{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,3]],"date-time":"2025-12-03T17:16:38Z","timestamp":1764782198584,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,7,5]],"date-time":"2019-07-05T00:00:00Z","timestamp":1562284800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["16KT0024"],"award-info":[{"award-number":["16KT0024"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001700","name":"Ministry of Education, Culture, Sports, Science and Technology","doi-asserted-by":"publisher","award":["Top Global University Project"],"award-info":[{"award-number":["Top Global University Project"]}],"id":[{"id":"10.13039\/501100001700","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004423","name":"Waseda University","doi-asserted-by":"publisher","award":["2019C-179","2019E-036"],"award-info":[{"award-number":["2019C-179","2019E-036"]}],"id":[{"id":"10.13039\/501100004423","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, symmetry analysis is extended to study nonlocal differential equations. In particular, two integrable nonlocal equations are investigated, the nonlocal nonlinear Schr\u00f6dinger equation and the nonlocal modified Korteweg\u2013de Vries equation. Based on general theory, Lie point symmetries are obtained and used to reduce these equations to nonlocal and local ordinary differential equations, separately; namely, one symmetry may allow reductions to both nonlocal and local equations, depending on how the invariant variables are chosen. For the nonlocal modified Korteweg\u2013de Vries equation, analogously to the local situation, all reduced local equations are integrable. We also define complex transformations to connect nonlocal differential equations and differential-difference equations.<\/jats:p>","DOI":"10.3390\/sym11070884","type":"journal-article","created":{"date-parts":[[2019,7,5]],"date-time":"2019-07-05T11:44:16Z","timestamp":1562327056000},"page":"884","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9255-8575","authenticated-orcid":false,"given":"Linyu","family":"Peng","sequence":"first","affiliation":[{"name":"Waseda Institute for Advanced Study, Waseda University, Tokyo 169-8050, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,5]]},"reference":[{"key":"ref_1","unstructured":"Ackerman, M., and Hermann, R. (1975). 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