{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T00:52:10Z","timestamp":1783644730048,"version":"3.55.0"},"reference-count":33,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T00:00:00Z","timestamp":1701129600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11971475"],"award-info":[{"award-number":["11971475"]}]},{"name":"National Natural Science Foundation of China","award":["12371256"],"award-info":[{"award-number":["12371256"]}]},{"name":"National Natural Science Foundation of China","award":["K202225"],"award-info":[{"award-number":["K202225"]}]},{"name":"National Natural Science Foundation of China","award":["K202316"],"award-info":[{"award-number":["K202316"]}]},{"name":"SuQian Sci&amp;Tech Program","award":["11971475"],"award-info":[{"award-number":["11971475"]}]},{"name":"SuQian Sci&amp;Tech Program","award":["12371256"],"award-info":[{"award-number":["12371256"]}]},{"name":"SuQian Sci&amp;Tech Program","award":["K202225"],"award-info":[{"award-number":["K202225"]}]},{"name":"SuQian Sci&amp;Tech Program","award":["K202316"],"award-info":[{"award-number":["K202316"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we consider how to find new exact solutions for nonlinear partial differential equations using the extended Kudryashov method. This method mainly uses the Riccati equation and the Bernoulli equation where there are some underdetermined constant parameters. And we also use the concept of symmetry to study its reduction equation, Lie transformation group, self-adjointness, and conservation laws. This paper mainly studies the Boussinesq class and the shallow water wave equation in (1 + 1) dimensions and tries to find new exact solutions and symmetry properties of them.<\/jats:p>","DOI":"10.3390\/sym15122122","type":"journal-article","created":{"date-parts":[[2023,11,28]],"date-time":"2023-11-28T11:43:16Z","timestamp":1701171796000},"page":"2122","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Exact Solutions of Nonlinear Partial Differential Equations Using the Extended Kudryashov Method and Some Properties"],"prefix":"10.3390","volume":"15","author":[{"given":"Jian","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"College of Mathematics, Suqian University, Suqian 223800, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Long","family":"Ju","sequence":"additional","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"Jiangsu Center for Applied Mathematics (CUMT), Xuzhou 221116, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shiyin","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"College of Mathematics, Suqian University, Suqian 223800, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yufeng","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"},{"name":"Jiangsu Center for Applied Mathematics (CUMT), Xuzhou 221116, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/S0375-9601(02)01775-9","article-title":"Periodic wave solutions to a coupled kdv equations with variable coefficients","volume":"308","author":"Zhou","year":"2003","journal-title":"Phys. Lett. A"},{"key":"ref_2","first-page":"88","article-title":"New exact jacobi elliptic function solutions for nonlinear equations using F-expansion method","volume":"2","author":"Seddeek","year":"2010","journal-title":"Stud. Math. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/0375-9601(93)90495-L","article-title":"Solitary wave solutions for some systems of coupled nonlinear equations","volume":"180","author":"Lu","year":"1993","journal-title":"Phys. Lett. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1201","DOI":"10.1016\/S0960-0779(03)00309-6","article-title":"Exact solitary wave solutions for a generalized kdv-burgers equation","volume":"19","author":"Hassan","year":"2004","journal-title":"Chaos Soliton Fract."},{"key":"ref_5","first-page":"150","article-title":"The sine-cosine and the tanh methods: Reliable tools for analytic treatment of nonlinear dispersive equations","volume":"173","author":"Wazwaz","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.jde.2022.05.003","article-title":"Soliton resolution for the complex short pulse equation with weighted Sobolev initial data in space-time solitonic regions","volume":"329","author":"Li","year":"2022","journal-title":"J. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1016\/j.joes.2022.02.012","article-title":"New solitary waves and exact solutions for the fifth-order nonlinear wave equation using two integration techniques","volume":"8","author":"Ahmed","year":"2023","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_8","first-page":"361","article-title":"Exact soliton solutions of the generalized evolution equation of wave dynamics","volume":"52","author":"Kudryashov","year":"1988","journal-title":"J. Appl. Math. Comp. Mec."