{"id":"https://openalex.org/W7213772147","doi":"https://doi.org/10.3842/umzh.v78i9-10.10162","title":"\u041b\u0456\u0457\u0432\u0441\u044c\u043a\u0456 \u0442\u0430 \u0442\u043e\u0447\u043a\u043e\u0432\u0456 \u0441\u0438\u043c\u0435\u0442\u0440\u0456\u0457 \u043c\u043e\u0434\u0435\u043b\u0435\u0439 \u041d\u0438\u0436\u043d\u0438\u043a\u0430","display_name":"\u041b\u0456\u0457\u0432\u0441\u044c\u043a\u0456 \u0442\u0430 \u0442\u043e\u0447\u043a\u043e\u0432\u0456 \u0441\u0438\u043c\u0435\u0442\u0440\u0456\u0457 \u043c\u043e\u0434\u0435\u043b\u0435\u0439 \u041d\u0438\u0436\u043d\u0438\u043a\u0430","publication_year":2026,"publication_date":"2026-09-20","ids":{"openalex":"https://openalex.org/W7213772147","doi":"https://doi.org/10.3842/umzh.v78i9-10.10162"},"language":"en","primary_location":{"id":"doi:10.3842/umzh.v78i9-10.10162","doi":"https://doi.org/10.3842/umzh.v78i9-10.10162","is_oa":true,"landing_page_url":"https://doi.org/10.3842/umzh.v78i9-10.10162","pdf_url":null,"source":{"id":"https://openalex.org/S4210234218","display_name":"Ukrains\u2019kyi Matematychnyi Zhurnal","issn_l":"1027-3190","issn":["1027-3190"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":["jufo-1"],"host_organization":"https://openalex.org/P4310322009","host_organization_name":"Institute of Mathematics of the National Academy of Sciences of Ukraine","host_organization_lineage":["https://openalex.org/P4310322009","https://openalex.org/P4310321802"],"host_organization_lineage_names":["Institute of Mathematics of the National Academy of Sciences of Ukraine","National Academy of Sciences of Ukraine"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Ukrains\u2019kyi Matematychnyi Zhurnal","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.3842/umzh.v78i9-10.10162","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048792031","display_name":"Oleksandra O. Vinnichenko","orcid":"https://orcid.org/0009-0007-8089-8162","observed_orcids":["https://orcid.org/0009-0007-8089-8162"]},"institutions":[{"id":"https://openalex.org/I190444043","display_name":"Institute of Mathematics","ror":"https://ror.org/04q3mp532","country_code":"UA","type":"facility","lineage":["https://openalex.org/I149851306","https://openalex.org/I190444043"]}],"countries":["UA"],"is_corresponding":false,"raw_author_name":"\u041e\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440\u0430 \u0412\u0456\u043d\u043d\u0456\u0447\u0435\u043d\u043a\u043e","raw_affiliation_strings":["\u0406\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u041d\u0410\u041d \u0423\u043a\u0440\u0430\u0457\u043d\u0438, \u041a\u0438\u0457\u0432"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"\u0406\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u041d\u0410\u041d \u0423\u043a\u0440\u0430\u0457\u043d\u0438, \u041a\u0438\u0457\u0432","institution_ids":["https://openalex.org/I190444043"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013946385","display_name":"Viacheslav BOIKO","orcid":"https://orcid.org/0000-0002-0842-7450","observed_orcids":["https://orcid.org/0000-0002-0842-7450"]},"institutions":[{"id":"https://openalex.org/I190444043","display_name":"Institute of Mathematics","ror":"https://ror.org/04q3mp532","country_code":"UA","type":"facility","lineage":["https://openalex.org/I149851306","https://openalex.org/I190444043"]},{"id":"https://openalex.org/I4210120823","display_name":"Kyiv Academic University","ror":"https://ror.org/02vrpj575","country_code":"UA","type":"education","lineage":["https://openalex.org/I4210120823"]}],"countries":["UA"],"is_corresponding":false,"raw_author_name":"\u0412\u044f\u0447\u0435\u0441\u043b\u0430\u0432 \u0411\u043e\u0439\u043a\u043e","raw_affiliation_strings":["\u0406\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u041d\u0410\u041d \u0423\u043a\u0440\u0430\u0457\u043d\u0438, \u041a\u0438\u0457\u0432;  \u041a\u0438\u0457\u0432\u0441\u044c\u043a\u0438\u0439 \u0430\u043a\u0430\u0434\u0435\u043c\u0456\u0447\u043d\u0438\u0439 \u0443\u043d\u0456\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442, \u041a\u0438\u0457\u0432"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"\u0406\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u041d\u0410\u041d \u0423\u043a\u0440\u0430\u0457\u043d\u0438, \u041a\u0438\u0457\u0432;  \u041a\u0438\u0457\u0432\u0441\u044c\u043a\u0438\u0439 \u0430\u043a\u0430\u0434\u0435\u043c\u0456\u0447\u043d\u0438\u0439 \u0443\u043d\u0456\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442, \u041a\u0438\u0457\u0432","institution_ids":["https://openalex.org/I190444043","https://openalex.org/I4210120823"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067443803","display_name":"Roman