{"id":"https://openalex.org/W3004806952","doi":"https://doi.org/10.3390/sym12060950","title":"Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions","display_name":"Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions","publication_year":2020,"publication_date":"2020-06-04","ids":{"openalex":"https://openalex.org/W3004806952","doi":"https://doi.org/10.3390/sym12060950","mag":"3004806952"},"language":"en","primary_location":{"id":"doi:10.3390/sym12060950","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12060950","pdf_url":"https://www.mdpi.com/2073-8994/12/6/950/pdf?version=1591263871","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/2073-8994/12/6/950/pdf?version=1591263871","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5040735583","display_name":"M.L. Gandarias","orcid":"https://orcid.org/0000-0001-8604-8272"},"institutions":[{"id":"https://openalex.org/I39147953","display_name":"Universidad de C\u00e1diz","ror":"https://ror.org/04mxxkb11","country_code":"ES","type":"education","lineage":["https://openalex.org/I39147953"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Mar\u00eda Luz Gandarias","raw_affiliation_strings":["Departamento de Matem\u00e1ticas, Universidad de C\u00e1diz, P.O. Box 40, 11510 Puerto Real, C\u00e1diz, Spain"],"raw_orcid":"https://orcid.org/0000-0001-8604-8272","affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1ticas, Universidad de C\u00e1diz, P.O. Box 40, 11510 Puerto Real, C\u00e1diz, Spain","institution_ids":["https://openalex.org/I39147953"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111515233","display_name":"Mar\u00eda Rosa Dur\u00e1n","orcid":"https://orcid.org/0009-0003-6201-7545"},"institutions":[{"id":"https://openalex.org/I39147953","display_name":"Universidad de C\u00e1diz","ror":"https://ror.org/04mxxkb11","country_code":"ES","type":"education","lineage":["https://openalex.org/I39147953"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Mar\u00eda Rosa Dur\u00e1n","raw_affiliation_strings":["Departamento de Matem\u00e1ticas, Universidad de C\u00e1diz, P.O. Box 40, 11510 Puerto Real, C\u00e1diz, Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1ticas, Universidad de C\u00e1diz, P.O. Box 40, 11510 Puerto Real, C\u00e1diz, Spain","institution_ids":["https://openalex.org/I39147953"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5015192518","display_name":"Chaudry Masood Khalique","orcid":"https://orcid.org/0000-0002-1986-4859"},"institutions":[{"id":"https://openalex.org/I27765905","display_name":"North-West University","ror":"https://ror.org/010f1sq29","country_code":"ZA","type":"education","lineage":["https://openalex.org/I27765905"]},{"id":"https://openalex.org/I80143920","display_name":"Shandong University of Science and Technology","ror":"https://ror.org/04gtjhw98","country_code":"CN","type":"education","lineage":["https://openalex.org/I80143920"]},{"id":"https://openalex.org/I87490745","display_name":"Azerbaijan University","ror":"https://ror.org/00g1psz15","country_code":"AZ","type":"education","lineage":["https://openalex.org/I87490745"]}],"countries":["AZ","CN","ZA"],"is_corresponding":false,"raw_author_name":"Chaudry Masood Khalique","raw_affiliation_strings":["College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China","Department of Mathematics and Informatics, Azerbaijan University, Jeyhun Hajibeyli str., 71, AZ1007 Baku, Azerbaijan","International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa"],"raw_orcid":"https://orcid.org/0000-0002-1986-4859","affiliations":[{"raw_affiliation_string":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China","institution_ids":["https://openalex.org/I80143920"]},{"raw_affiliation_string":"Department of Mathematics and Informatics, Azerbaijan University, Jeyhun Hajibeyli str., 71, AZ1007 Baku, Azerbaijan","institution_ids":["https://openalex.org/I87490745"]},{"raw_affiliation_string":"International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa","institution_ids":["https://openalex.org/I27765905"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5040735583"],"corresponding_institution_ids":["https://openalex.org/I39147953"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":1.7473,"has_fulltext":true,"cited_by_count":20,"citation_normalized_percentile":{"value":0.83870583,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"12","issue":"6","first_page":"950","last_page":"950"},"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":0.9984999895095825,"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/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.9983000159263611,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/conservation-law","display_name":"Conservation law","score":0.874585747718811},{"id":"https://openalex.org/keywords/dispersion","display_name":"Dispersion (optics)","score":0.5850939750671387},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5830016732215881},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5204201340675354},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4995088577270508},{"id":"https://openalex.org/keywords/traveling-wave","display_name":"Traveling wave","score":0.4849715232849121},{"id":"https://openalex.org/keywords/soliton","display_name":"Soliton","score":0.4673507511615753},{"id":"https://openalex.org/keywords/variational-principle","display_name":"Variational principle","score":0.4645751416683197},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.37897834181785583},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.3299727439880371},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.3283660113811493},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3206897974014282},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.06236457824707031}],"concepts":[{"id":"https://openalex.org/C3445786","wikidata":"https://www.wikidata.org/wiki/Q205805","display_name":"Conservation