{"id":"https://openalex.org/W4388265021","doi":"https://doi.org/10.3390/sym15111972","title":"Multiple Soliton Solutions for Coupled Modified Korteweg\u2013de Vries (mkdV) with a Time-Dependent Variable Coefficient","display_name":"Multiple Soliton Solutions for Coupled Modified Korteweg\u2013de Vries (mkdV) with a Time-Dependent Variable Coefficient","publication_year":2023,"publication_date":"2023-10-25","ids":{"openalex":"https://openalex.org/W4388265021","doi":"https://doi.org/10.3390/sym15111972"},"language":"en","primary_location":{"id":"doi:10.3390/sym15111972","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15111972","pdf_url":"https://www.mdpi.com/2073-8994/15/11/1972/pdf?version=1698230324","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/11/1972/pdf?version=1698230324","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5113088085","display_name":"Haroon D.S. Adam","orcid":null},"institutions":[{"id":"https://openalex.org/I47164929","display_name":"Najran University","ror":"https://ror.org/05edw4a90","country_code":"SA","type":"education","lineage":["https://openalex.org/I47164929"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Haroon D. S. Adam","raw_affiliation_strings":["Department of Basic Sciences, Najran University, P.O. Box 1988, Najran 61441, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Basic Sciences, Najran University, P.O. Box 1988, Najran 61441, Saudi Arabia","institution_ids":["https://openalex.org/I47164929"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5081185783","display_name":"Khalid I. A. Ahmed","orcid":"https://orcid.org/0009-0009-4298-4806"},"institutions":[{"id":"https://openalex.org/I47164929","display_name":"Najran University","ror":"https://ror.org/05edw4a90","country_code":"SA","type":"education","lineage":["https://openalex.org/I47164929"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Khalid I. A. Ahmed","raw_affiliation_strings":["Department of Basic Sciences, Najran University, P.O. Box 1988, Najran 61441, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Basic Sciences, Najran University, P.O. Box 1988, Najran 61441, Saudi Arabia","institution_ids":["https://openalex.org/I47164929"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110886540","display_name":"Mukhtar Yagoub Youssif","orcid":null},"institutions":[{"id":"https://openalex.org/I179331831","display_name":"Taif University","ror":"https://ror.org/014g1a453","country_code":"SA","type":"education","lineage":["https://openalex.org/I179331831"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Mukhtar Yagoub Youssif","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia","institution_ids":["https://openalex.org/I179331831"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5015681053","display_name":"Mar\u00edn Mar\u00edn","orcid":"https://orcid.org/0000-0003-1552-3763"},"institutions":[{"id":"https://openalex.org/I3123573182","display_name":"Transylvania University of Bra\u0219ov","ror":"https://ror.org/01cg9ws23","country_code":"RO","type":"education","lineage":["https://openalex.org/I3123573182"]},{"id":"https://openalex.org/I33800924","display_name":"Academia Oamenilor de \u0218tiin\u021b\u0103 din Rom\u00e2nia","ror":"https://ror.org/04ybnj478","country_code":"RO","type":"facility","lineage":["https://openalex.org/I33800924"]}],"countries":["RO"],"is_corresponding":true,"raw_author_name":"Marin Marin","raw_affiliation_strings":["Academy of Romanian Scientists, Ilfov Street, No. 3, 050045 Bucharest, Romania","Department of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Brasov, Romania"],"raw_orcid":"https://orcid.org/0000-0003-1552-3763","affiliations":[{"raw_affiliation_string":"Academy of Romanian Scientists, Ilfov Street, No. 3, 050045 Bucharest, Romania","institution_ids":["https://openalex.org/I33800924"]},{"raw_affiliation_string":"Department of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Brasov, Romania","institution_ids":["https://openalex.org/I3123573182"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5015681053"],"corresponding_institution_ids":["https://openalex.org/I3123573182","https://openalex.org/I33800924"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.7997,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.70521332,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":99},"biblio":{"volume":"15","issue":"11","first_page":"1972","last_page":"1972"},"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":0.9993000030517578,"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/T10988","display_name":"Advanced Fiber Laser