O. Popovych","orcid":"https://orcid.org/0000-0003-2403-1525","observed_orcids":["https://orcid.org/0000-0003-2403-1525"]},"institutions":[{"id":"https://openalex.org/I190444043","display_name":"Institute of Mathematics","ror":"https://ror.org/04q3mp532","country_code":"UA","type":"facility","lineage":["https://openalex.org/I149851306","https://openalex.org/I190444043"]},{"id":"https://openalex.org/I32366509","display_name":"Silesian University in Opava","ror":"https://ror.org/02w4f2z17","country_code":"CZ","type":"education","lineage":["https://openalex.org/I32366509"]}],"countries":["CZ","UA"],"is_corresponding":false,"raw_author_name":"\u0420\u043e\u043c\u0430\u043d \u041f\u043e\u043f\u043e\u0432\u0438\u0447","raw_affiliation_strings":["\u0406\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u041d\u0410\u041d \u0423\u043a\u0440\u0430\u0457\u043d\u0438, \u041a\u0438\u0457\u0432;  \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u0438\u0439 \u0456\u043d\u0441\u0442\u0438\u0442\u0443\u0442, \u0421\u0456\u043b\u0435\u0437\u044c\u043a\u0438\u0439 \u0443\u043d\u0456\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442, \u041e\u043f\u0430\u0432\u0430, \u0427\u0435\u0441\u044c\u043a\u0430 \u0420\u0435\u0441\u043f\u0443\u0431\u043b\u0456\u043a\u0430"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"\u0406\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u041d\u0410\u041d \u0423\u043a\u0440\u0430\u0457\u043d\u0438, \u041a\u0438\u0457\u0432;  \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u0438\u0439 \u0456\u043d\u0441\u0442\u0438\u0442\u0443\u0442, \u0421\u0456\u043b\u0435\u0437\u044c\u043a\u0438\u0439 \u0443\u043d\u0456\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442, \u041e\u043f\u0430\u0432\u0430, \u0427\u0435\u0441\u044c\u043a\u0430 \u0420\u0435\u0441\u043f\u0443\u0431\u043b\u0456\u043a\u0430","institution_ids":["https://openalex.org/I190444043","https://openalex.org/I32366509"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.65142354,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"78","issue":"9-10","first_page":"624","last_page":"672"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9961000084877014,"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":0.9961000084877014,"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/T11673","display_name":"Advanced Topics in Algebra","score":0.0007999999797903001,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T12589","display_name":"Differential Equations and Boundary Problems","score":0.00039999998989515007,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/point-symmetry","display_name":"point symmetry","score":0.9200000166893005},{"id":"https://openalex.org/keywords/contact-symmetries","display_name":"contact symmetries","score":0.9940000176429749},{"id":"https://openalex.org/keywords/lax-representation","display_name":"Lax representation","score":0.7540000081062317},{"id":"https://openalex.org/keywords/dispersive-partial-differential-equations","display_name":"dispersive equations","score":0.8410000205039978},{"id":"https://openalex.org/keywords/algebraic-methods","display_name":"algebraic methods","score":0.5180000066757202}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8090999722480774},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5410000085830688},{"id":"https://openalex.org/C187915474","wikidata":"https://www.wikidata.org/wiki/Q622679","display_name":"Lie group","level":2,"score":0.5349000096321106},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5321999788284302},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.5120000243186951},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.47209998965263367},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.46619999408721924},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.4357999861240387},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.39980000257492065},{"id":"https://openalex.org/C172829509","wikidata":"https://www.wikidata.org/wiki/Q4115796","display_name":"Lax pair","level":3,"score":0.3756999969482422},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.37059998512268066},{"id":"https://openalex.org/C143135876","wikidata":"https://www.wikidata.org/wiki/Q357300","display_name":"Lie