{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:04:28Z","timestamp":1760234668421,"version":"build-2065373602"},"reference-count":54,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,7]],"date-time":"2021-06-07T00:00:00Z","timestamp":1623024000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we propose the fractal (2 + 1)-dimensional Zakharov\u2013Kuznetsov equation based on He\u2019s fractal derivative for the first time. The fractal generalized variational formulation is established by using the semi-inverse method and two-scale fractal theory. The obtained fractal variational principle is important since it not only reveals the structure of the traveling wave solutions but also helps us study the symmetric theory. The finding of this paper will contribute to the study of symmetry in the fractal space.<\/jats:p>","DOI":"10.3390\/sym13061022","type":"journal-article","created":{"date-parts":[[2021,6,7]],"date-time":"2021-06-07T22:23:00Z","timestamp":1623104580000},"page":"1022","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Generalized Variational Principle for the Fractal (2 + 1)-Dimensional Zakharov\u2013Kuznetsov Equation in Quantum Magneto-Plasmas"],"prefix":"10.3390","volume":"13","author":[{"given":"Yan-Hong","family":"Liang","sequence":"first","affiliation":[{"name":"School of Qilu Transportation, Shandong University, Jinan 250061, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3905-0844","authenticated-orcid":false,"given":"Kang-Jia","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"104000","DOI":"10.1016\/j.geomphys.2020.104000","article-title":"On integrability of the higher-dimensional time fractional KdV-type equation","volume":"160","author":"Liu","year":"2021","journal-title":"J. Geom. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2150028","DOI":"10.1142\/S0218348X21500286","article-title":"Variational principles for fractal Whitham-Broer-Kaup Equations in Shallow Water","volume":"29","author":"Wang","year":"2021","journal-title":"Fractals"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"49","DOI":"10.2174\/1876402912666200123154001","article-title":"A Micro-channel Cooling Model for a Three-dimensional Integrated Circuit Considering Through-silicon Vias","volume":"13","author":"Wang","year":"2021","journal-title":"Micro Nanosyst."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Liu, J., Yang, X., Geng, L., and Fan, Y. (2021). Group analysis of the time fractional (3 + 1)-dimensional KdV-type equation. Fractals.","DOI":"10.1142\/S0218348X21501693"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/j.crma.2012.05.007","article-title":"A Note on the Cauchy problem for the 2D generalized Zakharov\u2013Kuznetsov equations","volume":"350","author":"Ribaud","year":"2012","journal-title":"C. R. Math."},{"key":"ref_6","first-page":"1421","article-title":"Generalized solitary and periodic wave solutions to a (2 + 1)-dimensional Zakharov\u2013Kuznetsov equation","volume":"217","author":"Aslan","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_7","first-page":"285","article-title":"On three-dimensional solitons","volume":"39","author":"Zakharov","year":"1974","journal-title":"Soviet Phys."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ghanbari, B., Yusuf, A., and Baleanu, D. (2019). The new exact solitary wave solutions and stability analysis for the (2 + 1)-dimensional Zakharov\u2013Kuznetsov equation. Adv. Differ. Equ., 49.","DOI":"10.1186\/s13662-019-1964-0"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.mcm.2011.01.049","article-title":"Exact solutions of the (2 + 1)-dimensional Zakharov\u2013Kuznetsov modified equal width equation using Lie group analysis","volume":"54","author":"Khalique","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.matcom.2020.07.005","article-title":"On group analysis of the time fractional extended (2 + 1)-dimensional Zakharov\u2013Kuznetsov equation in quantum magneto-plasmas","volume":"178","author":"Liu","year":"2020","journal-title":"Math. Comput. Simul."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wang, K.L., Wang, H., and Muhammad, H. (2021). A new perspective for two different types of fractal Zakharov-Kuznetsov models. Fractals.","DOI":"10.1142\/S0218348X21501681"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2851","DOI":"10.1016\/j.camwa.2018.01.014","article-title":"The new exact solitary and multi-soliton solutions for the (2 + 1)-dimensional Zakharov\u2013Kuznetsov equation","volume":"75","author":"Kuo","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1016\/j.cnsns.2006.10.007","article-title":"The extended tanh method for the Zakharov\u2013Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms","volume":"13","author":"Wazwaz","year":"2008","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"102959","DOI":"10.1016\/j.nonrwa.2019.06.009","article-title":"Regular solutions to initial-boundary value problems in a half-strip for two-dimensional Zakharov\u2013Kuznetsov equation","volume":"51","author":"Faminskii","year":"2020","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"760","DOI":"10.1016\/j.jmaa.2018.03.048","article-title":"Initial-boundary value problems in a rectangle for two-dimensional Zakharov\u2013Kuznetsov equation","volume":"463","author":"Faminskii","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Faminskii, A. (2007). Nonlocal well-posedness of the mixed problem for the Zakharov-Kuznetsov equation. J. Math. Sci., 147.","DOI":"10.1007\/s10958-007-0491-9"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"513","DOI":"10.22363\/2413-3639-2019-65-3-513-546","article-title":"On Inner Regularity of Solutions of Two-Dimensional Zakharov-Kuznetsov Equation","volume":"65","author":"Faminskii","year":"2019","journal-title":"Contemp. Math. Fundam. Dir."