Technologies","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/soliton","display_name":"Soliton","score":0.7199385166168213},{"id":"https://openalex.org/keywords/symbolic-computation","display_name":"Symbolic computation","score":0.7032287120819092},{"id":"https://openalex.org/keywords/bilinear-interpolation","display_name":"Bilinear interpolation","score":0.6980975270271301},{"id":"https://openalex.org/keywords/maple","display_name":"Maple","score":0.6492781043052673},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.6192858219146729},{"id":"https://openalex.org/keywords/matrix-similarity","display_name":"Matrix similarity","score":0.5763428807258606},{"id":"https://openalex.org/keywords/bilinear-transform","display_name":"Bilinear transform","score":0.5727605819702148},{"id":"https://openalex.org/keywords/variable-coefficient","display_name":"Variable coefficient","score":0.5548758506774902},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5404665470123291},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.532001793384552},{"id":"https://openalex.org/keywords/transformation","display_name":"Transformation (genetics)","score":0.5171403288841248},{"id":"https://openalex.org/keywords/bilinear-form","display_name":"Bilinear form","score":0.4640691876411438},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.43612900376319885},{"id":"https://openalex.org/keywords/similarity","display_name":"Similarity (geometry)","score":0.42565861344337463},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.32805711030960083},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.27588772773742676},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.24194878339767456},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.23861846327781677},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10670575499534607},{"id":"https://openalex.org/keywords/digital-filter","display_name":"Digital filter","score":0.06204870343208313}],"concepts":[{"id":"https://openalex.org/C87651913","wikidata":"https://www.wikidata.org/wiki/Q464949","display_name":"Soliton","level":3,"score":0.7199385166168213},{"id":"https://openalex.org/C110812573","wikidata":"https://www.wikidata.org/wiki/Q175515","display_name":"Symbolic computation","level":2,"score":0.7032287120819092},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.6980975270271301},{"id":"https://openalex.org/C2780414537","wikidata":"https://www.wikidata.org/wiki/Q42292","display_name":"Maple","level":2,"score":0.6492781043052673},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.6192858219146729},{"id":"https://openalex.org/C39739852","wikidata":"https://www.wikidata.org/wiki/Q254491","display_name":"Matrix similarity","level":3,"score":0.5763428807258606},{"id":"https://openalex.org/C109786386","wikidata":"https://www.wikidata.org/wiki/Q861804","display_name":"Bilinear transform","level":4,"score":0.5727605819702148},{"id":"https://openalex.org/C3018795394","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Variable coefficient","level":2,"score":0.5548758506774902},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5404665470123291},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.532001793384552},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.5171403288841248},{"id":"https://openalex.org/C8828549","wikidata":"https://www.wikidata.org/wiki/Q837924","display_name":"Bilinear form","level":2,"score":0.4640691876411438},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.43612900376319885},{"id":"https://openalex.org/C103278499","wikidata":"https://www.wikidata.org/wiki/Q254465","display_name":"Similarity (geometry)","level":3,"score":0.42565861344337463},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32805711030960083},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.27588772773742676},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.24194878339767456},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.23861846327781677},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10670575499534607},{"id":"https://openalex.org/C36390408","wikidata":"https://www.wikidata.org/wiki/Q1163067","display_name":"Digital filter","level":3,"score":0.06204870343208313},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C59822182","wikidata":"https://www.wikidata.org/wiki/Q441","display_name":"Botany","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C106131492","wikidata":"https://www.wikidata.org/wiki/Q3072260","display_name":"Filter (signal processing)","level":2,"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/C31972630","wikidata":"https://www.wikidata.org/wiki/Q844240","display_name":"Computer