{"id":"https://openalex.org/W4412893345","doi":"https://doi.org/10.3390/axioms14080590","title":"Exact Solutions and Soliton Transmission in Relativistic Wave Phenomena of Klein\u2013Fock\u2013Gordon Equation via Subsequent Sine-Gordon Equation Method","display_name":"Exact Solutions and Soliton Transmission in Relativistic Wave Phenomena of Klein\u2013Fock\u2013Gordon Equation via Subsequent Sine-Gordon Equation Method","publication_year":2025,"publication_date":"2025-07-29","ids":{"openalex":"https://openalex.org/W4412893345","doi":"https://doi.org/10.3390/axioms14080590"},"language":"en","primary_location":{"id":"doi:10.3390/axioms14080590","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14080590","pdf_url":null,"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://doi.org/10.3390/axioms14080590","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005308068","display_name":"Muhammad Uzair","orcid":"https://orcid.org/0000-0002-0721-738X"},"institutions":[{"id":"https://openalex.org/I172780181","display_name":"University of the Punjab","ror":"https://ror.org/011maz450","country_code":"PK","type":"education","lineage":["https://openalex.org/I172780181"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Uzair","raw_affiliation_strings":["Center for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan"],"affiliations":[{"raw_affiliation_string":"Center for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan","institution_ids":["https://openalex.org/I172780181"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013556216","display_name":"Ali H. Tedjani","orcid":null},"institutions":[{"id":"https://openalex.org/I240666556","display_name":"Imam Mohammad ibn Saud Islamic University","ror":"https://ror.org/05gxjyb39","country_code":"SA","type":"education","lineage":["https://openalex.org/I240666556"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Ali H. Tedjani","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia","institution_ids":["https://openalex.org/I240666556"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102906438","display_name":"Irfan Mahmood","orcid":"https://orcid.org/0000-0002-8835-0122"},"institutions":[{"id":"https://openalex.org/I172780181","display_name":"University of the Punjab","ror":"https://ror.org/011maz450","country_code":"PK","type":"education","lineage":["https://openalex.org/I172780181"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Irfan Mahmood","raw_affiliation_strings":["Center for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan"],"affiliations":[{"raw_affiliation_string":"Center for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan","institution_ids":["https://openalex.org/I172780181"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5002193142","display_name":"Ejaz Hussain","orcid":null},"institutions":[{"id":"https://openalex.org/I172780181","display_name":"University of the Punjab","ror":"https://ror.org/011maz450","country_code":"PK","type":"education","lineage":["https://openalex.org/I172780181"]}],"countries":["PK"],"is_corresponding":true,"raw_author_name":"Ejaz Hussain","raw_affiliation_strings":["Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan","institution_ids":["https://openalex.org/I172780181"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5002193142"],"corresponding_institution_ids":["https://openalex.org/I172780181"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":2.5123,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.89873086,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"14","issue":"8","first_page":"590","last_page":"590"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9998999834060669,"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":0.9998999834060669,"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":0.996999979019165,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9664999842643738,"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/sine-gordon-equation","display_name":"sine-Gordon equation","score":0.7426966428756714},{"id":"https://openalex.org/keywords/klein\u2013gordon-equation","display_name":"Klein\u2013Gordon equation","score":0.7038946151733398},{"id":"https://openalex.org/keywords/sine","display_name":"Sine","score":0.5823198556900024},{"id":"https://openalex.org/keywords/soliton","display_name":"Soliton","score":0.577064573764801},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.570636510848999},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.5132172107696533},{"id":"https://openalex.org/keywords/fock-space","display_name":"Fock space","score":0.5011482238769531},{"id":"https://openalex.org/keywords/transmission","display_name":"Transmission (telecommunications)","score":0.4178355634212494},{"id":"https://openalex.org/keywords/quantum-electrodynamics","display_name":"Quantum electrodynamics","score":0.38601866364479065},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.34215134382247925},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.24916917085647583},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06292456388473511},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.06033805012702942}],"concepts":[{"id":"https://openalex.org/C44444269","wikidata":"https://www.wikidata.org/wiki/Q2558473","display_name":"sine-Gordon equation","level":4,"score":0.7426966428756714},{"id":"https://openalex.org/C101682238","wikidata":"https://www.wikidata.org/wiki/Q868967","display_name":"Klein\u2013Gordon equation","level":3,"score":0.7038946151733398},{"id":"https://openalex.org/C186661526","wikidata":"https://www.wikidata.org/wiki/Q13647261","display_name":"Sine","level":2,"score":0.5823198556900024},{"id":"https://openalex.org/C87651913","wikidata":"https://www.wikidata.org/wiki/Q464949","display_name":"Soliton","level":3,"score":0.577064573764801},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.570636510848999},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.5132172107696533},{"id":"https://openalex.org/C114852677","wikidata":"https://www.wikidata.org/wiki/Q899044","display_name":"Fock