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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 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.

Introduction

It is well known that AKNS spectral equation [1] is an important linear problem, from which and its associated time evolution equations of eigenfunctions abundant nonlinear partial differential equations (PDEs) [2, 3] have been derived, such as the KdV equation, mKdV equation, NLS equation, Burgers equation, sinh-Gordon (sG) equation, loop soliton equations, Harry–Dym (HD) equation, Heisenberg ferromagnet (HF) equation, and the N-wave equations. Besides, from the same AKNS linear problem, isopsectral and non-isospectral hierarchies of nonlinear PDEs can be derived. Researchers often called such derived hierarchies the isospsectral AKNS (isAKNS) hierarchies and non-isospectral AKNS (nisAKNS) hierarchies, respectively. It was Ablowitz et al. [1] who first solved the isAKNS equations by IST. In 1998, Gesztesy and Ratnaseelan [4] obtained algebro-geometric solutions of the AKNS hierarchy based on elementary algebraic methods. In 2004, Ning et al. [5] solved the nisAKNS hierarchy by IST. In 2008, Yin et al. [6] solved the isAKNS hierarchy by its bilinear form. In 2012, Chen et al. [7] solved the isAKNS hierarchy by HBM. In 2017, Zhang and Gao [8] solved a generalized isAKNS hierarchy by HBM. In 2018, Zhang and Hong [9] solved a generalized isAKNS hierarchy by IST. In 2018, Zhang and Hong [10] solved a generalized nisAKNS hierarchy by IST. Recently, some nonlocal integrable evolution equations [11, 12] have been found from the symmetry reductions of the isAKNS hierarchy. In 2011, Wu and Zhang [13] constructed Hamiltonian structures of a fractional AKNS hierarchy by a generalized Tu formula. In 2020, Gao et al. [14] solved the (\(2+1\))-dimensional AKNS equation with conformable derivatives and a perturbation parameter by the sine-Gordon expansion method. However, to the best of our knowledge, there are no reports on HBM, IST, and fractal solutions for fractional AKNS equations.

The aim of this paper is to derive the stfisAKNS hierarchy

D t α ( u v ) = L m α ( u v ) (m=1,2,),
(1)

with the fractional order operator

L α = ( D x α + 2 u I x , a α v 2 u I x , a α u 2 v I x , a α v D x α 2 v I x , a α u ) ,
(2)

and the tfnisAKNS hierarchy

D t α ( u v ) = L m ( x u x v ) (m=1,2,),
(3)

with the following operator:

L = σ + 2 ( u v ) 1 ( v , u ) , σ = ( 1 0 0 1 ) , 1 = 1 2 ( x d x x + d x ) ,
(4)

and then solve the fractional order AKNS hierarchies (1) and (3) by extending HBM [15] and IST [1], respectively. Here, \(D_{x}^{\alpha } \) and \(I_{x,a}^{\alpha } \) represent the local fractional order partial derivative operator [16]

$$ D_{x}^{\alpha } \phi (x,t) = \Gamma (1 + \alpha )\lim _{\varepsilon \to 0}\frac{\phi (x + \varepsilon ,t) - \phi (x,t)}{\varepsilon ^{\alpha }}\quad (0 < \alpha \le 1) $$
(5)

and the local fractional integral operator [16]

$$ I_{x,a}^{\alpha } \varphi (x,t) = \frac{1}{\Gamma (1 + \alpha )} \int _{a}^{x} \varphi (\omega ,t) (\mathrm{d} \omega )^{\alpha }\quad (0 < \alpha \le 1) $$
(6)

for any nondifferentiable functions \(\phi (x,t)\) and \(\varphi (x,t)\) defined on a fractal set Ω, respectively. The concept of local fractional derivative was first proposed by Kolwankar and Gangal [17], and it has received continuous developments and extensive applications like [1825]. In addition, this article will show some obtained fractal solutions for more insights into novel nonlinearities hidden behind the fractional order models.

The rest of this article consists of four parts: Sect. 2 derives the stfisAKNS and tfnisAKNS hierarchies and gives three reductions for the fractional order nonlocal hierarchies of evolution equations; Sect. 3 constructs N-fractal solutions with Mittag-Leffler functions of the stfisAKNS hierarchy by considering two fractional order bilinear forms and shows the obtained one- and two-fractal solutions restricted to the Cantor set; Sect. 4 extends IST for constructing N-fractal solutions with Mittag-Leffler functions of the tfnisAKNS hierarchy and shows the obtained one-fractal solutions restricted to the Cantor set; Sect. 5 concludes this article.

