{"id":"https://openalex.org/W3088810264","doi":"https://doi.org/10.1142/s021812742050162x","title":"A New Type of Solitary Wave Solution of the mKdV Equation Under Singular Perturbations","display_name":"A New Type of Solitary Wave Solution of the mKdV Equation Under Singular Perturbations","publication_year":2020,"publication_date":"2020-09-15","ids":{"openalex":"https://openalex.org/W3088810264","doi":"https://doi.org/10.1142/s021812742050162x","mag":"3088810264"},"language":"en","primary_location":{"id":"doi:10.1142/s021812742050162x","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s021812742050162x","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/A5100448142","display_name":"Lijun Zhang","orcid":"https://orcid.org/0000-0001-5697-4611"},"institutions":[{"id":"https://openalex.org/I80143920","display_name":"Shandong University of Science and Technology","ror":"https://ror.org/04gtjhw98","country_code":"CN","type":"education","lineage":["https://openalex.org/I80143920"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lijun Zhang","raw_affiliation_strings":["Department of Mathematics, Shandong University of Science and Technology, Qingdao, Shandong 266590, P. R. China"],"raw_orcid":"https://orcid.org/0000-0001-5697-4611","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shandong University of Science and Technology, Qingdao, Shandong 266590, P. R. China","institution_ids":["https://openalex.org/I80143920"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030649439","display_name":"Maoan Han","orcid":"https://orcid.org/0000-0001-8792-1256"},"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":"Maoan Han","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"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040653613","display_name":"Mingji Zhang","orcid":"https://orcid.org/0000-0002-8249-4758"},"institutions":[{"id":"https://openalex.org/I207123951","display_name":"New Mexico Institute of Mining and Technology","ror":"https://ror.org/005p9kw61","country_code":"US","type":"education","lineage":["https://openalex.org/I207123951"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mingji Zhang","raw_affiliation_strings":["Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM, 87801, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM, 87801, USA","institution_ids":["https://openalex.org/I207123951"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5015192518","display_name":"Chaudry Masood Khalique","orcid":"https://orcid.org/0000-0002-1986-4859"},"institutions":[{"id":"https://openalex.org/I27765905","display_name":"North-West University","ror":"https://ror.org/010f1sq29","country_code":"ZA","type":"education","lineage":["https://openalex.org/I27765905"]},{"id":"https://openalex.org/I80143920","display_name":"Shandong University of Science and Technology","ror":"https://ror.org/04gtjhw98","country_code":"CN","type":"education","lineage":["https://openalex.org/I80143920"]}],"countries":["CN","ZA"],"is_corresponding":false,"raw_author_name":"Chaudry Masood Khalique","raw_affiliation_strings":["Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, South Africa","Department of Mathematics, Shandong University of Science and Technology, Qingdao, Shandong 266590, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, South Africa","institution_ids":["https://openalex.org/I27765905"]},{"raw_affiliation_string":"Department of Mathematics, Shandong University of Science and Technology, Qingdao, Shandong 266590, P. R. China","institution_ids":["https://openalex.org/I80143920"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.2705,"has_fulltext":false,"cited_by_count":32,"citation_normalized_percentile":{"value":0.88295253,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":97,"max":99},"biblio":{"volume":"30","issue":"11","first_page":"2050162","last_page":"2050162"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9987999796867371,"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/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.9894999861717224,"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/homoclinic-orbit","display_name":"Homoclinic orbit","score":0.900594174861908},{"id":"https://openalex.org/keywords/singular-perturbation","display_name":"Singular perturbation","score":0.7362909913063049},{"id":"https://openalex.org/keywords/breather","display_name":"Breather","score":0.6867193579673767},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5900036096572876},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5196872353553772},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.5109290480613708},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4895809590816498},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.4893178641796112},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.45674553513526917},{"id":"https://openalex.org/keywords/wave-equation","display_name":"Wave equation","score":0.44684261083602905},{"id":"https://openalex.org/keywords/periodic-wave","display_name":"Periodic wave","score":0.4113944172859192},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.39622125029563904},{"id":"https://openalex.org/keywords/traveling-wave","display_name":"Traveling wave","score":0.32835233211517334},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.1960342824459076},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07186335325241089}],"concepts":[{"id":"https://openalex.org/C200581526","wikidata":"https://www.wikidata.org/wiki/Q690494","display_name":"Homoclinic