{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T00:26:55Z","timestamp":1786580815851,"version":"build-2736575974"},"reference-count":35,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,7,6]],"date-time":"2026-07-06T00:00:00Z","timestamp":1783296000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009059","name":"Pacific Institute for the Mathematical Sciences","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100009059","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2024-04308"],"award-info":[{"award-number":["RGPIN-2024-04308"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.cnsns.2026.110510","type":"journal-article","created":{"date-parts":[[2026,7,7]],"date-time":"2026-07-07T15:25:54Z","timestamp":1783437954000},"page":"110510","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"P2","title":["Invariant reduction for partial differential equations. IV: Symmetries that rescale geometric structures"],"prefix":"10.1016","volume":"163","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3190-6465","authenticated-orcid":false,"given":"Kostya","family":"Druzhkov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9647-3683","authenticated-orcid":false,"given":"Alexey","family":"Shevyakov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cnsns.2026.110510_bib0001","series-title":"Group analysis of differential equations","author":"Ovsiannikov","year":"1982"},{"key":"10.1016\/j.cnsns.2026.110510_bib0002","article-title":"Applications of Lie groups to differential equations","volume":"107","author":"Olver","year":"1993"},{"key":"10.1016\/j.cnsns.2026.110510_bib0003","article-title":"Symmetries and differential equations","volume":"81","author":"Bluman","year":"1989"},{"key":"10.1016\/j.cnsns.2026.110510_bib0004","article-title":"Applications of symmetry methods to partial differential equations","volume":"168","author":"Bluman","year":"2010"},{"key":"10.1016\/j.cnsns.2026.110510_bib0005","series-title":"Introduction to symmetry analysis","author":"Cantwell","year":"2002"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110510_bib0006","first-page":"608","article-title":"Double reduction of PDEs from the association of symmetries with conservation laws with applications","volume":"184","author":"Sj\u00f6berg","year":"2007","journal-title":"Appl Math Comput"},{"key":"10.1016\/j.cnsns.2026.110510_bib0007","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2020.105349","article-title":"Symmetry multi-reduction method for partial differential equations with conservation laws","volume":"91","author":"Anco","year":"2020","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110510_bib0008","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1353\/ajm.1997.0015","article-title":"Symmetry reduction of variational bicomplexes and the principle of symmetric criticality","volume":"119","author":"Anderson","year":"1997","journal-title":"Am J Math"},{"key":"10.1016\/j.cnsns.2026.110510_bib0009","doi-asserted-by":"crossref","unstructured":"Druzhkov K., Cheviakov A.. Invariant reduction for partial differential equations. II: the general framework. 2025. arXiv preprint arXiv: 2501.09313.","DOI":"10.2139\/ssrn.6792400"},{"issue":"5","key":"10.1016\/j.cnsns.2026.110510_bib0010","doi-asserted-by":"crossref","DOI":"10.1088\/1361-6544\/adcfe8","article-title":"Invariant reduction for partial differential equations: I. Conservation laws and systems with two independent variables","volume":"38","author":"Druzhkov","year":"2025","journal-title":"Nonlinearity"},{"key":"10.1016\/j.cnsns.2026.110510_bib0011","unstructured":"Druzhkov K.. Invariant reduction for partial differential equations. III: Poisson brackets. 2025. arXiv preprint arXiv: 2507.08213."},{"key":"10.1016\/j.cnsns.2026.110510_bib0012","series-title":"Symmetries and conservation laws for differential equations of mathematical physics","volume":"182","year":"1999"},{"key":"10.1016\/j.cnsns.2026.110510_bib0013","first-page":"003","article-title":"The profinite dimensional manifold structure of formal solution spaces of formally integrable PDEs","volume":"13","author":"G\u00fcneysu","year":"2017","journal-title":"SIGMA Symmetry Integr Geom: Method Appl"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110510_bib0014","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0022-247X(84)90071-4","article-title":"The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory, II. The nonlinear theory","volume":"100","author":"Vinogradov","year":"1984","journal-title":"J Math Anal Appl"},{"key":"10.1016\/j.cnsns.2026.110510_bib0015","doi-asserted-by":"crossref","DOI":"10.1016\/j.geomphys.2023.104848","article-title":"Lagrangian formalism and the intrinsic geometry of PDEs","volume":"189","author":"Druzhkov","year":"2023","journal-title":"J Geom Phys"},{"key":"10.1016\/j.cnsns.2026.110510_bib0016","doi-asserted-by":"crossref","DOI":"10.1016\/j.geomphys.2024.105143","article-title":"Internal Lagrangians of PDEs as variational principles","volume":"199","author":"Druzhkov","year":"2024","journal-title":"J Geom Phys"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110510_bib0017","doi-asserted-by":"crossref","first-page":"365","DOI":"10.3934\/jgm.2013.5.365","article-title":"Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms","volume":"5","author":"Suris","year":"2013","journal-title":"J