vision","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym15111972","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15111972","pdf_url":"https://www.mdpi.com/2073-8994/15/11/1972/pdf?version=1698230324","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:7a9ee6596b3844dc891fa31042a7cf3f","is_oa":false,"landing_page_url":"https://doaj.org/article/7a9ee6596b3844dc891fa31042a7cf3f","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 11, p 1972 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym15111972","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15111972","pdf_url":"https://www.mdpi.com/2073-8994/15/11/1972/pdf?version=1698230324","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320323502","display_name":"Najran University","ror":"https://ror.org/05edw4a90"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4388265021.pdf","grobid_xml":"https://content.openalex.org/works/W4388265021.grobid-xml"},"referenced_works_count":40,"referenced_works":["https://openalex.org/W1964115476","https://openalex.org/W1964204275","https://openalex.org/W1979427603","https://openalex.org/W1985049458","https://openalex.org/W1996820401","https://openalex.org/W2005031767","https://openalex.org/W2009752891","https://openalex.org/W2037482896","https://openalex.org/W2050639649","https://openalex.org/W2079628965","https://openalex.org/W2118308885","https://openalex.org/W2125294736","https://openalex.org/W2136228658","https://openalex.org/W2276891719","https://openalex.org/W2290910650","https://openalex.org/W2516842535","https://openalex.org/W2560959562","https://openalex.org/W2776850568","https://openalex.org/W2885808410","https://openalex.org/W2890477977","https://openalex.org/W2902590634","https://openalex.org/W2960743217","https://openalex.org/W3013855405","https://openalex.org/W3025489219","https://openalex.org/W3038739517","https://openalex.org/W3042159028","https://openalex.org/W3176573568","https://openalex.org/W3195428251","https://openalex.org/W3200975995","https://openalex.org/W3210089652","https://openalex.org/W4200161442","https://openalex.org/W4280643398","https://openalex.org/W4281291912","https://openalex.org/W4296901123","https://openalex.org/W4309728357","https://openalex.org/W4312109213","https://openalex.org/W4327777657","https://openalex.org/W4366824603","https://openalex.org/W4367299947","https://openalex.org/W6803372614"],"related_works":["https://openalex.org/W3186157483","https://openalex.org/W2055618379","https://openalex.org/W2006965949","https://openalex.org/W3149344120","https://openalex.org/W2090764660","https://openalex.org/W2197675418","https://openalex.org/W2050356154","https://openalex.org/W4389478944","https://openalex.org/W1973033761","https://openalex.org/W2027783819"],"abstract_inverted_index":{"In":[0],"this":[1,139],"manuscript,":[2],"we":[3,31],"implement":[4,51],"analytical":[5,58],"multiple":[6,34],"soliton":[7,11,149],"wave":[8,12,35,38],"and":[9,27,37,121,134,147],"singular":[10,39],"solutions":[13,41,59],"for":[14,42,60,77,126,181],"coupled":[15,43,62],"mKdV":[16,44,63],"with":[17,45,73],"a":[18,46,71,96,161],"time-dependent":[19,47],"variable":[20,48],"coefficient.":[21,49],"Based":[22],"on":[23,171],"the":[24,52,61,90,104,113,117,143,157,168,172,175],"similarity":[25],"transformation":[26],"Hirota":[28,53,94],"bilinear":[29,54,91],"technique,":[30],"construct":[32],"both":[33],"kink":[36,40],"We":[50,110],"technique":[55,98],"to":[56,87],"compute":[57],"system.":[64],"Such":[65],"calculations":[66],"are":[67],"made":[68],"by":[69,155],"using":[70],"software":[72],"symbolic":[74],"computation":[75],"software,":[76],"instance,":[78],"Maple.":[79],"Recently":[80],"some":[81],"researchers":[82],"used":[83,102,125,131],"Maple":[84],"in":[85,103,132,167,174,186],"order":[86],"show":[88],"that":[89,116,142],"method":[92],"of":[93,106,138,145,160,178,184],"is":[95,119],"straightforward":[97],"which":[99],"can":[100,122,151],"be":[101,123,152],"approach":[105],"differential,":[107],"nonlinear":[108],"models.":[109],"analyzed":[111],"whether":[112],"experiments":[114],"proved":[115],"procedure":[118],"effective":[120],"successfully":[124],"many":[127],"other":[128],"mathematical":[129],"models":[130],"physics":[133],"engineering.":[135],"The":[136,165],"results":[137],"study":[140],"display":[141],"profiles":[144],"multiple-kink":[146],"singular-kink":[148],"types":[150],"efficiently":[153],"controlled":[154],"selecting":[156],"particular":[158],"form":[159],"similar":[162],"time":[163,179],"variable.":[164],"changes":[166,173],"solitons":[169],"based":[170],"arbitrary":[176],"function":[177],"allows":[180],"more":[182],"applications":[183],"them":[185],"applied":[187],"sciences.":[188]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2023-11-03T00:00:00"}