orbit","level":4,"score":0.900594174861908},{"id":"https://openalex.org/C165160513","wikidata":"https://www.wikidata.org/wiki/Q7524249","display_name":"Singular perturbation","level":2,"score":0.7362909913063049},{"id":"https://openalex.org/C74455749","wikidata":"https://www.wikidata.org/wiki/Q5254599","display_name":"Breather","level":3,"score":0.6867193579673767},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5900036096572876},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5196872353553772},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.5109290480613708},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4895809590816498},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4893178641796112},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.45674553513526917},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.44684261083602905},{"id":"https://openalex.org/C2993172089","wikidata":"https://www.wikidata.org/wiki/Q37172","display_name":"Periodic wave","level":3,"score":0.4113944172859192},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.39622125029563904},{"id":"https://openalex.org/C2988173416","wikidata":"https://www.wikidata.org/wiki/Q37172","display_name":"Traveling wave","level":2,"score":0.32835233211517334},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.1960342824459076},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07186335325241089},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s021812742050162x","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s021812742050162x","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/G2195079576","display_name":null,"funder_award_id":"11431008","funder_id":"https://openalex.org/F4320315254","funder_display_name":"Innovative Research Group Project of the National Natural Science Foundation of China"},{"id":"https://openalex.org/G2355614857","display_name":"\u57fa\u4e8e\u5b9a\u6027\u548c\u5206\u5c94\u7406\u8bba\u5206\u6790\u7684\u9ad8\u9636\u975e\u7ebf\u6027\u6ce2\u65b9\u7a0b\u52a8\u529b\u5b66\u7814\u7a76","funder_award_id":"11672270","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5151787008","display_name":null,"funder_award_id":"11771296","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320315254","display_name":"Innovative Research Group Project of the National Natural Science Foundation of China","ror":null},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W99189181","https://openalex.org/W100484353","https://openalex.org/W1480223776","https://openalex.org/W1552987673","https://openalex.org/W1966646678","https://openalex.org/W1990958245","https://openalex.org/W1996316724","https://openalex.org/W2001958656","https://openalex.org/W2016767879","https://openalex.org/W2016830029","https://openalex.org/W2028798193","https://openalex.org/W2064757593","https://openalex.org/W2073061323","https://openalex.org/W2075729358","https://openalex.org/W2083157056","https://openalex.org/W2089876099","https://openalex.org/W2094380752","https://openalex.org/W2224899793","https://openalex.org/W2339032597","https://openalex.org/W2507381567","https://openalex.org/W2911985954","https://openalex.org/W2929773227","https://openalex.org/W4238297087","https://openalex.org/W4294967943"],"related_works":["https://openalex.org/W2356279728","https://openalex.org/W1967698464","https://openalex.org/W2005794525","https://openalex.org/W3088810264","https://openalex.org/W3130706592","https://openalex.org/W2019676381","https://openalex.org/W2067948123","https://openalex.org/W2083157056","https://openalex.org/W2163513648","https://openalex.org/W2383535339"],"abstract_inverted_index":{"In":[0],"this":[1],"work,":[2],"we":[3],"examine":[4],"the":[5,10,88,104,116,120,125],"solitary":[6,37,67,122],"wave":[7,38,46,55,68,107,117],"solutions":[8,39],"of":[9,22,36,40,66,87,103,119],"mKdV":[11,41],"equation":[12],"with":[13,128],"small":[14,58],"singular":[15,24,59],"perturbations.":[16,60],"Our":[17,30],"analysis":[18],"is":[19,92],"a":[20,63,83],"combination":[21],"geometric":[23],"perturbation":[25],"theory":[26],"and":[27,48,73,99,124,131],"Melnikov\u2019s":[28],"method.":[29],"result":[31],"shows":[32],"that":[33],"two":[34],"families":[35],"equation,":[42],"having":[43],"arbitrary":[44],"positive":[45],"speeds":[47,56,118],"infinite":[49],"boundary":[50],"limits,":[51],"persist":[52],"for":[53],"selected":[54],"after":[57],"More":[61],"importantly,":[62],"new":[64],"type":[65],"solution":[69],"possessing":[70],"both":[71,129],"valley":[72,130],"peak,":[74],"named":[75],"as":[76],"breather":[77],"in":[78],"physics,":[79],"which":[80],"corresponds":[81],"to":[82,113],"big":[84],"homoclinic":[85],"loop":[86],"associated":[89],"dynamical":[90],"system":[91],"observed.":[93],"It":[94],"reveals":[95],"an":[96],"exotic":[97],"phenomenon":[98],"exhibits":[100],"rich":[101],"dynamics":[102],"perturbed":[105],"nonlinear":[106],"equation.":[108],"Numerical":[109],"simulations":[110],"are":[111],"performed":[112],"further":[114],"detect":[115],"persistent":[121],"waves":[123],"nontrivial":[126],"one":[127],"peak.":[132]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":7},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":6}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}