{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T10:06:17Z","timestamp":1780567577233,"version":"3.54.1"},"reference-count":41,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,11]],"date-time":"2025-01-11T00:00:00Z","timestamp":1736553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12371256"],"award-info":[{"award-number":["12371256"]}]},{"name":"National Natural Science Foundation of China","award":["11971475"],"award-info":[{"award-number":["11971475"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we generalize the 2 + 1-dimensional Gardner (2DG) equation to three spatial dimensions, i.e., 3 + 1 and 3 + 2 dimensions, and construct the solutions of the Cauchy problems and Lax pairs for the Gardner equation in three spatial dimensions via a novel non-local d-bar formalism. Several new long derivative operators Dx, Dy and Dt are introduced to study the initial value problems for the Gardner equation in three spatial dimensions. It follows that Propositions 1 and 3 summarize the main results of this paper.<\/jats:p>","DOI":"10.3390\/sym17010102","type":"journal-article","created":{"date-parts":[[2025,1,13]],"date-time":"2025-01-13T04:01:52Z","timestamp":1736740912000},"page":"102","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Solutions of Cauchy Problems for the Gardner Equation in Three Spatial Dimensions"],"prefix":"10.3390","volume":"17","author":[{"given":"Yufeng","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Technology and Data, Yantai Nanshan University, Yantai 265713, China"},{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Linlin","family":"Gui","sequence":"additional","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Binlu","family":"Feng","sequence":"additional","affiliation":[{"name":"College of Technology and Data, Yantai Nanshan University, Yantai 265713, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2197","DOI":"10.1088\/0266-5611\/22\/6\/017","article-title":"Inverse scattering transform for the Camassa-Holm equation","volume":"22","author":"Constantin","year":"2006","journal-title":"Inverse Problem."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1090\/proc\/15174","article-title":"Inverse scattering and soliton solutions of nonlocal reverse-spacetime nonlinear Schr\u00f6dinger equations","volume":"149","author":"Ma","year":"2021","journal-title":"J. Proc. Am. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1103\/PhysRevLett.19.1095","article-title":"Method for solving the Korteweg-de Vries equation","volume":"19","author":"Gardner","year":"1967","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1002\/sapm1974534249","article-title":"The inverse scattering transform-Fourier analysis for nonlinear problems","volume":"53","author":"Ablowitz","year":"1974","journal-title":"Stud. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/BF00420705","article-title":"Three-dimensional model of relativistic-invariant field theory, integrable by the inverse scattering transform","volume":"5","author":"Manakov","year":"1981","journal-title":"Lett. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1103\/PhysRevLett.31.125","article-title":"Nonlinear-evolution equations of physical significance","volume":"31","author":"Ablowitz","year":"1973","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ablowitz, M.J., and Clarkson, P.A. (1991). Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press.","DOI":"10.1017\/CBO9780511623998"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1002\/sapm1983693211","article-title":"On the inverse scatering of the time dependent Schr\u00f6dinger equation and the associated KPI equation","volume":"69","author":"Fokas","year":"1983","journal-title":"Stud. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1143\/JPSJ.34.1289","article-title":"The modified Korteweg-de Vries equation","volume":"34","author":"Wadati","year":"1973","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2559","DOI":"10.1088\/0951-7715\/23\/10\/012","article-title":"Inverse scattering transform for the Degasperis-Procesi equation","volume":"23","author":"Constantin","year":"2010","journal-title":"Nonlinearity"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1185","DOI":"10.1103\/PhysRevLett.35.1185","article-title":"Nonlinear evolution equations in two and three dimensions","volume":"35","author":"Ablowitz","year":"1975","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1103\/PhysRevLett.51.7","article-title":"Method of Solution for a Class of Multidimensional Nonlinear Evolution Equations","volume":"51","author":"Fokas","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5663","DOI":"10.1093\/imrn\/rnx051","article-title":"Existence of global solutions to the derivative NLS equation with the inverse scattering transform method","volume":"18","author":"Pelinovsky","year":"2018","journal-title":"Int. Math. Res. Notices"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1479","DOI":"10.1016\/j.chaos.2012.08.010","article-title":"Solving the non-isospectral Ablowitz-Ladik hierarchy via the inverse scattering transform and reductions","volume":"45","author":"Li","year":"2012","journal-title":"Chaos Solitons Fractals"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/0167-2789(86)90184-3","article-title":"The D-bar approach to inverse scattering and nonlinear equations","volume":"18","author":"Beals","year":"1986","journal-title":"Phys. D"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2879","DOI":"10.1090\/proc\/15716","article-title":"The \u2202-dressing method for the (2+1)-dimensional Jimbo-Miwa equation","volume":"150","author":"Chai","year":"2022","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_17","first-page":"9","article-title":"Scattering, transformations spectrales et equations d\u2019evolution nonlineare. I. Seminaire Goulaouic-Meyer-Schwartz","volume":"22","author":"Beals","year":"1981","journal-title":"Ec. Polytech. Palaiseau"},{"key":"ref_18","first-page":"8","article-title":"Scattering, transformations spectrales et equations d\u2019evolution nonlineare. II. Seminaire Goulaouic-Meyer-Schwartz","volume":"21","author":"Beals","year":"1982","journal-title":"Ec. Polytech. Palaiseau"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1103\/PhysRevLett.51.3","article-title":"Inverse scattering of first-order systems in the plane related to nonlinear multidimensional equations","volume":"51","author":"Fokas","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3617","DOI":"10.1088\/0305-4470\/29\/13\/027","article-title":"The application of the \u2202-dressing method to some integrable (2+1)-dimensional nonlinear equations","volume":"29","author":"Dubrovsky","year":"1996","journal-title":"J. Phys. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"L537","DOI":"10.1088\/0305-4470\/21\/10\/001","article-title":"Nonlocal \u2202-problem and (2 + 1)-dimensional soliton equations","volume":"21","author":"Bogdanov","year":"1988","journal-title":"J. Phys. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1088\/0266-5611\/5\/2\/002","article-title":"Linear spectral problems, nonlinear equations and the \u2202-method","volume":"5","author":"Beals","year":"1989","journal-title":"Inverse Prob."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"013516","DOI":"10.1063\/1.4788665","article-title":"Discussion on integrable properties for higher-dimensional variable-coefficient nonlinear partial differential equations","volume":"54","author":"Zhang","year":"2013","journal-title":"J. Math. Phys."},{"key":"ref_24","first-page":"408","article-title":"Symmetry properties and explicit solutions of some nonlinear differential and fractional equations","volume":"337","author":"Zhang","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1635","DOI":"10.1007\/s00366-021-01297-8","article-title":"Learning nonlinear dynamics with behavior ordinary\/partial\/system of the differential equations: Looking through the lens of orthogonal neural networks","volume":"38","author":"Omidi","year":"2021","journal-title":"Eng. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1016\/j.jmaa.2008.07.047","article-title":"Controllability of nonlinear integro-differential third order dispersion system","volume":"348","author":"Chalishajar","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1016\/j.jmaa.2006.10.084","article-title":"Exact controllability of the nonlinear third-order dispersion equation","volume":"332","author":"George","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"190201","DOI":"10.1103\/PhysRevLett.96.190201","article-title":"Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions","volume":"96","author":"Fokas","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1111\/j.1467-9590.2009.00437.x","article-title":"Kadomtsev-Petviashvili equation revisited and integrability in 4 + 2 and 3 + 1","volume":"122","author":"Fokas","year":"2009","journal-title":"Stud. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"091413","DOI":"10.1063\/1.5032110","article-title":"Complexification and integrability in multidimensions","volume":"59","author":"Fokas","year":"2018","journal-title":"J. Math. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2093","DOI":"10.1088\/0951-7715\/20\/9\/005","article-title":"Nonlinear Fourier transforms, integrability and nonlocality in multidimensions","volume":"20","author":"Fokas","year":"2007","journal-title":"Nonlinearity"},{"key":"ref_32","unstructured":"Faddeev, L.D. (1960). The Inverse Problem in the Quantum Theory of Scattering, New York University."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Zhang, Y.F., and Gui, L.L. (2024). Solutions of Cauchy Problems for the Caudrey-Dodd-Gibbon-Kotera-Sawada equation in three spatial and two temporal dimensions. Axioms, 14.","DOI":"10.3390\/axioms14010011"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"20220074","DOI":"10.1098\/rspa.2022.0074","article-title":"Integrable nonlinear evolution equations in three spatial dimensions","volume":"478","author":"Fokas","year":"2022","journal-title":"Proc. R. Soc. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/s00033-009-0017-z","article-title":"Integrable decompositions for the (2 + 1)-dimensional Gardner equation","volume":"61","author":"Xu","year":"2010","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1088\/0951-7715\/14\/6\/302","article-title":"Decomposition of the (2 + 1)-dimensional Gardner equation and its quasi-periodic solutions","volume":"14","author":"Geng","year":"2001","journal-title":"Nonlinearity"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1016\/j.jmaa.2006.08.021","article-title":"On the (2 + 1)-dimensional Gardner equation: Determinant solutions and pfaffianization (ENG)","volume":"330","author":"Yu","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1051\/mmnp\/2024004","article-title":"(3 + 1)-Dimensional Gardner Equation Deformed from (1+1)-Dimensional Gardner Equation and its Conservation Laws","volume":"19","author":"Jin","year":"2024","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_39","unstructured":"Chadan, K., Colton, D., Pivrinta, L., and Rundell, W. (1987). An Introduction to Inverse Scattering and Inverse Spectral Problems, Society for Industrial and Applied Mathematics."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2150405","DOI":"10.1142\/S0217984921504054","article-title":"Conservation laws of some multi-component integrable systems","volume":"24","author":"Gui","year":"2021","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.amc.2007.06.002","article-title":"New solutions of distinct physical structures to high-dimensional nonlinear evolution equations","volume":"196","author":"Wazwaz","year":"2008","journal-title":"Appl. Math. Comp."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/102\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:26:49Z","timestamp":1759919209000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/102"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,11]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010102"],"URL":"https:\/\/doi.org\/10.3390\/sym17010102","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,11]]}}}