theory","level":5,"score":0.3594000041484833},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.352400004863739},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.3472000062465668},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.3407999873161316},{"id":"https://openalex.org/C58910974","wikidata":"https://www.wikidata.org/wiki/Q1017106","display_name":"Adjoint representation","level":3,"score":0.31380000710487366},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.30300000309944153},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.289000004529953},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2874999940395355},{"id":"https://openalex.org/C186489112","wikidata":"https://www.wikidata.org/wiki/Q763735","display_name":"Representation of a Lie group","level":3,"score":0.2825999855995178},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.27880001068115234},{"id":"https://openalex.org/C23917780","wikidata":"https://www.wikidata.org/wiki/Q50698","display_name":"Algebraic equation","level":3,"score":0.2599000036716461},{"id":"https://openalex.org/C203946495","wikidata":"https://www.wikidata.org/wiki/Q1017106","display_name":"Adjoint representation of a Lie algebra","level":4,"score":0.2581999897956848}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3842/umzh.v78i9-10.10162","doi":"https://doi.org/10.3842/umzh.v78i9-10.10162","is_oa":true,"landing_page_url":"https://doi.org/10.3842/umzh.v78i9-10.10162","pdf_url":null,"source":{"id":"https://openalex.org/S4210234218","display_name":"Ukrains\u2019kyi Matematychnyi Zhurnal","issn_l":"1027-3190","issn":["1027-3190"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":["jufo-1"],"host_organization":"https://openalex.org/P4310322009","host_organization_name":"Institute of Mathematics of the National Academy of Sciences of Ukraine","host_organization_lineage":["https://openalex.org/P4310322009","https://openalex.org/P4310321802"],"host_organization_lineage_names":["Institute of Mathematics of the National Academy of Sciences of Ukraine","National Academy of Sciences of Ukraine"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Ukrains\u2019kyi Matematychnyi Zhurnal","raw_type":"journal-article"},{"id":"pmh:oai:ojs2.umj-ojs.imath.kiev.ua:article/10162","doi":null,"is_oa":true,"landing_page_url":"https://umj.imath.kiev.ua/index.php/umj/article/view/10162","pdf_url":null,"source":{"id":"https://openalex.org/S4210234218","display_name":"Ukrains\u2019kyi Matematychnyi Zhurnal","issn_l":"1027-3190","issn":["1027-3190"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":["jufo-1"],"host_organization":"https://openalex.org/P4310322009","host_organization_name":"Institute of Mathematics of the National Academy of Sciences of Ukraine","host_organization_lineage":["https://openalex.org/P4310322009","https://openalex.org/P4310321802"],"host_organization_lineage_names":["Institute of Mathematics of the National Academy of Sciences of Ukraine","National Academy of Sciences of Ukraine"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"1027-3190","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:arXiv.org:2609.29353","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29353","pdf_url":"https://arxiv.org/pdf/2609.29353","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.29353","doi":"https://doi.org/10.48550/arxiv.2609.29353","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.29353","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.3842/umzh.v78i9-10.10162","doi":"https://doi.org/10.3842/umzh.v78i9-10.10162","is_oa":true,"landing_page_url":"https://doi.org/10.3842/umzh.v78i9-10.10162","pdf_url":null,"source":{"id":"https://openalex.org/S4210234218","display_name":"Ukrains\u2019kyi Matematychnyi Zhurnal","issn_l":"1027-3190","issn":["1027-3190"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":["jufo-1"],"host_organization":"https://openalex.org/P4310322009","host_organization_name":"Institute of Mathematics of the National Academy of Sciences of Ukraine","host_organization_lineage":["https://openalex.org/P4310322009","https://openalex.org/P4310321802"],"host_organization_lineage_names":["Institute of Mathematics of the National Academy of Sciences of Ukraine","National Academy of Sciences of Ukraine"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Ukrains\u2019kyi Matematychnyi Zhurnal","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G5206891976","display_name":null,"funder_award_id":"2025.07","funder_id":"https://openalex.org/F4320318013","funder_display_name":"National Research Foundation of Ukraine"}],"funders":[{"id":"https://openalex.org/F4320318013","display_name":"National Research Foundation of