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"He, J.H., and Sun, C. (2019). A variational principle for a thin film equation. J. Math. Chem., 57.","DOI":"10.1007\/s10910-019-01063-8"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.aml.2016.08.008","article-title":"Generalized equilibrium equations for shell derived from a generalized variational principle","volume":"64","author":"He","year":"2017","journal-title":"Appl. Math. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.aml.2017.04.008","article-title":"Hamilton\u2019s principle for dynamical elasticity","volume":"72","author":"He","year":"2017","journal-title":"Appl. Math. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Wang, K.L. (2020). Variational principle for nonlinear oscillator arising in a fractal nano\/microelectromechanical system. Math. Methods Appl. Sci.","DOI":"10.1002\/mma.6726"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/j.aml.2018.05.008","article-title":"A remark on Samuelson\u2019s variational principle in economics","volume":"84","author":"Wu","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1073\/pnas.67.3.1477","article-title":"Law of conservation of the capital-output ratio","volume":"67","author":"Samuelson","year":"1970","journal-title":"Proc. Natl. Acad. Sci. Appl. Math. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1836","DOI":"10.1016\/j.chaos.2008.07.034","article-title":"Allometric scaling laws in biology and physics","volume":"41","author":"He","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"He, J.H. (2012). Asymptotic methods for solitary solutions and compactons. Abstr. Appl. Anal., 916793.","DOI":"10.1155\/2012\/916793"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1939","DOI":"10.1016\/j.nonrwa.2008.02.031","article-title":"Variational approach to the Benjamin Ono equation","volume":"10","author":"Tao","year":"2009","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2050024","DOI":"10.1142\/S0218348X20500243","article-title":"A fractal variational theory for one-dimensional compressible flow in a microgravity space","volume":"28","author":"He","year":"2020","journal-title":"Fractals"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2150059","DOI":"10.1142\/S0218348X21500596","article-title":"A novel approach for fractal Burgers-BBM equation and its variational principle","volume":"29","author":"Wang","year":"2021","journal-title":"Fractals"},{"key":"ref_29","first-page":"735","article-title":"Variational principle for the generalized KdV-burgers equation with fractal derivatives for shallow water waves","volume":"6","author":"He","year":"2020","journal-title":"J. Appl. Comput. Mech."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"166377","DOI":"10.1109\/ACCESS.2020.3022798","article-title":"The fractional Sallen-Key filter described by local fractional derivative","volume":"8","author":"Wang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3244","DOI":"10.1016\/j.aej.2020.08.049","article-title":"A a-order R-L high-pass filter modeled by local fractional derivative","volume":"59","author":"Wang","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2050031","DOI":"10.1142\/S0218348X20500310","article-title":"On a High-pass filter described by local fractional derivative","volume":"28","author":"Wang","year":"2020","journal-title":"Fractals"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Wang, K.L. (2020). He\u2019s frequency formulation for fractal nonlinear oscillator arising in a microgravity space. Numer. Methods Partial. Differ. Equ.","DOI":"10.1002\/num.22584"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"5776","DOI":"10.1002\/mma.6319","article-title":"Hadamard type local fractional integral inequalities for generalized harmonically convex functions and applications","volume":"43","author":"Sun","year":"2020","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4669","DOI":"10.1016\/j.aej.2020.08.024","article-title":"The transient analysis for zero-input response of fractal RC circuit based on local fractional derivative","volume":"59","author":"Wang","year":"2020","journal-title":"Alexandria Eng. J."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1140\/epjp\/s13360-020-00891-x","article-title":"A new fractional nonlinear singular heat conduction model for the human head considering the effect of febrifuge","volume":"135","author":"Wang","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Attia, R.A.M., Baleanu, D., Lu, D., Mostafa, M.A.K., and El-Sayed, A. (2021). Computational and numerical simulations for the deoxyribonucleic acid (DNA) model. Discret. Contin. Dyn. Syst.