Derivations of the fractional order AKNS hierarchies

Based on the Lax scheme [26], this section derives the stfisAKNS and tfnisAKNS hierarchies (1) and (3).

Fractional order isospectral AKNS hierarchy

Theorem 1

Suppose

A= I x , a α (v,u) ( B C ) 1 2 ( 2 i α k ) m ,
(7)

where \(\mathrm{i}^{\alpha } \) is the necessary formal imaginary number unit [16] to connect the relationships between Mittag-Leffler functions and trigonometric functions defined in a fractal set Ω. Then the stfisAKNS hierarchy (1) can be derived from the fractional order zero curvature equation, i.e., the fractional order compatibility condition

$$ D_{t}^{\alpha } U - D_{x}^{\alpha } V + [U,V] = 0, \qquad [U,V] \equiv UV - VU $$
(8)

of the following fractional order linear spectral problem:

D x α F=UF,U= ( i α k u v i α k ) ,F= ( F 1 F 2 ) ,
(9)

and its associated time-fractional order evolution equation

D t α F=VF,V= ( A B C A ) .
(10)

Here, \(u = u(x,t)\) and \(v = v(x,t)\) and their local fractional order derivatives with respect to x and t are all nondifferentiable functions, \(\mathrm{i}^{\alpha } k\) is the spectral parameter being independent of x and t, while A, B, and C are all undetermined local fractional differentiable functions of x, t, u, v, and k.

Proof

On the one hand, from Eq. (9) we obtain the α-order local fractional derivative with respect to t:

$$ D_{t}^{\alpha } \bigl(D_{x}^{\alpha } F \bigr) = \bigl(D_{t}^{\alpha } U\bigr)F + U \bigl(D_{t}^{\alpha } F\bigr). $$
(11)

On the other hand, the local fractional derivative of Eq. (10) with respect to x gives

$$ D_{x}^{\alpha } \bigl(D_{t}^{\alpha } F \bigr) = \bigl(D_{x}^{\alpha } V\bigr)F + V\bigl(D_{x}^{\alpha } F\bigr). $$
(12)

Since u, v, A, B, and C are all local fractional differentiable functions, F is local fractional continuous (see Definition 1.2 of [25]), and then we have \(D_{x}^{\alpha } (D_{t}^{\alpha } F) = D_{t}^{\alpha } (D_{x}^{\alpha } F)\) from Theorem 3 in [25]. Thus, Eqs. (9)–(12) lead to

$$ \bigl(D_{t}^{\alpha } V + UV\bigr)F = \bigl(D_{x}^{\alpha } V + VU\bigr)F, $$
(13)

which is namely Eq. (8) by using the arbitrariness of F.

Substituting the matrices U and V of Eqs. (9) and (10) into Eq. (8) yields

$$\begin{aligned}& D_{x}^{\alpha } A = uC - vB, \end{aligned}$$
(14)
$$\begin{aligned}& D_{t}^{\alpha } u = D_{x}^{\alpha } B + 2 \mathrm{i}^{\alpha } kB + 2uA,\qquad D_{t}^{\alpha } v = D_{x}^{\alpha } C - 2\mathrm{i}^{\alpha } kC - 2vA. \end{aligned}$$
(15)

In view of Eq. (7), we rewrite Eqs. (14) and (15) as

D t α ( u v ) = L α ( B C ) 2 i α k ( B C ) + ( 2 i α k ) m ( u v ) .
(16)

Further supposing

( B C ) = l = 1 m ( b l c l ) ( 2 i α k ) m l
(17)

and substituting it into Eq. (16), then comparing the coefficients of the same powers of \(2\mathrm{i}^{\alpha } k\) in Eq. (17), we have

( 2 i α k ) 0 : D t m α ( u v ) = L α ( b m c m ) ,
(18)
( 2 i α k ) m + 1 l : ( b l c l ) = L α ( b l 1 c l 1 ) (l=2,,m),
(19)
( 2 i α k ) m : ( b 1 c 1 ) = ( u v ) .
(20)

Equations (19) and (20) give

( b m c m ) = L ( m 1 ) α ( u v ) .
(21)

