{"id":"https://openalex.org/W3107944481","doi":"https://doi.org/10.3390/sym12121987","title":"Self-Adjointness and Conservation Laws of Frobenius Type Equations","display_name":"Self-Adjointness and Conservation Laws of Frobenius Type Equations","publication_year":2020,"publication_date":"2020-12-01","ids":{"openalex":"https://openalex.org/W3107944481","doi":"https://doi.org/10.3390/sym12121987","mag":"3107944481"},"language":"en","primary_location":{"id":"doi:10.3390/sym12121987","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12121987","pdf_url":"https://www.mdpi.com/2073-8994/12/12/1987/pdf?version=1606803140","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/2073-8994/12/12/1987/pdf?version=1606803140","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100386396","display_name":"Haifeng Wang","orcid":"https://orcid.org/0000-0002-1058-7111"},"institutions":[{"id":"https://openalex.org/I25757504","display_name":"China University of Mining and Technology","ror":"https://ror.org/01xt2dr21","country_code":"CN","type":"education","lineage":["https://openalex.org/I25757504"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Haifeng Wang","raw_affiliation_strings":["School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China","institution_ids":["https://openalex.org/I25757504"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100459702","display_name":"Yufeng Zhang","orcid":null},"institutions":[{"id":"https://openalex.org/I25757504","display_name":"China University of Mining and Technology","ror":"https://ror.org/01xt2dr21","country_code":"CN","type":"education","lineage":["https://openalex.org/I25757504"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yufeng Zhang","raw_affiliation_strings":["School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China","institution_ids":["https://openalex.org/I25757504"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5100459702"],"corresponding_institution_ids":["https://openalex.org/I25757504"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.1299,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.45454545,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":93},"biblio":{"volume":"12","issue":"12","first_page":"1987","last_page":"1987"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":1.0,"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":1.0,"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":0.9983000159263611,"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9922999739646912,"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/mathematics","display_name":"Mathematics","score":0.7358211278915405},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.6301676034927368},{"id":"https://openalex.org/keywords/frobenius-theorem","display_name":"Frobenius theorem (differential topology)","score":0.5912683606147766},{"id":"https://openalex.org/keywords/burgers-equation","display_name":"Burgers' equation","score":0.5698028802871704},{"id":"https://openalex.org/keywords/korteweg\u2013de-vries-equation","display_name":"Korteweg\u2013de Vries equation","score":0.5537760853767395},{"id":"https://openalex.org/keywords/conservation-law","display_name":"Conservation law","score":0.4968927204608917},{"id":"https://openalex.org/keywords/frobenius-group","display_name":"Frobenius group","score":0.4767266809940338},{"id":"https://openalex.org/keywords/homogeneous-space","display_name":"Homogeneous space","score":0.4719654321670532},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.44986677169799805},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4217774271965027},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.36718541383743286},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2653237283229828},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.20167455077171326},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09388798475265503},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.05644825100898743},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.049562275409698486}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7358211278915405},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.6301676034927368},{"id":"https://openalex.org/C86177461","wikidata":"https://www.wikidata.org/wiki/Q837551","display_name":"Frobenius theorem (differential topology)","level":5,"score":0.5912683606147766},{"id":"https://openalex.org/C129747778","wikidata":"https://www.wikidata.org/wiki/Q1014918","display_name":"Burgers' equation","level":3,"score":0.5698028802871704},{"id":"https://openalex.org/C146630112","wikidata":"https://www.wikidata.org/wiki/Q601796","display_name":"Korteweg\u2013de Vries equation","level":3,"score":0.5537760853767395},{"id":"https://openalex.org/C3445786","wikidata":"https://www.wikidata.org/wiki/Q205805","display_name":"Conservation law","level":2,"score":0.4968927204608917},{"id":"https://openalex.org/C127641909","wikidata":"https://www.wikidata.org/wiki/Q1457165","display_name":"Frobenius