Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions
Abstract Ablowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral pr...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2021-04-01
|
| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13662-021-03374-0 |
| _version_ | 1818561532109258752 |
|---|---|
| author | Bo Xu Yufeng Zhang Sheng Zhang |
| author_facet | Bo Xu Yufeng Zhang Sheng Zhang |
| author_sort | Bo Xu |
| collection | DOAJ |
| description | Abstract Ablowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies. |
| first_indexed | 2024-12-14T00:52:04Z |
| format | Article |
| id | doaj.art-e926a2d50b5d470a98beb07143114958 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-14T00:52:04Z |
| publishDate | 2021-04-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-e926a2d50b5d470a98beb071431149582022-12-21T23:23:47ZengSpringerOpenAdvances in Difference Equations1687-18472021-04-012021112710.1186/s13662-021-03374-0Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functionsBo Xu0Yufeng Zhang1Sheng Zhang2School of Mathematics, China University of Mining and TechnologySchool of Mathematics, China University of Mining and TechnologySchool of Mathematical Sciences, Bohai UniversityAbstract Ablowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies.https://doi.org/10.1186/s13662-021-03374-0Fractional order isospectral AKNS hierarchyFractional order non-isospectral AKNS hierarchyLocal fractional order partial derivativeN-fractal solutions with Mittag-Leffler functionsHirota bilinear methodInverse scattering transform |
| spellingShingle | Bo Xu Yufeng Zhang Sheng Zhang Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions Advances in Difference Equations Fractional order isospectral AKNS hierarchy Fractional order non-isospectral AKNS hierarchy Local fractional order partial derivative N-fractal solutions with Mittag-Leffler functions Hirota bilinear method Inverse scattering transform |
| title | Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions |
| title_full | Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions |
| title_fullStr | Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions |
| title_full_unstemmed | Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions |
| title_short | Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions |
| title_sort | fractional isospectral and non isospectral akns hierarchies and their analytic methods for n fractal solutions with mittag leffler functions |
| topic | Fractional order isospectral AKNS hierarchy Fractional order non-isospectral AKNS hierarchy Local fractional order partial derivative N-fractal solutions with Mittag-Leffler functions Hirota bilinear method Inverse scattering transform |
| url | https://doi.org/10.1186/s13662-021-03374-0 |
| work_keys_str_mv | AT boxu fractionalisospectralandnonisospectralaknshierarchiesandtheiranalyticmethodsfornfractalsolutionswithmittaglefflerfunctions AT yufengzhang fractionalisospectralandnonisospectralaknshierarchiesandtheiranalyticmethodsfornfractalsolutionswithmittaglefflerfunctions AT shengzhang fractionalisospectralandnonisospectralaknshierarchiesandtheiranalyticmethodsfornfractalsolutionswithmittaglefflerfunctions |