Finally, the substitution of Eq. (21) into Eq. (18) leads to the stfisAKNS hierarchy (1). □

Fractional order non-isospectral AKNS hierarchy

Theorem 2

Let the spectral parameter ik satisfy

$$ D_{t}^{\alpha } (\mathrm{i}k) = \frac{1}{2}(2 \mathrm{i}k)^{m}, $$
(22)

and suppose

A= 1 (v,u) ( B C ) 1 2 ( 2 i k ) m x,
(23)

then the time-fractional order zero curvature equation

$$ D_{t}^{\alpha } U - V_{x} + [U,V] = 0, $$
(24)

i.e., the time-fractional order compatibility condition of the following linear spectral problem:

F x =UF,U= ( i k u v i k ) ,F= ( F 1 F 2 ) ,
(25)

and its associated time-fractional order evolution Eq. (10) generate the tfnisAKNS hierarchy (3). Here, \(u = u(x,t)\), \(v = v(x,t)\) and their integer order derivatives with respect to x are all smooth functions, but \(u = u(x,t)\), \(v = v(x,t)\) and their fractional order derivatives with respect to t are all nondifferentiable functions.

Proof

In a similar way to the proof of Theorem 1, with the help of the matrices U and V of Eqs. (25) and (10), we convert Eq. (24) into

$$\begin{aligned}& A_{x} = uC - vB - \frac{1}{2}(2\mathrm{i}k)^{m}, \end{aligned}$$
(26)
$$\begin{aligned}& D_{t}^{\alpha } u = B_{x} + 2\mathrm{i}kB + 2uA, \qquad D_{t}^{\alpha } v = C_{x} - 2\mathrm{i}kC - 2vA, \end{aligned}$$
(27)

which can be written as

D t α ( u v ) = L α ( B C ) 2ik ( B C ) + ( 2 i k ) m ( x u x v )
(28)

by using Eq. (23). Introducing [3]

( B C ) = l = 1 m ( b l c l ) ( 2 i k ) m l
(29)

and substituting it into Eq. (28), then comparing the coefficients of the same powers of \(2\mathrm{i}k\) in Eq. (28) yield

( 2 i k ) 0 : D t α ( u v ) =L ( b m c m ) ,
(30)
( 2 i k ) m + 1 l : ( b l c l ) =L ( b l 1 c l 1 ) (l=2,,m),
(31)
( 2 i k ) m : ( b 1 c 1 ) = ( x u x v ) .
(32)

Then we have

( b m c m ) = L m 1 ( x u x v ) ,
(33)

and hence we reach the tfnisAKNS hierarchy (3). □

Nonlocal and reverse-t reductions of stfisAKNS hierarchy

Conveniently, we rewrite the stfisAKNS hierarchy (1) as the following equivalent form:

D t m + 1 α ( u v ) = K m α = ( K 1 , m α K 2 , m α ) = L α D t m α ( u v ) (m=1,2,)
(34)

by introducing the variables \(t_{1} = x,t_{2},t_{3}, \ldots \) and treating u and v as two infinite-dimensional functions of these variables.

As special cases of Eq. (34), we would like to give three reductions. The first one is the flnNLS hierarchy

$$ \mathrm{i}^{\alpha } u_{t_{2s}}^{\alpha } = K_{2s}^{\alpha ,flnNLS} = - K_{1,2s}^{\alpha } |_{(\text{36})} \quad (s = 1,2, \ldots ) $$
(35)

reduced from Eq. (34) with transformation

$$ v(x,t) = \delta u^{*}(\sigma x,t), \quad \delta ,\sigma = \pm 1, t \to \mathrm{i}^{\alpha } t, t \in \mathbb{R}, $$
(36)

the representative of which is the flnNLS equation

$$ \mathrm{i}^{\alpha } D_{t_{2}}^{\alpha } u = K_{2}^{\alpha ,flnNLS} = D_{x}^{2\alpha } u - 2 \delta u^{2}u^{*}(\delta x,t). $$
(37)