Ukraine","ror":"https://ror.org/02gmp5156"},{"id":"https://openalex.org/F4320320847","display_name":"Science Foundation Ireland","ror":"https://ror.org/0271asj38"},{"id":"https://openalex.org/F4320321691","display_name":"Universit\u00e4t Wien","ror":"https://ror.org/03prydq77"},{"id":"https://openalex.org/F4320326543","display_name":"Slezsk\u00e1 Univerzita v Opav\u011b","ror":"https://ror.org/02w4f2z17"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213772147.pdf","grobid_xml":"https://content.openalex.org/works/W7213772147.grobid-xml"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W129094198","https://openalex.org/W238445507","https://openalex.org/W1494563177","https://openalex.org/W1506397612","https://openalex.org/W1549461434","https://openalex.org/W1627889290","https://openalex.org/W1694770817","https://openalex.org/W1878138668","https://openalex.org/W2001570681","https://openalex.org/W2012019547","https://openalex.org/W2031357596","https://openalex.org/W2047062875","https://openalex.org/W2050051867","https://openalex.org/W2071180590","https://openalex.org/W2073715002","https://openalex.org/W2075485449","https://openalex.org/W2085617542","https://openalex.org/W2099641188","https://openalex.org/W2106147467","https://openalex.org/W2114772071","https://openalex.org/W2164373883","https://openalex.org/W2167929317","https://openalex.org/W2760165836","https://openalex.org/W2903385450","https://openalex.org/W3175396305","https://openalex.org/W4246618248","https://openalex.org/W4292170110","https://openalex.org/W4391919778","https://openalex.org/W4391983556","https://openalex.org/W4400368823","https://openalex.org/W4406068202","https://openalex.org/W4415032573"],"related_works":["https://openalex.org/W7214313931","https://openalex.org/W4413449434","https://openalex.org/W4415032573","https://openalex.org/W4391983556","https://openalex.org/W4400368823","https://openalex.org/W2037519758","https://openalex.org/W2088719683","https://openalex.org/W2021469538","https://openalex.org/W2116333132","https://openalex.org/W4406030880"],"abstract_inverted_index":{"UDC":[0],"517.95:512.81":[1],"We":[2],"consider":[3],"a":[4,32],"hierarchy":[5],"of":[6,41,87,131,189,195,205,211],"models":[7],"that":[8,138],"can":[9,56,78],"be":[10],"obtained":[11],"from":[12],"the":[13,52,99,111,118,122,128,132,152,166,173,177,190,196,206],"general":[14],"Nyzhnyk":[15,53,200],"system":[16],"by":[17,149],"imposing":[18],"conditions":[19],"on":[20,172],"parameters,":[21],"introducing":[22],"potentials":[23],"or":[24,48,85,94,101],"pseudopotentials,":[25],"performing":[26],"limiting":[27],"processes":[28],"with":[29],"respect":[30],"to":[31],"scaling":[33],"parameter,":[34],"applying":[35],"differential":[36,83],"substitutions,":[37],"and":[38,60,63,66,74,124,143,168,179,186,202],"interpreting":[39],"parts":[40],"independent":[42],"and/or":[43],"dependent":[44],"variables":[45],"as":[46,68,70,90,92],"complex":[47],"real.":[49],"Therefore,":[50],"among":[51],"models,":[54,181],"one":[55],"distinguish":[57,80],"between":[58,140,144,151,176],"symmetric":[59,119,167,198],"asymmetric,":[61],"dispersive":[62,100,180],"dispersion\\-less,":[64],"standard":[65],"modified,":[67],"well":[69,91],"real,":[71],"complex,":[72],"mixed,":[73],"specific":[75,169],"models.":[76,154],"One":[77],"also":[79],"single":[81],"partial":[82],"equations":[84],"systems":[86],"such":[88],"equations,":[89],"linear":[93,213],"nonlinear":[95],"Lax":[96,214],"representations":[97],"in":[98,117,161,165],"dispersionless":[102,178],"cases,":[103],"respectively.":[104],"For":[105],"each":[106],"specified":[107],"model,":[108],"we":[109,182],"compute":[110],"maximal":[112,191,207],"Lie":[113,192,208],"invariance":[114,193,209],"pseudoalgebra":[115,194,210],"and,":[116],"case,":[120],"find":[121],"point-":[123],"contact-symmetry":[125],"pseudogroups":[126,146],"using":[127],"megaideal-based":[129],"version":[130],"algebraic":[133],"method.":[134],"It":[135],"is":[136],"shown":[137],"relations":[139,150],"these":[141,145,163],"pseudoalgebras":[142,164],"are":[147,158],"induced":[148],"corresponding":[153],"Defining":[155],"(finite-dimensional)":[156],"subalgebras":[157,188,204],"singled":[159],"out":[160],"all":[162],"cases.":[170],"Based":[171],"established":[174],"correspondences":[175],"completely":[183],"classify":[184],"one-":[185],"two-dimensional":[187],"(dispersive":[197],"potential)":[199],"equation":[201],"one-dimensional":[203],"its":[212],"representation.":[215]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-21T00:00:00"}