-S.","DOI":"10.3934\/dcdss.2021018"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.rinp.2018.06.011","article-title":"Fractal calculus and its geometrical explanation","volume":"10","author":"He","year":"2018","journal-title":"Results Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"659","DOI":"10.2298\/TSCI200127065H","article-title":"New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle","volume":"24","author":"He","year":"2020","journal-title":"Therm. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2150062","DOI":"10.1142\/S0218348X21500626","article-title":"A new fractal transform frequency formulation for fractal nonlinear oscillators","volume":"29","author":"Wang","year":"2021","journal-title":"Fractals"},{"key":"ref_41","first-page":"1","article-title":"On a variational principle for the fractal Wu\u2013Zhang system arising in shallow water","volume":"12","author":"Liang","year":"2021","journal-title":"GEM-Int. J. Geomath."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1515\/TJJ.1997.14.1.23","article-title":"Semi-Inverse Method of Establishing Generalized Variational Principles for Fluid Mechanics with Emphasis on Turbomachinery Aerodynamics","volume":"14","author":"He","year":"1997","journal-title":"Int. J. Turbo Jet Engines"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Wang, K.J., Wang, G.D., and Zhu, H.W. (2021). A new perspective on the study of the fractal coupled Boussinesq-Burger equation in shallow water. Fractals.","DOI":"10.1142\/S0218348X2150122X"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"45001","DOI":"10.1088\/1572-9494\/abdea1","article-title":"On the new exact traveling wave solutions of the time-space fractional strain wave equation in microstructured solids via the variational method","volume":"73","author":"Wang","year":"2021","journal-title":"Commun. Theor. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2150075","DOI":"10.1142\/S0218348X21500754","article-title":"Variational principle and approximate solution for the fractal generalized Benjamin-Bona-Mahony-Burgers equation in fluid mechanics","volume":"3","author":"Wang","year":"2021","journal-title":"Fractals"},{"key":"ref_46","first-page":"95","article-title":"A Family of Variational Principles for Compressible Rotational Blade-to-Blade Flow Using Semi-Inverse Method","volume":"15","author":"He","year":"1998","journal-title":"Int. J. Turbo Jet Engines"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Wang, K.J., and Wang, G.D. (2021). Variational principle, solitary and periodic wave solutions of the fractal modified equal width equation in plasma physics. Fractals.","DOI":"10.1142\/S0218348X21501152"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2150044","DOI":"10.1142\/S0218348X21500444","article-title":"Variational principle and approximate solution for the generalized Burgers-Huxley equation with fractal derivative","volume":"29","author":"Wang","year":"2021","journal-title":"Fractals"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Wang, K.J., and Wang, G.D. (2020). He\u2019s variational method for the time-space fractional nonlinear Drinfeld-Sokolov-Wilson system. Math. Methods Appl. Sci.","DOI":"10.1002\/mma.7200"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"44002","DOI":"10.1209\/0295-5075\/132\/44002","article-title":"A variational principle for the (3 + 1)-dimensional extended quantum Zakharov-Kuznetsov equation in plasma physics","volume":"132","author":"Wang","year":"2020","journal-title":"EPL"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"104375","DOI":"10.1016\/j.rinp.2021.104375","article-title":"Constructions of new abundant traveling wave solutions for system of the ion sound and Langmuir waves by the variational direct method","volume":"26","year":"2021","journal-title":"Results Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"5617","DOI":"10.1002\/mma.7135","article-title":"Solitary and periodic wave solutions of the generalized fourth order boussinesq equation via He\u2019s variational methods","volume":"44","author":"Wang","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"103666","DOI":"10.1016\/j.rinp.2020.103666","article-title":"Periodic solution of the (2 + 1)-dimensional nonlinear electrical transmission line equation via variational method","volume":"20","author":"Wang","year":"2021","journal-title":"Results Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"103031","DOI":"10.1016\/j.rinp.2020.103031","article-title":"Variational principle and periodic solution of the Kundu\u2013Mukherjee\u2013Naskar equation","volume":"17","author":"He","year":"2020","journal-title":"Results Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1022\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:11:35Z","timestamp":1760163095000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1022"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,7]]},"references-count":54,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["sym13061022"],"URL":"https:\/\/doi.org\/10.3390\/sym13061022","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,6,7]]}}}