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.

Ablowitz-Kaup-Newell-Segur（AKNS）线性谱问题催生了许多重要的非线性数学物理方程，包括非局部方程。本文通过首次为AKNS谱问题及其伴随方程配备局部分数阶偏导数，推导了两个分数阶AKNS层次结构，这在文献中尚未报道。一种是时空分数阶等谱AKNS（stfisAKNS）层次结构，其中的三个简化生成分数阶局部和非局部非线性Schrödinger（flnNLS）以及经修改的Kortweg-de Vries（fmKdV）层次结构以及反向tNLS（frtNLS）层次结构，另一个是时间分数阶非等光谱AKNS（tfnisAKNS）层次结构。通过将stfisAKNS层次转换为两个分数双线性形式并从与tfnisAKNS层次相对应的分数散射数据中重建电位，通过Hirota双线性方法（HBM）获得了三对具有Mittag-Leffler函数的新颖N分数解的统一公式。和逆散射变换（IST）。限于Cantor集，一些数字在任何地方都是连续的，但不可微分的一和二分形解直接显示在图中。更有意义的是，构造N值得探索的问题从局部分数阶导数的角度出发，以stfisAKNS和tfnisAKNS层次结构为例，求解了HBM和IST的孤子方程层次结构的分形解。此外，本文表明，HBM和IST可用于构造其他孤子方程层次结构的N分形解。