law","level":2,"score":0.874585747718811},{"id":"https://openalex.org/C177562468","wikidata":"https://www.wikidata.org/wiki/Q182893","display_name":"Dispersion (optics)","level":2,"score":0.5850939750671387},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5830016732215881},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5204201340675354},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4995088577270508},{"id":"https://openalex.org/C2988173416","wikidata":"https://www.wikidata.org/wiki/Q37172","display_name":"Traveling wave","level":2,"score":0.4849715232849121},{"id":"https://openalex.org/C87651913","wikidata":"https://www.wikidata.org/wiki/Q464949","display_name":"Soliton","level":3,"score":0.4673507511615753},{"id":"https://openalex.org/C9259290","wikidata":"https://www.wikidata.org/wiki/Q745215","display_name":"Variational principle","level":2,"score":0.4645751416683197},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.37897834181785583},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.3299727439880371},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.3283660113811493},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3206897974014282},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.06236457824707031},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym12060950","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12060950","pdf_url":"https://www.mdpi.com/2073-8994/12/6/950/pdf?version=1591263871","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:dnet:rodin_______::e7022d3d42c004a9a94a8e144c942898","is_oa":true,"landing_page_url":"http://hdl.handle.net/10498/23582","pdf_url":"http://hdl.handle.net/10498/23582","source":{"id":"https://openalex.org/S4306402641","display_name":"LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Cient\u00edficas)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4383465926","host_organization_name":"LA Referencia","host_organization_lineage":["https://openalex.org/I4383465926"],"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 2020, 12(6), 950","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:doaj.org/article:c310479d7da54041abe3c4e086954b63","is_oa":true,"landing_page_url":"https://doaj.org/article/c310479d7da54041abe3c4e086954b63","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 6, p 950 (2020)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym12060950","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12060950","pdf_url":"https://www.mdpi.com/2073-8994/12/6/950/pdf?version=1591263871","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":0.7200000286102295,"display_name":"Life in Land","id":"https://metadata.un.org/sdg/15"}],"awards":[{"id":"https://openalex.org/G809214183","display_name":null,"funder_award_id":"FQM-201","funder_id":"https://openalex.org/F4320326754","funder_display_name":"Junta de Andaluc\u00eda"}],"funders":[{"id":"https://openalex.org/F4320323194","display_name":"North-West University","ror":"https://ror.org/010f1sq29"},{"id":"https://openalex.org/F4320326754","display_name":"Junta de Andaluc\u00eda","ror":"https://ror.org/01jem9c82"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3004806952.pdf","grobid_xml":"https://content.openalex.org/works/W3004806952.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W1506397612","https://openalex.org/W1546226987","https://openalex.org/W1573927207","https://openalex.org/W2032355643","https://openalex.org/W2034063402","https://openalex.org/W2050474001","https://openalex.org/W2062561585","https://openalex.org/W2063907161","https://openalex.org/W2085493421","https://openalex.org/W2124875637","https://openalex.org/W2166619939","https://openalex.org/W2191923789","https://openalex.org/W2404293814","https://openalex.org/W2619436764","https://openalex.org/W2762737161","https://openalex.org/W2797256054","https://openalex.org/W2912266620","https://openalex.org/W2963822786","https://openalex.org/W2963906946","https://openalex.org/W2994975089","https://openalex.org/W3101707560","https://openalex.org/W3103118624","https://openalex.org/W3106014882","https://openalex.org/W4229672964","https://openalex.org/W4249311024","https://openalex.org/W6735000620"],"related_works":["https://openalex.org/W1995118522","https://openalex.org/W64916076","https://openalex.org/W2090020940","https://openalex.org/W1072895066","https://openalex.org/W4364366001","https://openalex.org/W2565229889","https://openalex.org/W2682605843","https://openalex.org/W2530750042","https://openalex.org/W2373699085","https://openalex.org/W1988088155"],"abstract_inverted_index":{"In":[0,88],"this":[1,89],"article,":[2],"we":[3,38,49,76],"investigate":[4],"two":[5,12,86,114],"types":[6],"of":[7],"double":[8,110],"dispersion":[9,22,111],"equations":[10,23],"in":[11,17,32,113],"different":[13],"dimensions,":[14],"which":[15,54],"arise":[16],"several":[18],"physical":[19],"applications.":[20],"Double":[21],"are":[24,94],"derived":[25],"to":[26,59,70,96],"describe":[27],"long":[28],"nonlinear":[29],"wave":[30],"evolution":[31],"a":[33,72,98,102],"thin":[34],"hyperelastic":[35],"rod.":[36],"Firstly,":[37],"obtain":[39,77,97],"conservation":[40,61,92],"laws":[41,62,93],"for":[42,84,108],"both":[43],"these":[44,85],"equations.":[45,87],"To":[46],"do":[47],"this,":[48],"employ":[50],"the":[51,68,91,109],"multiplier":[52],"method,":[53],"is":[55,106],"an":[56],"efficient":[57],"method":[58],"derive":[60],"as":[63],"it":[64],"does":[65],"not":[66],"require":[67],"PDEs":[69],"admit":[71],"variational":[73],"principle.":[74],"Secondly,":[75],"travelling":[78,82],"waves":[79,83],"and":[80],"line":[81,103],"process,":[90],"used":[95],"triple":[99],"reduction.":[100],"Finally,":[101],"soliton":[104],"solution":[105],"found":[107],"equation":[112],"dimensions.":[115]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":2}],"updated_date":"2026-08-10T07:06:39.071514","created_date":"2025-10-10T00:00:00"}