The second one is the frtNLS hierarchy

$$ u_{t_{2s}}^{\alpha } = K_{2s}^{\alpha ,frtNLS} = K_{1,2s}^{\alpha } |_{(\text{39})} \quad (s = 1,2, \ldots ) $$
(38)

reduced from Eq. (34) with transformation

$$ v(x,t) = \delta u(\sigma x, - t),\quad \delta ,\sigma = \pm 1, $$
(39)

one representative of which is the frtNLS equation

$$ D_{t_{2}}^{\alpha } u = K_{2}^{\alpha ,NLS} = - D_{x}^{2\alpha } u + 2\delta u^{2}u(\delta x, - t). $$
(40)

And the last one is the fmKdV hierarchy

$$ u_{t_{2s + 1}}^{\alpha } = K_{2s + 1}^{\alpha ,fmKdV} = K_{1,2s + 1}^{\alpha } |_{(\text{42})} \quad (s = 1,2, \ldots ) $$
(41)

reduced from Eq. (34) with transformation

$$ v(x,t) = \delta u(\sigma x,\sigma t), \quad \delta ,\sigma = \pm 1, $$
(42)

one representative of which is the fmKdV equation

$$ D_{t_{3}}^{\alpha } u = K_{3}^{\alpha ,fmKdV} = D_{x}^{3\alpha } u - 6\delta u\bigl(D_{x}^{\alpha } u\bigr)u(\delta x,\sigma t). $$
(43)

HBM for N-fractal solutions of the stfisAKNS hierarchy

To construct N-fractal solutions, we first derive the fractional bilinear forms of the stfiAKNS hierarchy (34). We then employ two reduced fractional bilinear forms to construct N-fractal solutions.

Fractional bilinear forms

Theorem 3

Letting

$$ u = \frac{g}{f}, \qquad v = \frac{h}{f}, $$
(44)

then the stfiAKNS hierarchy (34) can be bilinearized as

$$\begin{aligned}& \bigl(H_{t_{m + 1}}^{\alpha } + H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)g \cdot f = fgI_{x,0}^{\alpha } \biggl\{ D_{t_{m}}^{\alpha } \biggl[ \frac{1}{f^{2}} \bigl(H_{x}^{2\alpha } f \cdot f + 2gh\bigr) \biggr] \biggr\} , \end{aligned}$$
(45)
$$\begin{aligned}& \bigl(H_{t_{m + 1}}^{\alpha } - H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)h \cdot f = - fhI_{x,0}^{\alpha } \biggl\{ D_{t_{m}}^{\alpha } \biggl[ \frac{1}{f^{2}} \bigl(H_{x}^{2\alpha } f \cdot f + 2gh\bigr) \biggr] \biggr\} , \end{aligned}$$
(46)

where \(H_{t_{m + 1}}^{\alpha }\), \(H_{t_{m}}^{\alpha }\), and \(H_{x}^{\alpha } \) are local fractional versions [27] of the Hirota bilinear operator [15].

Proof

We write the component forms of Eq. (34) as

$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } u + D_{t_{m},x}^{2\alpha } u = 2uI_{x,0}^{\alpha } \bigl[D_{t_{m}}^{\alpha } (uv)\bigr], \end{aligned}$$
(47)
$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } v - D_{t_{m},x}^{2\alpha } v = - 2vI_{x,0}^{\alpha } \bigl[D_{t_{m}}^{\alpha } (uv)\bigr]. \end{aligned}$$
(48)

In view of Eq. (44), we have

$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } u = \frac{D_{t_{m + 1}}^{\alpha } gD_{x}^{\alpha } f - gD_{t_{m + 1}}^{\alpha } f}{f^{2}}, \end{aligned}$$
(49)
$$\begin{aligned}& D_{t_{m},x}^{2\alpha } u = \frac{(D_{t_{m + 1},x}^{2\alpha } g)f - D_{t_{m + 1}}^{\alpha } gD_{x}^{\alpha } f - D_{x}^{\alpha } gD_{t_{m + 1}}^{\alpha } f - gD_{t_{m + 1},x}^{2\alpha } f}{f^{2}} + \frac{2gD_{x}^{\alpha } fD_{t_{m + 1}}^{\alpha } f}{f^{3}}, \\ \end{aligned}$$
(50)
$$\begin{aligned}& 2uI_{x,0}^{\alpha } \bigl[D_{t_{m}}^{\alpha } (uv)\bigr] = \frac{2g}{f}I_{x,0}^{\alpha } D_{t_{m}}^{\alpha } \frac{gh}{f^{2}}. \end{aligned}$$
(51)

