{"id":"https://openalex.org/W3005545283","doi":"https://doi.org/10.1142/s0218127420500042","title":"Bifurcations and Exact Solutions for a Class of MKdV Equations with the Conformable Fractional Derivative via Dynamical System Method","display_name":"Bifurcations and Exact Solutions for a Class of MKdV Equations with the Conformable Fractional Derivative via Dynamical System Method","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3005545283","doi":"https://doi.org/10.1142/s0218127420500042","mag":"3005545283"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127420500042","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127420500042","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","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/A5101513231","display_name":"Jianli Liang","orcid":"https://orcid.org/0000-0002-8777-146X"},"institutions":[{"id":"https://openalex.org/I119045251","display_name":"Huaqiao University","ror":"https://ror.org/03frdh605","country_code":"CN","type":"education","lineage":["https://openalex.org/I119045251"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jianli Liang","raw_affiliation_strings":["School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China","institution_ids":["https://openalex.org/I119045251"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021875923","display_name":"Longkun Tang","orcid":"https://orcid.org/0000-0002-1354-239X"},"institutions":[{"id":"https://openalex.org/I119045251","display_name":"Huaqiao University","ror":"https://ror.org/03frdh605","country_code":"CN","type":"education","lineage":["https://openalex.org/I119045251"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Longkun Tang","raw_affiliation_strings":["School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China"],"raw_orcid":"https://orcid.org/0000-0002-1354-239X","affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China","institution_ids":["https://openalex.org/I119045251"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5010932652","display_name":"Yonghui Xia","orcid":"https://orcid.org/0000-0001-8918-3509"},"institutions":[{"id":"https://openalex.org/I135237710","display_name":"Zhejiang Normal University","ror":"https://ror.org/01vevwk45","country_code":"CN","type":"education","lineage":["https://openalex.org/I135237710"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yonghui Xia","raw_affiliation_strings":["Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China"],"raw_orcid":"https://orcid.org/0000-0001-8918-3509","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China","institution_ids":["https://openalex.org/I135237710"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100661899","display_name":"Yi Zhang","orcid":"https://orcid.org/0000-0001-7764-4953"},"institutions":[{"id":"https://openalex.org/I135237710","display_name":"Zhejiang Normal University","ror":"https://ror.org/01vevwk45","country_code":"CN","type":"education","lineage":["https://openalex.org/I135237710"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yi Zhang","raw_affiliation_strings":["Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China","institution_ids":["https://openalex.org/I135237710"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.5422,"has_fulltext":false,"cited_by_count":24,"citation_normalized_percentile":{"value":0.73452559,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"30","issue":"01","first_page":"2050004","last_page":"2050004"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"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"}},{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9997000098228455,"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9957000017166138,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/conformable-matrix","display_name":"Conformable matrix","score":0.8640621304512024},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.751166582107544},{"id":"https://openalex.org/keywords/ode","display_name":"Ode","score":0.7121322751045227},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5779527425765991},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5663514137268066},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.5514454245567322},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5210503935813904},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.5053172707557678},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.4520227909088135},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.4518411457538605},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.43778687715530396},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.42564278841018677},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4226109981536865},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.37222951650619507},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.35982954502105713},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.35243701934814453},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.16842910647392273},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11253836750984192},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07127818465232849}],"concepts":[{"id":"https://openalex.org/C86072612","wikidata":"https://www.wikidata.org/wiki/Q5160239","display_name":"Conformable