{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T08:10:11Z","timestamp":1773389411435,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T00:00:00Z","timestamp":1756857600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union under the REFRESH-Research Excellence For Region Sustainability and High-tech Industries","award":["CZ .10.03.01\/00\/22_003\/0000048"],"award-info":[{"award-number":["CZ .10.03.01\/00\/22_003\/0000048"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study presents the complete analysis of a (2 + 1)-dimensional nonlinear wave-type partial differential equation with anisotropic power-law nonlinearities and a general power-law source term, which arises in physical domains such as fluid dynamics, nonlinear acoustics, and wave propagation in elastic media, yet their symmetry properties and exact solution structures remain largely unexplored for arbitrary nonlinearity exponents. To fill this gap, a complete Lie symmetry classification of the equation is performed for arbitrary values of m and n, providing all admissible symmetry generators. These generators are then employed to systematically reduce the PDE to ordinary differential equations, enabling the construction of exact analytical solutions. Traveling wave and soliton solutions are derived using Jacobi elliptic function and sine-cosine methods, revealing rich nonlinear dynamics and wave patterns under anisotropic conditions. Additionally, conservation laws associated with variational symmetries are obtained via Noether\u2019s theorem, yielding invariant physical quantities such as energy-like integrals. The results extend the existing literature by providing, for the first time, a full symmetry classification for arbitrary m and n, new families of soliton and traveling wave solutions in multidimensional settings, and associated conserved quantities. The findings contribute both computationally and theoretically to the study of nonlinear wave phenomena in multidimensional cases, extending the catalog of exact solutions and conserved dynamics of a broad class of nonlinear partial differential equations.<\/jats:p>","DOI":"10.3390\/sym17091445","type":"journal-article","created":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T12:07:32Z","timestamp":1756901252000},"page":"1445","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Lie Symmetries, Solitary Waves, and Noether Conservation Laws for (2 + 1)-Dimensional Anisotropic Power-Law Nonlinear Wave Systems"],"prefix":"10.3390","volume":"17","author":[{"given":"Samina","family":"Samina","sequence":"first","affiliation":[{"name":"General Education Centre, Quanzhou University of Information Engineering, Quanzhou 362000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5024-866X","authenticated-orcid":false,"given":"Hassan","family":"Almusawa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences, Jazan University, Jazan 45142, Saudi Arabia"}]},{"given":"Faiza","family":"Arif","sequence":"additional","affiliation":[{"name":"Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6747-425X","authenticated-orcid":false,"given":"Adil","family":"Jhangeer","sequence":"additional","affiliation":[{"name":"IT4-Innovations, VSB-Technical University of Ostrava, 70800 Ostrava-Poruba, Czech Republic"},{"name":"Center for Theoretical Physics, Khazar University, 41 Mehseti Str., Baku AZ1096, Azerbaijan"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/0020-7462(91)90014-K","article-title":"Group properties of utt\u2212uxmuxx = f(u)","volume":"26","author":"Arrigo","year":"1991","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1023\/A:1026062615145","article-title":"On implicit constitutive theories","volume":"48","author":"Rajagopal","year":"2003","journal-title":"Appl. Math."},{"key":"ref_3","first-page":"437","article-title":"Nonlinear acoustics and generalized Burgers equations","volume":"40","author":"Jordan","year":"2004","journal-title":"Wave Motion"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1023\/A:1014832724750","article-title":"An exact solution for anti-plane shear waves in a nonlinear elastic medium","volume":"64","author":"Leblond","year":"2001","journal-title":"J. Elast."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Murray, J.D. (2002). Mathematical Biology I: An Introduction, Springer.","DOI":"10.1007\/b98868"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Kapitula, T., and Promislow, K. (2013). Spectral and Dynamical Stability of Nonlinear Waves, Springer.","DOI":"10.1007\/978-1-4614-6995-7"},{"key":"ref_7","unstructured":"Volpert, A.I., Volpert, V.A., and Volpert, V.A. (1994). Traveling Wave Solutions of Parabolic Systems, American Mathematical Society (AMS)."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Olver, P.J. (1993). Applications of Lie Groups to Differential Equations, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4612-4350-2"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ibragimov, N.H. (1985). Transformation Groups Applied to Mathematical Physics, Reidel.","DOI":"10.1007\/978-94-009-5243-0"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bluman, G.W., and Kumei, S. (1989). Symmetries and Differential Equations, Springer.","DOI":"10.1007\/978-1-4757-4307-4"},{"key":"ref_11","unstructured":"Ghanbari, B., and Nadjafikhah, M. (2021). Lie group classification and exact solutions of a nonlinear wave equation. Commun. Nonlinear Sci. Numer. Simul., 93."},{"key":"ref_12","unstructured":"Khan, M.A., and Imran, M. (2022). Lie symmetry analysis and conservation laws of generalized Korteweg\u2013de Vries type equations. Mathematics, 10."},{"key":"ref_13","first-page":"405","article-title":"Lie symmetries and similarity reductions of a two-dimensional Burgers-type equation","volume":"21","author":"Vali","year":"2020","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_14","first-page":"2351","article-title":"Lie symmetry analysis of a nonlinear evolution equation in optics","volume":"112","author":"Yurushev","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_15","unstructured":"Ahmad, I., and Bhatti, A. (2024). Exact solutions of nonlinear PDEs using Jacobi elliptic functions and Lie symmetry reductions. Appl. Math. Comput., 451."},{"key":"ref_16","first-page":"85","article-title":"Symmetries, reductions, and conserved quantities in nonlinear field equations","volume":"154","author":"Debnath","year":"2025","journal-title":"Stud. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Samina, S., Arif, F., Jhangeer, A., and Wali, S. (2025). Lie Group Classification, Symmetry Reductions, and Conservation Laws of a Monge\u2013Amp\u00e8re Equation. Symmetry, 17.","DOI":"10.3390\/sym17030355"},{"key":"ref_18","unstructured":"Singhal, A., Joshi, U., and Arora, R. (2025). Lie Symmetry Analysis, Parametric Reduction and Conservation Laws of (3+1) Dimensional Nonlinear Dispersive Soliton Equation. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gou, T., Hajaiej, H., and Stefanov, A.G. (2023). On the solitary waves for anisotropic nonlinear Schr\u00f6dinger models on the plane. Eur. J. Math., 9.","DOI":"10.1007\/s40879-023-00647-8"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1016\/j.chaos.2005.06.004","article-title":"Two reliable methods for solving variants of the KdV equation with compact and noncompact structures","volume":"28","author":"Wazwaz","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Byrd, P.F., and Friedman, M.D. (1971). Handbook of Elliptic Integrals for Engineers and Scientists, Springer.","DOI":"10.1007\/978-3-642-65138-0"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1080\/00411457108231446","article-title":"Invariant Variation Problems","volume":"1","author":"Noether","year":"1971","journal-title":"Transp. Theory Stat. Phys."},{"key":"ref_23","unstructured":"Enflo, B.O., and Hedberg, C.M. (2002). Theory of Nonlinear Acoustics in Fluids, Springer."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Blackstock, D.T., and Hamilton, M.F. (1998). Nonlinear Acoustics, Academic Press.","DOI":"10.1002\/9780470172513.ch17"},{"key":"ref_25","unstructured":"Whitham, G.B. (2011). Linear and Nonlinear Waves, John Wiley & Sons."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1445\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:38:38Z","timestamp":1760035118000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1445"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,3]]},"references-count":25,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["sym17091445"],"URL":"https:\/\/doi.org\/10.3390\/sym17091445","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,3]]}}}