Using Eqs. (49)–(51), we convert the left- and right-hand sides of Eq. (47) into

$$\begin{aligned}& \mathrm{LHS} = \frac{D_{t_{m + 1}}^{\alpha } gD_{x}^{\alpha } f - gD_{t_{m + 1}}^{\alpha } f}{f^{2}} \\& \hphantom{\mathrm{SHD} =}{} + \frac{(D_{t_{m + 1},x}^{2\alpha } g)f - D_{t_{m + 1}}^{\alpha } gD_{x}^{\alpha } f - D_{x}^{\alpha } gD_{t_{m + 1}}^{\alpha } f + gD_{t_{m + 1},x}^{2\alpha } f}{f^{2}}, \end{aligned}$$
(52)
$$\begin{aligned}& \mathrm{RHS} = \frac{2gD_{t_{m + 1},x}^{2\alpha } f}{f^{2}} - \frac{2gD_{x}^{\alpha } fD_{t_{m + 1}}^{\alpha } f}{f^{3}} + \frac{2g}{f}I_{x,0}^{\alpha } D_{t_{m}}^{\alpha } \frac{gh}{f^{2}} \\& \hphantom{\mathrm{RHD}} = \frac{2g}{f}I_{x,0}^{\alpha } \biggl\{ D_{t_{m}}^{\alpha } \biggl[ \frac{(D_{x}^{2\alpha } f)f - (D_{x}^{\alpha } f)^{2} + gh}{f^{2}} \biggr] \biggr\} . \end{aligned}$$
(53)

With the help of Eqs. (52) and (53), we can rewrite Eq. (47) as Eq. (45) by employing the fractional order Hirota bilinear operators. In a similar way we reach Eq. (48). Thus, we finish the proof. □

Generally, if we construct N-fractal solutions of the stfisANKS hierarchy (34) directly by using Eqs. (45) and (46), the calculation is relatively complex and difficult. Inspired by the work [7], this section employs two reductions of the fractional bilinear forms (45) and (46) to construct fractional N-fractal solutions. The one adopts the constraint \(H_{x}^{2\alpha } f \cdot f + 2gh = 0\), and the other weakens this constraint by letting \(H_{x}^{2\alpha } f \cdot f + 2gh = a^{2}f^{2}\), here a is a constant.

First reduced bilinear forms and N-fractal solutions

We set

$$ H_{x}^{2\alpha } f \cdot f + 2gh = 0, $$
(54)

then the fractional order bilinear forms (45) and (46) become

$$\begin{aligned}& \bigl(H_{t_{m + 1}}^{\alpha } + H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)g \cdot f = 0, \end{aligned}$$
(55)
$$\begin{aligned}& \bigl(H_{t_{m + 1}}^{\alpha } - H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)h \cdot f = 0. \end{aligned}$$
(56)

For the fractional one-fractal solutions, we assume that

$$\begin{aligned}& f = 1 + \varepsilon ^{2}f^{(2)} + \varepsilon ^{4}f^{(4)} + \cdots + \varepsilon ^{2j}f^{(2j)} + \cdots , \end{aligned}$$
(57)
$$\begin{aligned}& g = \varepsilon ^{1}g^{(1)} + \varepsilon ^{3}g^{(3)} + \cdots + \varepsilon ^{2j + 1}g^{(2j + 1)} + \cdots , \end{aligned}$$
(58)
$$\begin{aligned}& h = \varepsilon ^{1}h^{(1)} + \varepsilon ^{3}h^{(3)} + \cdots + \varepsilon ^{2j + 1}h^{(2j + 1)} + \cdots , \end{aligned}$$
(59)

and substitute Eqs. (57)–(59) into Eqs. (54)–(56). Collecting the coefficients of the same order of ε yields a system of fractional differential equations as follows:

