Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation
In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we e...
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2024-08-01
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| author | Sheng Zhang Hongwei Li Bo Xu |
| author_facet | Sheng Zhang Hongwei Li Bo Xu |
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| description | In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional <i>n</i>-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media. |
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| language | English |
| publishDate | 2024-08-01 |
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| spelling | doaj-art-1207a3edd4a4422691c2a768a814fb1b2025-08-20T01:55:31ZengMDPI AGFractal and Fractional2504-31102024-08-018952010.3390/fractalfract8090520Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type EquationSheng Zhang0Hongwei Li1Bo Xu2School of Mathematical Sciences, Bohai University, Jinzhou 121013, ChinaSchool of Mathematical Sciences, Bohai University, Jinzhou 121013, ChinaSchool of Educational Sciences, Bohai University, Jinzhou 121013, ChinaIn the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional <i>n</i>-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media.https://www.mdpi.com/2504-3110/8/9/520inverse scattering integrabilityvariable-coefficient fractional-order KdV-type equationinverse scattering transformfractional soliton solution |
| spellingShingle | Sheng Zhang Hongwei Li Bo Xu Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation Fractal and Fractional inverse scattering integrability variable-coefficient fractional-order KdV-type equation inverse scattering transform fractional soliton solution |
| title | Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation |
| title_full | Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation |
| title_fullStr | Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation |
| title_full_unstemmed | Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation |
| title_short | Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation |
| title_sort | inverse scattering integrability and fractional soliton solutions of a variable coefficient fractional order kdv type equation |
| topic | inverse scattering integrability variable-coefficient fractional-order KdV-type equation inverse scattering transform fractional soliton solution |
| url | https://www.mdpi.com/2504-3110/8/9/520 |
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