{"id":"https://openalex.org/W4307703013","doi":"https://doi.org/10.3390/sym14112259","title":"On New Hamiltonian Structures of Two Integrable Couplings","display_name":"On New Hamiltonian Structures of Two Integrable Couplings","publication_year":2022,"publication_date":"2022-10-27","ids":{"openalex":"https://openalex.org/W4307703013","doi":"https://doi.org/10.3390/sym14112259"},"language":"en","primary_location":{"id":"doi:10.3390/sym14112259","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14112259","pdf_url":"https://www.mdpi.com/2073-8994/14/11/2259/pdf?version=1668053249","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/2073-8994/14/11/2259/pdf?version=1668053249","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100345684","display_name":"Yu Liu","orcid":"https://orcid.org/0000-0001-9686-1726"},"institutions":[{"id":"https://openalex.org/I113940042","display_name":"Shanghai University","ror":"https://ror.org/006teas31","country_code":"CN","type":"education","lineage":["https://openalex.org/I113940042"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yu Liu","raw_affiliation_strings":["Department of Mathematics, Shanghai University, Shanghai 200444, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shanghai University, Shanghai 200444, China","institution_ids":["https://openalex.org/I113940042"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100327090","display_name":"Jin Liu","orcid":"https://orcid.org/0009-0002-3977-0216"},"institutions":[{"id":"https://openalex.org/I113940042","display_name":"Shanghai University","ror":"https://ror.org/006teas31","country_code":"CN","type":"education","lineage":["https://openalex.org/I113940042"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jin Liu","raw_affiliation_strings":["Department of Mathematics, Shanghai University, Shanghai 200444, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shanghai University, Shanghai 200444, China","institution_ids":["https://openalex.org/I113940042"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5069105446","display_name":"Da\u2010jun Zhang","orcid":"https://orcid.org/0000-0003-3691-4165"},"institutions":[{"id":"https://openalex.org/I113940042","display_name":"Shanghai University","ror":"https://ror.org/006teas31","country_code":"CN","type":"education","lineage":["https://openalex.org/I113940042"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Da-jun Zhang","raw_affiliation_strings":["Department of Mathematics, Shanghai University, Shanghai 200444, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shanghai University, Shanghai 200444, China","institution_ids":["https://openalex.org/I113940042"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5069105446"],"corresponding_institution_ids":["https://openalex.org/I113940042"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.2504,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.51982265,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"14","issue":"11","first_page":"2259","last_page":"2259"},"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9847999811172485,"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"}},{"id":"https://openalex.org/T12748","display_name":"Molecular spectroscopy and chirality","score":0.9735000133514404,"subfield":{"id":"https://openalex.org/subfields/1607","display_name":"Spectroscopy"},"field":{"id":"https://openalex.org/fields/16","display_name":"Chemistry"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/integrable-system","display_name":"Integrable system","score":0.9378217458724976},{"id":"https://openalex.org/keywords/hamiltonian","display_name":"Hamiltonian (control theory)","score":0.7711050510406494},{"id":"https://openalex.org/keywords/poisson-bracket","display_name":"Poisson bracket","score":0.6811802387237549},{"id":"https://openalex.org/keywords/superintegrable-hamiltonian-system","display_name":"Superintegrable Hamiltonian system","score":0.6019395589828491},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5795835256576538},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.561492383480072},{"id":"https://openalex.org/keywords/hamiltonian-system","display_name":"Hamiltonian system","score":0.5421434640884399},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.508592963218689},{"id":"https://openalex.org/keywords/poisson-distribution","display_name":"Poisson distribution","score":0.5039340853691101},{"id":"https://openalex.org/keywords/poisson-manifold","display_name":"Poisson manifold","score":0.4942643642425537},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4735245704650879},{"id":"https://openalex.org/keywords/covariant-hamiltonian-field-theory","display_name":"Covariant Hamiltonian field theory","score":0.44189220666885376},{"id":"https://openalex.org/keywords/lie-algebra","display_name":"Lie algebra","score":0.07522013783454895},{"id":"https://openalex.org/keywords/symplectic-geometry","display_name":"Symplectic geometry","score":0.039157480001449585}],"concepts":[{"id":"https://openalex.org/C200741047","wikidata":"https://www.wikidata.org/wiki/Q1957758","display_name":"Integrable system","level":2,"score":0.9378217458724976},{"id":"https://openalex.org/C130787639","wikidata":"https://www.wikidata.org/wiki/Q5645293","display_name":"Hamiltonian (control