$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } g^{(1)} + D_{t_{m},x}^{2\alpha } g^{(1)} = 0, \\ \end{aligned}$$
(60)
$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } h^{(1)} - D_{t_{m},x}^{\alpha } h^{(1)} = 0, \end{aligned}$$
(61)
$$\begin{aligned}& D_{x}^{2\alpha } f^{(2)} + g^{(1)}h^{(1)} = 0, \end{aligned}$$
(62)
$$\begin{aligned}& \bigl(D_{t_{m + 1}}^{\alpha } + D_{t_{m},x}^{\alpha } \bigr)g^{(3)} = - \bigl(H_{t_{m + 1}}^{\alpha } + H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)g^{(1)}f^{(2)}, \end{aligned}$$
(63)
$$\begin{aligned}& \bigl(D_{t_{m + 1}}^{\alpha } - D_{t_{m},x}^{\alpha } \bigr)h^{(3)} = - \bigl(H_{t_{m + 1}}^{\alpha } - H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)h^{(1)}f^{(2)}, \end{aligned}$$
(64)
$$\begin{aligned}& 2D_{x}^{2\alpha } f^{(4)} = - H_{x}^{2\alpha } f^{(2)}f^{(2)} - 2\bigl(g^{(1)}h^{(3)} + g^{(3)}h^{(1)}\bigr), \end{aligned}$$
(65)
$$\begin{aligned}& \bigl(D_{t_{m + 1}}^{\alpha } + D_{t_{m},x}^{\alpha } \bigr)g^{(5)} = - \bigl(H_{t_{m + 1}}^{\alpha } + H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr) \bigl(g^{(1)}f^{(4)} + g^{(3)}f^{(2)} \bigr), \end{aligned}$$
(66)
$$\begin{aligned}& \bigl(D_{t_{m + 1}}^{\alpha } - D_{t_{m},x}^{\alpha } \bigr)h^{(5)} = - \bigl(H_{t_{m + 1}}^{\alpha } - H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr) \bigl(h^{(1)}f^{(4)} + h^{(3)}f^{(2)} \bigr), \end{aligned}$$
(67)
$$\begin{aligned}& 2D_{x}^{2\alpha } f^{(6)} = - H_{x}^{2\alpha } f^{(2)}f^{(4)} - 2\bigl(g^{(1)}h^{(5)} + g^{(3)}h^{(3)} + g^{(5)}h^{(1)} \bigr), \\& \vdots \end{aligned}$$
(68)

and so forth. Letting

$$\begin{aligned}& g^{(1)} = E_{\alpha } \bigl(\xi _{1}^{\alpha } \bigr), \qquad \xi _{1} = k_{1}x + \sum _{m = 1}^{\infty } w_{1,m + 1}t_{m + 1} + \xi _{1}^{(0)}, \end{aligned}$$
(69)
$$\begin{aligned}& h^{(1)} = E_{\alpha } \bigl(\eta _{1}^{\alpha } \bigr),\qquad \eta _{1} = p_{1}x + \sum _{m = 1}^{\infty } q_{1,m + 1}t_{m + 1} + \eta _{1}^{(0)}, \end{aligned}$$
(70)

from Eqs. (60) and (61) we have

$$\begin{aligned}& w_{1,m + 1}^{\alpha } = - k_{1}^{\alpha } w_{1,m}^{\alpha },\qquad w_{1,1}^{\alpha } = k_{1}^{\alpha }, \end{aligned}$$
(71)
$$\begin{aligned}& q_{1,m + 1}^{\alpha } = p_{1}^{\alpha } q_{1,m}^{\alpha },\qquad q_{1,1}^{\alpha } = p_{1}^{\alpha }. \end{aligned}$$
(72)

In view of Eq. (62), we suppose that

$$ f^{(2)} = E_{\alpha } \bigl(\xi _{1}^{\alpha } + \eta _{1}^{\alpha } + \theta _{13}^{\alpha } \bigr), $$
(73)

where \(\theta _{13}^{\alpha } \) is a constant determined later and the Mittag-Leffler function is defined on a fractal set Ω [16]:

$$ E_{\alpha } \bigl(\mu ^{\alpha } \bigr) = \sum _{k = 0}^{\infty } \frac{\mu ^{k\alpha }}{\Gamma (1 + k\alpha )}. $$
(74)

Then Eqs. (69), (70), and (73) hint

$$ E_{\alpha } \bigl(\theta _{13}^{\alpha } \bigr) = - \frac{1}{(k_{1} + k_{2})^{2}}. $$
(75)

