{"id":"https://openalex.org/keywords/nonlinear-schr\u00f6dinger-equation","display_name":"Nonlinear Schr\u00f6dinger equation","score":0.5543500781059265},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.511618435382843},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.46414366364479065},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.42851340770721436},{"id":"https://openalex.org/keywords/schr\u00f6dinger-equation","display_name":"Schr\u00f6dinger equation","score":0.40658053755760193},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.38923180103302},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.36297959089279175},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3278372287750244},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.09521418809890747}],"concepts":[{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.6926348209381104},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6326242089271545},{"id":"https://openalex.org/C9259290","wikidata":"https://www.wikidata.org/wiki/Q745215","display_name":"Variational principle","level":2,"score":0.5546647906303406},{"id":"https://openalex.org/C83774755","wikidata":"https://www.wikidata.org/wiki/Q2579914","display_name":"Nonlinear Schr\u00f6dinger equation","level":3,"score":0.5543500781059265},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.511618435382843},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.46414366364479065},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42851340770721436},{"id":"https://openalex.org/C63036615","wikidata":"https://www.wikidata.org/wiki/Q165498","display_name":"Schr\u00f6dinger equation","level":2,"score":0.40658053755760193},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.38923180103302},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.36297959089279175},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3278372287750244},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.09521418809890747}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms14050376","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14050376","pdf_url":"https://www.mdpi.com/2075-1680/14/5/376/pdf?version=1747400500","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:4248d72606fc451a83a0b88a1201ae7c","is_oa":true,"landing_page_url":"https://doaj.org/article/4248d72606fc451a83a0b88a1201ae7c","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 14, Iss 5, p 376 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms14050376","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14050376","pdf_url":"https://www.mdpi.com/2075-1680/14/5/376/pdf?version=1747400500","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4410439834.pdf","grobid_xml":"https://content.openalex.org/works/W4410439834.grobid-xml"},"referenced_works_count":51,"referenced_works":["https://openalex.org/W1983411149","https://openalex.org/W2060356946","https://openalex.org/W2108663617","https://openalex.org/W2314956446","https://openalex.org/W2510725858","https://openalex.org/W2586576485","https://openalex.org/W2606848891","https://openalex.org/W2769725909","https://openalex.org/W2775378605","https://openalex.org/W2776525081","https://openalex.org/W2786642484","https://openalex.org/W2807734377","https://openalex.org/W2951237654","https://openalex.org/W3008460648","https://openalex.org/W3008882901","https://openalex.org/W3047240004","https://openalex.org/W3080792345","https://openalex.org/W3093892710","https://openalex.org/W3111156163","https://openalex.org/W3113228700","https://openalex.org/W3115380054","https://openalex.org/W3123659041","https://openalex.org/W3136055026","https://openalex.org/W3137404460","https://openalex.org/W3158583231","https://openalex.org/W3167186227","https://openalex.org/W3200411489","https://openalex.org/W4205502199","https://openalex.org/W4221051089","https://openalex.org/W4282912738","https://openalex.org/W4294572825","https://openalex.org/W4313328568","https://openalex.org/W4313361855","https://openalex.org/W4377098825","https://openalex.org/W4381853670","https://openalex.org/W4387479821","https://openalex.org/W4390115805","https://openalex.org/W4390741204","https://openalex.org/W4391850757","https://openalex.org/W4393086241","https://openalex.org/W4399744224","https://openalex.org/W4403931006","https://openalex.org/W4405265942","https://openalex.org/W4406808058","https://openalex.org/W4407630428","https://openalex.org/W4407947910","https://openalex.org/W4408055770","https://openalex.org/W4408721530","https://openalex.org/W4410109383","https://openalex.org/W4410244133","https://openalex.org/W6786636533"],"related_works":["https://openalex.org/W3082846906","https://openalex.org/W2972268228","https://openalex.org/W4386695727","https://openalex.org/W3123182488","https://openalex.org/W4308570214","https://openalex.org/W4391810631","https://openalex.org/W2245214771","https://openalex.org/W2073973926","https://openalex.org/W2333053900","https://openalex.org/W1975562467"],"abstract_inverted_index":{"The":[0,55,66,98],"well-known":[1],"nonlinear":[2],"Schr\u00f6dinger":[3],"equation":[4],"(NLSE)":[5],"plays":[6],"a":[7,28],"crucial":[8],"role":[9],"in":[10,17,48,77,87,103,118],"describing":[11],"the":[12,25,35,39,45,52,62,71,82,91,113,119,127,132],"temporal":[13],"evolution":[14],"of":[15,69,112,115,131],"disturbances":[16],"marginally":[18],"stable":[19],"or":[20],"unstable":[21],"media.":[22],"However,":[23],"when":[24],"media":[26],"is":[27,41,58,75,90,106],"fractal":[29,36,46,63,72,83,99,120],"form,":[30],"it":[31],"becomes":[32],"ineffective.":[33],"Thus,":[34],"modification":[37],"to":[38,60,96,108],"NLSE":[40],"presented":[42,76],"based":[43],"on":[44],"derivative":[47],"this":[49,88,104],"work":[50],"for":[51,126],"first":[53,92],"time.":[54],"semi-inverse":[56],"method":[57],"employed":[59],"establish":[61],"variational":[64,73,84,100,133],"principle.":[65],"entire":[67],"process":[68],"deriving":[70],"principle":[74,85,101],"detail.":[78],"To":[79],"our":[80,110],"knowledge,":[81],"mentioned":[86],"article":[89],"exploration":[93,130],"and":[94,122,129],"report":[95],"date.":[97],"established":[102],"paper":[105],"expected":[107],"deepen":[109],"understanding":[111],"essence":[114],"physical":[116],"phenomena":[117],"space":[121],"offer":[123],"new":[124],"ideas":[125],"application":[128],"approaches.":[134]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":12}],"updated_date":"2026-03-13T16:22:10.518609","created_date":"2025-10-10T00:00:00"}