theory)","level":2,"score":0.7711050510406494},{"id":"https://openalex.org/C188845816","wikidata":"https://www.wikidata.org/wiki/Q1052775","display_name":"Poisson bracket","level":3,"score":0.6811802387237549},{"id":"https://openalex.org/C177715462","wikidata":"https://www.wikidata.org/wiki/Q7643453","display_name":"Superintegrable Hamiltonian system","level":4,"score":0.6019395589828491},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5795835256576538},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.561492383480072},{"id":"https://openalex.org/C121770821","wikidata":"https://www.wikidata.org/wiki/Q2072471","display_name":"Hamiltonian system","level":2,"score":0.5421434640884399},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.508592963218689},{"id":"https://openalex.org/C100906024","wikidata":"https://www.wikidata.org/wiki/Q205692","display_name":"Poisson distribution","level":2,"score":0.5039340853691101},{"id":"https://openalex.org/C197375991","wikidata":"https://www.wikidata.org/wiki/Q1464936","display_name":"Poisson manifold","level":3,"score":0.4942643642425537},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4735245704650879},{"id":"https://openalex.org/C109798219","wikidata":"https://www.wikidata.org/wiki/Q25304038","display_name":"Covariant Hamiltonian field theory","level":3,"score":0.44189220666885376},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.07522013783454895},{"id":"https://openalex.org/C168619227","wikidata":"https://www.wikidata.org/wiki/Q2190991","display_name":"Symplectic geometry","level":2,"score":0.039157480001449585},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14112259","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14112259","pdf_url":"https://www.mdpi.com/2073-8994/14/11/2259/pdf?version=1668053249","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:31301345f4f84eb8afad5219a28902e2","is_oa":true,"landing_page_url":"https://doaj.org/article/31301345f4f84eb8afad5219a28902e2","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":"Symmetry, Vol 14, Iss 11, p 2259 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/11/2259/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14112259","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/sym14112259","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14112259","pdf_url":"https://www.mdpi.com/2073-8994/14/11/2259/pdf?version=1668053249","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":[{"id":"https://openalex.org/G1912760790","display_name":null,"funder_award_id":"12126343","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2603578701","display_name":null,"funder_award_id":"11875040","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8667829189","display_name":null,"funder_award_id":"12126352","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"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4307703013.pdf","grobid_xml":"https://content.openalex.org/works/W4307703013.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W180396170","https://openalex.org/W617771690","https://openalex.org/W1524373516","https://openalex.org/W1526681407","https://openalex.org/W1625897192","https://openalex.org/W1963842652","https://openalex.org/W1964997672","https://openalex.org/W1972996770","https://openalex.org/W1976346027","https://openalex.org/W1985498208","https://openalex.org/W2000079624","https://openalex.org/W2004706791","https://openalex.org/W2015208000","https://openalex.org/W2022028964","https://openalex.org/W2032083457","https://openalex.org/W2034899601","https://openalex.org/W2043531989","https://openalex.org/W2047132046","https://openalex.org/W2048917368","https://openalex.org/W2060502162","https://openalex.org/W2064396384","https://openalex.org/W2066696215","https://openalex.org/W2093785095","https://openalex.org/W2095923974","https://openalex.org/W2118575947","https://openalex.org/W2145187298","https://openalex.org/W2766163233","https://openalex.org/W2800022937","https://openalex.org/W2975558964","https://openalex.org/W3163361310","https://openalex.org/W6607367260","https://openalex.org/W6633114417","https://openalex.org/W6654067825","https://openalex.org/W6664400418"],"related_works":["https://openalex.org/W2091190549","https://openalex.org/W2084004466","https://openalex.org/W3104129854","https://openalex.org/W1796527113","https://openalex.org/W1987526249","https://openalex.org/W4226215642","https://openalex.org/W3123072352","https://openalex.org/W1999694081","https://openalex.org/W4367020943","https://openalex.org/W3104718465"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"present":[4],"new":[5,37,51],"Hamiltonian":[6,54],"operators":[7],"for":[8,35],"the":[9,13,17,36,44,50],"integrable":[10],"couplings":[11],"of":[12],"Ablowitz\u2013Kaup\u2013Newell\u2013Segur":[14],"hierarchy":[15],"and":[16,38,52],"Kaup\u2013Newell":[18],"hierarchy.":[19],"The":[20,30],"corresponding":[21],"Hamiltonians":[22,40],"allow":[23],"nontrivial":[24],"degeneration.":[25],"Multi-Hamiltonian":[26],"structures":[27],"are":[28],"investigated.":[29],"involutive":[31],"property":[32],"is":[33],"proven":[34],"known":[39,53],"with":[41],"respect":[42],"to":[43],"two":[45],"Poisson":[46],"brackets":[47],"defined":[48],"by":[49],"operators.":[55]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