group","level":2,"score":0.4767266809940338},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.4719654321670532},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.44986677169799805},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4217774271965027},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.36718541383743286},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2653237283229828},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.20167455077171326},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09388798475265503},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.05644825100898743},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.049562275409698486},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"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/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.0},{"id":"https://openalex.org/C203934109","wikidata":"https://www.wikidata.org/wiki/Q7322952","display_name":"Ricci-flat manifold","level":4,"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":3,"locations":[{"id":"doi:10.3390/sym12121987","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12121987","pdf_url":"https://www.mdpi.com/2073-8994/12/12/1987/pdf?version=1606803140","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:8ee46d27ba9549c8954e42c7af76eb57","is_oa":true,"landing_page_url":"https://doaj.org/article/8ee46d27ba9549c8954e42c7af76eb57","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 12, Iss 12, p 1987 (2020)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/12/12/1987/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym12121987","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym12121987","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12121987","pdf_url":"https://www.mdpi.com/2073-8994/12/12/1987/pdf?version=1606803140","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":[{"score":0.7400000095367432,"id":"https://metadata.un.org/sdg/15","display_name":"Life in Land"}],"awards":[{"id":"https://openalex.org/G6678499769","display_name":null,"funder_award_id":"No.11971475","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/F4320321605","display_name":"Government of Jiangsu Province","ror":"https://ror.org/004svx814"},{"id":"https://openalex.org/F4320323154","display_name":"China University of Mining and Technology","ror":"https://ror.org/01xt2dr21"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3107944481.pdf","grobid_xml":"https://content.openalex.org/works/W3107944481.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W1573927207","https://openalex.org/W1626180465","https://openalex.org/W1977560113","https://openalex.org/W2001748301","https://openalex.org/W2007984120","https://openalex.org/W2020164266","https://openalex.org/W2029110043","https://openalex.org/W2030669316","https://openalex.org/W2043465319","https://openalex.org/W2053257495","https://openalex.org/W2067522280","https://openalex.org/W2075965386","https://openalex.org/W2076605545","https://openalex.org/W2085413857","https://openalex.org/W2085846913","https://openalex.org/W2102681519","https://openalex.org/W2236164843","https://openalex.org/W2261252858","https://openalex.org/W2269337355","https://openalex.org/W2473628137","https://openalex.org/W2757048151","https://openalex.org/W2804897682","https://openalex.org/W2889760844","https://openalex.org/W2916299170","https://openalex.org/W2983670644","https://openalex.org/W2998524017","https://openalex.org/W3047094800","https://openalex.org/W3101725055","https://openalex.org/W3123123291","https://openalex.org/W6636553930","https://openalex.org/W6781446500"],"related_works":["https://openalex.org/W2369511497","https://openalex.org/W2319082512","https://openalex.org/W2351208265","https://openalex.org/W2048471423","https://openalex.org/W2101741420","https://openalex.org/W2091193375","https://openalex.org/W1519781192","https://openalex.org/W2368637410","https://openalex.org/W2582546171","https://openalex.org/W2110627003"],"abstract_inverted_index":{"The":[0,37],"Frobenius":[1,6,13,16,20,51,82,86],"KDV":[2,83],"equation":[3,8,18,23,88],"and":[4,11,19,54,65,84],"the":[5,12,35,48,58,66,78,81,85,93],"KP":[7,87],"are":[9,24,40,62,70],"introduced,":[10],"Kompaneets":[14],"equation,":[15],"Burgers":[17],"Harry":[21],"Dym":[22],"constructed":[25],"by":[26],"taking":[27],"values":[28],"in":[29,34,89],"a":[30],"commutative":[31],"subalgebra":[32],"Z2\u03b5":[33],"paper.":[36],"five":[38],"equations":[39,61,69],"selected":[41],"as":[42],"examples":[43],"to":[44,91],"help":[45],"us":[46],"study":[47],"self-adjointness":[49],"of":[50,80],"type":[52],"equations,":[53],"we":[55,76],"show":[56],"that":[57,75],"first":[59],"two":[60],"quasi":[63],"self-adjoint":[64],"last":[67],"three":[68],"nonlinear":[71],"self-adjointness.":[72],"It":[73],"follows":[74],"give":[77],"symmetries":[79],"order":[90],"construct":[92],"corresponding":[94],"conservation":[95],"laws.":[96]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}