space","level":2,"score":0.5011482238769531},{"id":"https://openalex.org/C761482","wikidata":"https://www.wikidata.org/wiki/Q118093","display_name":"Transmission (telecommunications)","level":2,"score":0.4178355634212494},{"id":"https://openalex.org/C3079626","wikidata":"https://www.wikidata.org/wiki/Q234881","display_name":"Quantum electrodynamics","level":1,"score":0.38601866364479065},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.34215134382247925},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.24916917085647583},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06292456388473511},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.06033805012702942},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms14080590","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14080590","pdf_url":null,"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:b0b6617102e74ea3a9299cb36bf42f4e","is_oa":true,"landing_page_url":"https://doaj.org/article/b0b6617102e74ea3a9299cb36bf42f4e","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 8, p 590 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms14080590","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14080590","pdf_url":null,"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":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":50,"referenced_works":["https://openalex.org/W1982780449","https://openalex.org/W2000121183","https://openalex.org/W2038917479","https://openalex.org/W2043097044","https://openalex.org/W2047313333","https://openalex.org/W2065991528","https://openalex.org/W2078262241","https://openalex.org/W2177804662","https://openalex.org/W2188622247","https://openalex.org/W2593706619","https://openalex.org/W2796107341","https://openalex.org/W2899859709","https://openalex.org/W2916258464","https://openalex.org/W2964062462","https://openalex.org/W3005282209","https://openalex.org/W3078278502","https://openalex.org/W3092954222","https://openalex.org/W3093364223","https://openalex.org/W3096795810","https://openalex.org/W3128846435","https://openalex.org/W3162402424","https://openalex.org/W3164318037","https://openalex.org/W3165370155","https://openalex.org/W3171897412","https://openalex.org/W4200156200","https://openalex.org/W4206565580","https://openalex.org/W4281691402","https://openalex.org/W4285600288","https://openalex.org/W4292829755","https://openalex.org/W4293385776","https://openalex.org/W4296360172","https://openalex.org/W4308741688","https://openalex.org/W4309230957","https://openalex.org/W4324278931","https://openalex.org/W4382395484","https://openalex.org/W4385147143","https://openalex.org/W4387469998","https://openalex.org/W4396828479","https://openalex.org/W4400851295","https://openalex.org/W4401409722","https://openalex.org/W4401502436","https://openalex.org/W4402996390","https://openalex.org/W4404851628","https://openalex.org/W4406786645","https://openalex.org/W4408272903","https://openalex.org/W4408290539","https://openalex.org/W4409174883","https://openalex.org/W4410380558","https://openalex.org/W4410858582","https://openalex.org/W4411260440"],"related_works":["https://openalex.org/W2132611405","https://openalex.org/W1991976129","https://openalex.org/W2377364503","https://openalex.org/W1979238274","https://openalex.org/W1991052464","https://openalex.org/W2041243052","https://openalex.org/W3213781763","https://openalex.org/W2071701494","https://openalex.org/W2072912408","https://openalex.org/W118516215"],"abstract_inverted_index":{"This":[0],"study":[1],"explores":[2],"the":[3,43,47,59,77,94,111,125,128,141,152,157,176,185],"(1+1)-dimensional":[4,129],"Klein\u2013Fock\u2013Gordon":[5,18,130],"equation,":[6],"a":[7,22,33,64,162],"distinct":[8],"third-order":[9],"nonlinear":[10,179,191],"differential":[11],"equation":[12,19,61,159],"of":[13,46,72,79,97,113,127,143,156,178,187],"significant":[14],"theoretical":[15,26],"interest.":[16],"The":[17,70,149],"(KFGE)":[20],"plays":[21],"pivotal":[23],"role":[24],"in":[25,76,190],"physics,":[27],"modeling":[28],"high-energy":[29],"particles":[30],"and":[31,63,86,89,106,120,146,154,161,193],"providing":[32,168],"fundamental":[34],"framework":[35],"for":[36,50,136],"simulating":[37],"relativistic":[38],"wave":[39],"phenomena.":[40],"To":[41,109],"find":[42],"exact":[44],"solution":[45],"proposed":[48],"model,":[49],"this":[51],"purpose,":[52],"we":[53,116,139],"utilized":[54],"two":[55],"effective":[56],"techniques,":[57],"including":[58],"sine-Gordon":[60,158],"method":[62,160],"new":[65,163],"extended":[66,164],"direct":[67,165],"algebraic":[68,166],"method.":[69],"novelty":[71],"these":[73,114],"approaches":[74],"lies":[75],"form":[78,96],"different":[80,95],"solutions":[81,170],"such":[82],"as":[83],"hyperbolic,":[84],"trigonometric,":[85],"rational":[87],"functions,":[88],"their":[90,147],"graphical":[91],"representations":[92],"demonstrate":[93,140],"solitons":[98],"like":[99],"kink":[100],"solitons,":[101,103,105],"bright":[102],"dark":[104],"periodic":[107],"waves.":[108],"illustrate":[110],"characteristics":[112],"solutions,":[115],"provide":[117],"two-dimensional,":[118],"three-dimensional,":[119],"contour":[121],"plots":[122],"that":[123,171],"visualize":[124],"magnitude":[126],"equation.":[131],"By":[132],"selecting":[133],"suitable":[134],"values":[135],"physical":[137],"parameters,":[138],"diversity":[142],"soliton":[144,188],"structures":[145],"behaviors.":[148],"results":[150,182],"highlighted":[151],"effectiveness":[153],"versatility":[155],"method,":[167],"analytical":[169],"deepen":[172],"our":[173],"insight":[174],"into":[175],"dynamics":[177],"models.":[180],"These":[181],"contribute":[183],"to":[184],"advancement":[186],"theory":[189],"optics":[192],"mathematical":[194],"physics.":[195]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}