Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions
A.S. Fokas has obtained integrable nonlinear partial differential equations (PDEs) in 4 + 2 dimensions by complexifying the independent variables. In this work, the complexification of the independent variables of the 2 + 1-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation yields the 4...
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| author | Yufeng Zhang Linlin Gui |
| author_facet | Yufeng Zhang Linlin Gui |
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| description | A.S. Fokas has obtained integrable nonlinear partial differential equations (PDEs) in 4 + 2 dimensions by complexifying the independent variables. In this work, the complexification of the independent variables of the 2 + 1-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation yields the 4 + 2 integrable extension of the CDGKS equation. Then, by transforming two temporal variables, the CDGKS equation in three dimensions is reduced, and the Lax pairs of the corresponding equations are given. Finally, the solutions of Cauchy problems for the CDGKS equation in three spatial and two temporal dimensions are constructed by introducing a novel nonlocal <i>d</i>-bar formalism, in which several new long derivative operators, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>x</mi></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>y</mi></msub></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>t</mi></msub></semantics></math></inline-formula>, are constructed for the study of the initial value problem for the CDGKS equation. Some significant propositions and results are presented in this paper. |
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| institution | Kabale University |
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| publishDate | 2024-12-01 |
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| spelling | doaj-art-295a9054abc84645a4ff74acf09d207d2025-01-24T13:22:08ZengMDPI AGAxioms2075-16802024-12-011411110.3390/axioms14010011Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal DimensionsYufeng Zhang0Linlin Gui1College of Technology and Data, Yantai Nanshan University, Yantai 265713, ChinaSchool of Mathematics, China University of Mining and Technology, Xuzhou 221116, ChinaA.S. Fokas has obtained integrable nonlinear partial differential equations (PDEs) in 4 + 2 dimensions by complexifying the independent variables. In this work, the complexification of the independent variables of the 2 + 1-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation yields the 4 + 2 integrable extension of the CDGKS equation. Then, by transforming two temporal variables, the CDGKS equation in three dimensions is reduced, and the Lax pairs of the corresponding equations are given. Finally, the solutions of Cauchy problems for the CDGKS equation in three spatial and two temporal dimensions are constructed by introducing a novel nonlocal <i>d</i>-bar formalism, in which several new long derivative operators, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>x</mi></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>y</mi></msub></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>t</mi></msub></semantics></math></inline-formula>, are constructed for the study of the initial value problem for the CDGKS equation. Some significant propositions and results are presented in this paper.https://www.mdpi.com/2075-1680/14/1/11CDGKS equationhigh-dimensionalCauchy problem<i>d</i>-bar |
| spellingShingle | Yufeng Zhang Linlin Gui Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions Axioms CDGKS equation high-dimensional Cauchy problem <i>d</i>-bar |
| title | Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions |
| title_full | Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions |
| title_fullStr | Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions |
| title_full_unstemmed | Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions |
| title_short | Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions |
| title_sort | solutions of cauchy problems for the caudrey dodd gibbon kotera sawada equation in three spatial and two temporal dimensions |
| topic | CDGKS equation high-dimensional Cauchy problem <i>d</i>-bar |
| url | https://www.mdpi.com/2075-1680/14/1/11 |
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