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/0375-9601(91)90481-M","article-title":"On types of nonlinear nonintegrable equations with exact solutions","volume":"155","author":"Kudryashov","year":"1991","journal-title":"Phys. Lett. A"},{"key":"ref_10","first-page":"396","article-title":"Extended simplest equation method for nonlinear differential equations","volume":"205","author":"Kudryashov","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2248","DOI":"10.1016\/j.cnsns.2011.10.016","article-title":"One method for finding exact solutions of nonlinear differential equations","volume":"17","author":"Kudryashov","year":"2012","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1002\/mma.1349","article-title":"Modified Kudryashov method for finding exact solitary wave solutions of higher-order nonlinear equations","volume":"34","author":"Kabir","year":"2011","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.rinp.2014.07.006","article-title":"The exp(\u2212\u03d5(\u03b7))-expansion method with application in the (1 + 1)-dimensional classical boussinesq equations","volume":"4","author":"Roshid","year":"2014","journal-title":"Results Phys."},{"key":"ref_14","first-page":"8840","article-title":"N-soliton solutions for shallow water waves equations in (1 + 1) and (2 + 1) dimensions","volume":"217","author":"Wazwaz","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1007\/BF02776223","article-title":"Symmetries and differential equations","volume":"28","author":"Bluman","year":"1980","journal-title":"Lett. Nuovo C."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Bluman, G., Cheviakov, A.F., and Anco, S.C. (2010). Applications of Symmetry Methods to Partial Differential Equations, Springer.","DOI":"10.1007\/978-0-387-68028-6"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.geomphys.2014.05.028","article-title":"Ordinary differential equations described by their Lie symmetry algebra","volume":"85","author":"Gianni","year":"2014","journal-title":"J. Geom. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Olver, P.J. (1993). Applications of Lie Groups to Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-4350-2"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1012","DOI":"10.1088\/0253-6102\/57\/6\/13","article-title":"Lie algebras and integrable systems","volume":"57","author":"Zhang","year":"2012","journal-title":"Commun. Theor. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1007\/s11082-023-05070-7","article-title":"Investigation of solitons and conservation laws in an inhomogeneous optical fiber through a generalized derivative nonlinear Schr\u00f6dinger equation with quintic nonlinearity","volume":"55","author":"Wafaa","year":"2023","journal-title":"Opt. Quant. Electron."},{"key":"ref_21","first-page":"7","article-title":"Triki-Biswas model: Its symmetry reduction, Nucci\u2019s reduction and conservation laws","volume":"37","author":"Akbulut","year":"2022","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_22","unstructured":"Cook, P., Roytburd, V., and Tulin, M. (1996). Mathematics Is for Solving Problems, SIAM."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s13538-022-01195-4","article-title":"Dynamical Behavior of the Solutions of Coupled Boussinesq-Burgers Equations Occurring at the Seaside Beaches","volume":"52","author":"Kumar","year":"2022","journal-title":"Braz. J. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"16655","DOI":"10.3934\/math.2023852","article-title":"Analytical solutions for nonlinear systems using Nucci\u2019s reduction approach and generalized projective Riccati equations","volume":"8","author":"Hashemi","year":"2023","journal-title":"AIMS Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"102272","DOI":"10.1016\/j.rinp.2019.102272","article-title":"Aljahdaly, Some applications of the modified (G\u2032\/G2)-expansion method in mathematical physics","volume":"13","author":"Noufe","year":"2019","journal-title":"Results Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1017\/S0956792598003477","article-title":"Integrating factors and first integrals for ordinary differential equations","volume":"9","author":"Anco","year":"1998","journal-title":"Eur. J. Appl. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2869","DOI":"10.1103\/PhysRevLett.78.2869","article-title":"Direct construction of conservation laws from field equations","volume":"78","author":"Anco","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1002\/mma.4091","article-title":"Conservation laws and double reduction of (2 + 1) dimensional Calogero-Bogoyavlenskii-Schiff equation","volume":"40","author":"San","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1016\/j.jmaa.2005.11.012","article-title":"Integrating factors, adjoint equations and lagrangians","volume":"318","author":"Ibragimov","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/j.jmaa.2006.10.078","article-title":"A new conservation theorem","volume":"333","author":"Ibragimov","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.cnsns.2014.11.010","article-title":"Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations","volume":"23","author":"Gazizov","year":"2015","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.nonrwa.2017.08.005","article-title":"Conservation laws and non-invariant solutions of anisotropic wave equations with a source","volume":"40","author":"Ibragimov","year":"2018","journal-title":"Nonlinear Anal-Real"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"432002","DOI":"10.1088\/1751-8113\/44\/43\/432002","article-title":"Nonlinear self-adjointness and conservation laws","volume":"44","author":"Ibragimov","year":"2011","journal-title":"J. Phys. A-Math. 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