mKdV Equation on Time Scales: Darboux Transformation and <i>N</i>-Soliton Solutions
In this paper, the spectral problem of the mKdV equation satisfying the compatibility condition on time scales is directly constructed. By using the zero-curvature equation on time scales, the mKdV equation on time scales is obtained. When <inline-formula><math xmlns="http://www.w3.org...
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2024-08-01
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| author | Baojian Jin Yong Fang Xue Sang |
| author_facet | Baojian Jin Yong Fang Xue Sang |
| author_sort | Baojian Jin |
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| description | In this paper, the spectral problem of the mKdV equation satisfying the compatibility condition on time scales is directly constructed. By using the zero-curvature equation on time scales, the mKdV equation on time scales is obtained. When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, the equation degenerates to the classical mKdV equation. Then, the single-soliton, two-soliton, and <i>N</i>-soliton solutions of the mKdV equation under the zero boundary condition on time scales are presented via employing the Darboux transformation (DT). Particularly, we obtain the corresponding single-soliton solutions expressed using the Cayley exponential function on four different time scales (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">Z</mi></semantics></math></inline-formula>, <i>q</i>-discrete, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">C</mi></semantics></math></inline-formula>). |
| format | Article |
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| language | English |
| publishDate | 2024-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-3715adef3d6045629acdb8db3cd3cd632025-08-20T01:56:10ZengMDPI AGAxioms2075-16802024-08-0113957810.3390/axioms13090578mKdV Equation on Time Scales: Darboux Transformation and <i>N</i>-Soliton SolutionsBaojian Jin0Yong Fang1Xue Sang2College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, ChinaCollege of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, ChinaCollege of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, ChinaIn this paper, the spectral problem of the mKdV equation satisfying the compatibility condition on time scales is directly constructed. By using the zero-curvature equation on time scales, the mKdV equation on time scales is obtained. When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, the equation degenerates to the classical mKdV equation. Then, the single-soliton, two-soliton, and <i>N</i>-soliton solutions of the mKdV equation under the zero boundary condition on time scales are presented via employing the Darboux transformation (DT). Particularly, we obtain the corresponding single-soliton solutions expressed using the Cayley exponential function on four different time scales (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">Z</mi></semantics></math></inline-formula>, <i>q</i>-discrete, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">C</mi></semantics></math></inline-formula>).https://www.mdpi.com/2075-1680/13/9/578modified Korteweg–de Vries equationtime scalesCayley exponential functionDarboux transformation |
| spellingShingle | Baojian Jin Yong Fang Xue Sang mKdV Equation on Time Scales: Darboux Transformation and <i>N</i>-Soliton Solutions Axioms modified Korteweg–de Vries equation time scales Cayley exponential function Darboux transformation |
| title | mKdV Equation on Time Scales: Darboux Transformation and <i>N</i>-Soliton Solutions |
| title_full | mKdV Equation on Time Scales: Darboux Transformation and <i>N</i>-Soliton Solutions |
| title_fullStr | mKdV Equation on Time Scales: Darboux Transformation and <i>N</i>-Soliton Solutions |
| title_full_unstemmed | mKdV Equation on Time Scales: Darboux Transformation and <i>N</i>-Soliton Solutions |
| title_short | mKdV Equation on Time Scales: Darboux Transformation and <i>N</i>-Soliton Solutions |
| title_sort | mkdv equation on time scales darboux transformation and i n i soliton solutions |
| topic | modified Korteweg–de Vries equation time scales Cayley exponential function Darboux transformation |
| url | https://www.mdpi.com/2075-1680/13/9/578 |
| work_keys_str_mv | AT baojianjin mkdvequationontimescalesdarbouxtransformationandinisolitonsolutions AT yongfang mkdvequationontimescalesdarbouxtransformationandinisolitonsolutions AT xuesang mkdvequationontimescalesdarbouxtransformationandinisolitonsolutions |