matrix","level":2,"score":0.8640621304512024},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.751166582107544},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.7121322751045227},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5779527425765991},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5663514137268066},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.5514454245567322},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5210503935813904},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.5053172707557678},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.4520227909088135},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4518411457538605},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.43778687715530396},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.42564278841018677},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4226109981536865},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37222951650619507},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.35982954502105713},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.35243701934814453},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.16842910647392273},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11253836750984192},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07127818465232849},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218127420500042","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127420500042","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G319631038","display_name":null,"funder_award_id":"11671176","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4447944634","display_name":null,"funder_award_id":"2018J01001","funder_id":"https://openalex.org/F4320321878","funder_display_name":"Natural Science Foundation of Fujian Province"},{"id":"https://openalex.org/G5136151406","display_name":null,"funder_award_id":"11871251","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8022640692","display_name":null,"funder_award_id":"LY15A010007","funder_id":"https://openalex.org/F4320338464","funder_display_name":"Natural Science Foundation of Zhejiang Province"},{"id":"https://openalex.org/G8094506775","display_name":null,"funder_award_id":"61573004","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321878","display_name":"Natural Science Foundation of Fujian Province","ror":null},{"id":"https://openalex.org/F4320338464","display_name":"Natural Science Foundation of Zhejiang Province","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1131478117","https://openalex.org/W1966553738","https://openalex.org/W1992271136","https://openalex.org/W1994828676","https://openalex.org/W2008052025","https://openalex.org/W2023107300","https://openalex.org/W2046881861","https://openalex.org/W2051483432","https://openalex.org/W2063067239","https://openalex.org/W2066637244","https://openalex.org/W2076383796","https://openalex.org/W2079127676","https://openalex.org/W2082275537","https://openalex.org/W2084797093","https://openalex.org/W2086379987","https://openalex.org/W2089766654","https://openalex.org/W2091234113","https://openalex.org/W2220254234","https://openalex.org/W2221458659","https://openalex.org/W2554884698","https://openalex.org/W2575265544","https://openalex.org/W2583712254","https://openalex.org/W2756500969","https://openalex.org/W2772331899","https://openalex.org/W2790253100","https://openalex.org/W2802276519","https://openalex.org/W2929773227","https://openalex.org/W4289981527"],"related_works":["https://openalex.org/W2792191619","https://openalex.org/W3158217983","https://openalex.org/W4248103533","https://openalex.org/W3046588275","https://openalex.org/W3085259849","https://openalex.org/W3046886605","https://openalex.org/W3049703542","https://openalex.org/W3008631921","https://openalex.org/W4313364931","https://openalex.org/W4302307873"],"abstract_inverted_index":{"In":[0,19],"2014,":[1],"Khalil":[2],"et":[3],"al.":[4],"[2014]":[5],"proposed":[6,71],"the":[7,16,24,33,46,55,67,79,91,99,106,110,120,126,132,135,141,158,164],"conformable":[8,47,165],"fractional":[9,48,100,142,166],"derivative,":[10],"which":[11,62,96],"obeys":[12],"chain":[13],"rule":[14],"and":[15],"Leibniz":[17],"rule.":[18],"this":[20,151],"paper,":[21],"motivated":[22],"by":[23],"monograph":[25],"of":[26,41,58,140],"Jibin":[27],"Li":[28],"[Li,":[29],"2013],":[30],"we":[31,89],"study":[32],"exact":[34,121],"traveling":[35,80],"wave":[36,81],"solutions":[37,122],"for":[38],"a":[39],"class":[40],"third-order":[42],"MKdV":[43],"equations":[44,162],"with":[45,125,163],"derivative.":[49,167],"Our":[50],"approach":[51],"is":[52,63,138,148],"based":[53],"on":[54,98,109],"bifurcation":[56],"theory":[57],"planar":[59],"dynamical":[60],"systems,":[61],"much":[64],"different":[65],"from":[66],"simplest":[68],"equation":[69],"method":[70,152],"in":[72,130],"[Chen":[73],"&amp;":[74],"Jiang,":[75],"2018].":[76],"By":[77],"employing":[78],"transformation":[82],"[Formula:":[83,86,102,112,117,144],"see":[84,87,103,113,118,145],"text]":[85],"text],":[88,104,119],"reduce":[90],"PDE":[92],"to":[93,156],"an":[94],"ODE":[95,137],"depends":[97,108],"order":[101,111,143],"then":[105],"analysis":[107],"text].":[114,146],"Moreover,":[115],"as":[116],"are":[123],"consistent":[124],"integer":[127],"PDE.":[128],"However,":[129],"all":[131],"existing":[133],"papers,":[134],"reduced":[136],"independent":[139],"It":[147],"believed":[149],"that":[150],"can":[153],"be":[154],"applicable":[155],"solve":[157],"other":[159],"nonlinear":[160],"differential":[161]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