Geom Mech"},{"key":"10.1016\/j.cnsns.2026.110510_bib0018","series-title":"On the Lagrangian structure of integrable hierarchies","first-page":"347","author":"Suris","year":"2016"},{"issue":"1\u20134","key":"10.1016\/j.cnsns.2026.110510_bib0019","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1002\/sapm1948271220","article-title":"On two-dimensional non-steady motion of a slender body in a compressible fluid","volume":"27","author":"Lin","year":"1948","journal-title":"J Math Phys"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110510_bib0020","first-page":"538","article-title":"On the theory of nonstationary transonic flows","volume":"185","author":"Mamontov","year":"1969","journal-title":"Dokl Acad Nauk SSSR"},{"key":"10.1016\/j.cnsns.2026.110510_bib0021","series-title":"CRC handbook of Lie group analysis of differential equations","volume":"I","year":"1994"},{"issue":"11","key":"10.1016\/j.cnsns.2026.110510_bib0022","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.1007\/s10483-011-1514-6","article-title":"Self-similar solutions to Lin-Reissner-Tsien equation","volume":"32","author":"Haussermann","year":"2011","journal-title":"Appl Math Mech"},{"key":"10.1016\/j.cnsns.2026.110510_bib0023","unstructured":"Krasil\u2019shchik J., Verbovetsky A.. Homological methods in equations of mathematical physics. 1998. arXiv preprint arXiv:math\/9808130."},{"issue":"1","key":"10.1016\/j.cnsns.2026.110510_bib0024","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1006\/jcph.1996.0130","article-title":"Efficient implementation of weighted ENO schemes","volume":"126","author":"Jiang","year":"1996","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110510_bib0025","series-title":"Riemann solvers and numerical methods for fluid dynamics: a practical introduction","author":"Toro","year":"2009"},{"key":"10.1016\/j.cnsns.2026.110510_bib0026","series-title":"Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems","author":"LeVeque","year":"2007"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110510_bib0027","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/0021-9991(88)90177-5","article-title":"Efficient implementation of essentially non-oscillatory shock-capturing schemes","volume":"77","author":"Shu","year":"1988","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110510_bib0028","series-title":"Strong stability preserving Runge\u2013Kutta and multistep time discretizations","author":"Gottlieb","year":"2011"},{"key":"10.1016\/j.cnsns.2026.110510_bib0029","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s11005-019-01240-5","article-title":"Variational symmetries and Lagrangian multiforms","volume":"110","author":"Sleigh","year":"2020","journal-title":"Lett Math Phys"},{"key":"10.1016\/j.cnsns.2026.110510_bib0030","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s11005-025-02016-w","article-title":"Lagrangian multiforms and dispersionless integrable systems","volume":"115","author":"Ferapontov","year":"2025","journal-title":"Lett Math Phys"},{"key":"10.1016\/j.cnsns.2026.110510_bib0031","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s11005-023-01766-9","article-title":"Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems","volume":"114","author":"Caudrelier","year":"2024","journal-title":"Lett Math Phys"},{"key":"10.1016\/j.cnsns.2026.110510_bib0032","series-title":"Contemporary mathematics","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1090\/conm\/788\/15822","article-title":"Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles","volume":"788","author":"Grigoriev","year":"2023"},{"issue":"33","key":"10.1016\/j.cnsns.2026.110510_bib0033","doi-asserted-by":"crossref","DOI":"10.1088\/1751-8121\/ad65a3","article-title":"Presymplectic minimal models of local gauge theories","volume":"57","author":"Dneprov","year":"2024","journal-title":"J Phys A: Math Theor"},{"issue":"8","key":"10.1016\/j.cnsns.2026.110510_bib0034","doi-asserted-by":"crossref","DOI":"10.1063\/5.0004372","article-title":"Invariant conservation law-preserving discretizations of linear and nonlinear wave equations","volume":"61","author":"Cheviakov","year":"2020","journal-title":"J Math Phys"},{"key":"10.1016\/j.cnsns.2026.110510_bib0035","doi-asserted-by":"crossref","DOI":"10.1098\/rspa.2024.0159","article-title":"Approximate conservation laws of partial differential equations","volume":"481","author":"Cheviakov","year":"2025","journal-title":"Proc R Soc A"}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426008646?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426008646?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T23:27:50Z","timestamp":1786577270000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1007570426008646"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":35,"alternative-id":["S1007570426008646"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110510","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Invariant reduction for partial differential equations. IV: Symmetries that rescale geometric structures","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110510","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 The Author(s). Published by Elsevier B.V.","name":"copyright","label":"Copyright"}],"article-number":"110510"}}