Substituting Eqs. (69)–(74) into Eqs. (63)–(65) and setting

$$ g^{(3)} = h^{(3)} = f^{(4)} = \cdots = 0, $$
(76)

we can see that Eq. (61) and the equations behind all hold. In this case, we write

$$ f_{1} = 1 + E_{\alpha } \bigl(\xi _{1}^{\alpha } + \eta _{1}^{\alpha } + \theta _{13}^{\alpha } \bigr), \qquad g_{1} = E_{\alpha } \bigl(\xi _{1}^{\alpha } \bigr),\qquad h_{1} = E_{\alpha } \bigl(\eta _{1}^{\alpha } \bigr), $$
(77)

and hence we obtain the fractional one-fractal solutions of the stfisAKNS hierarchy (34):

$$ u = \frac{E_{\alpha } (\xi _{1}^{\alpha } )}{1 + E_{\alpha } (\xi _{1}^{\alpha } + \eta _{1}^{\alpha } + \theta _{13}^{\alpha } )},\qquad v = \frac{E_{\alpha } (\eta _{1}^{\alpha } )}{1 + E_{\alpha } (\xi _{1}^{\alpha } + \eta _{1}^{\alpha } + \theta _{13}^{\alpha } )}. $$
(78)

The one-fractal solutions (78) restricted to the Cantor set are shown in Figs. 13, where the parameters \(k_{1} = \mathrm{i}\) and \(p_{1} = 0.5\).

Figure 1
figure1

Space-time structures of one-fractal solutions (78) with \(\alpha = \ln 2/\ln 3\).

Figure 2
figure2

Profiles along the \(t_{2}\) axis of one-fractal solutions (78) with \(\alpha = \ln 2/\ln 3\).

Figure 3
figure3

Profiles along the x axis of one-fractal solutions (78) with \(\alpha = \ln 2/\ln 3\).

For the two-fractal solutions, we assume that

$$\begin{aligned}& g^{(1)} = E_{\alpha } \bigl(\xi _{1}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{2}^{\alpha } \bigr), \qquad \xi _{i} = k_{i}x + \sum _{m = 1}^{\infty } w_{i,m + 1}t_{m + 1} + \xi _{i}^{(0)} \quad (i = 1,2), \end{aligned}$$
(79)
$$\begin{aligned}& h^{(1)} = E_{\alpha } \bigl(\eta _{1}^{\alpha } \bigr) + E_{\alpha } \bigl(\eta _{2}^{\alpha } \bigr), \qquad \eta _{i} = p_{i}x + \sum _{m = 1}^{\infty } q_{i,m + 1}t_{m + 1} + \eta _{i}^{(0)} \quad (i = 1,2), \end{aligned}$$
(80)

and substitute Eqs. (79) and (80) into Eqs. (60) and (61). Then the resulting equations show

$$\begin{aligned}& w_{2,m + 1}^{\alpha } = - k_{2}^{\alpha } w_{2,m}^{\alpha },\qquad w_{2,1}^{\alpha } = k_{2}^{\alpha }, \end{aligned}$$
(81)
$$\begin{aligned}& q_{2,m + 1}^{\alpha } = p_{2}^{\alpha } q_{2,m}^{\alpha },\qquad q_{2,1}^{\alpha } = p_{2}^{\alpha }. \end{aligned}$$
(82)

In view of Eq. (57), we further let

$$\begin{aligned} f^{(2)} =& E_{\alpha } \bigl(\xi _{1}^{\alpha } + \eta _{1}^{\alpha } + \theta _{13}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{1}^{\alpha } + \eta _{2}^{\alpha } + \theta _{14}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{2}^{\alpha } + \eta _{1}^{\alpha } + \theta _{23}^{\alpha } \bigr) \\ &{} + E_{\alpha } \bigl(\xi _{2}^{\alpha } + \eta _{2}^{\alpha } + \theta _{24}^{\alpha } \bigr), \end{aligned}$$
(83)

where \(\theta _{14}^{\alpha }\), \(\theta _{23}^{\alpha }\), and \(\theta _{24}^{\alpha } \) are constants. Substituting Eq. (83) into Eq. (62) yields

$$ E_{\alpha } \bigl(\theta _{14}^{\alpha } \bigr) = - \frac{1}{(k_{1} + l_{2})^{2}},\qquad E_{\alpha } \bigl(\theta _{23}^{\alpha } \bigr) = - \frac{1}{(k_{2} + l_{1})^{2}},\qquad E_{\alpha } \bigl(\theta _{24}^{\alpha } \bigr) = - \frac{1}{(k_{2} + l_{2})^{2}}. $$
(84)

Taking into consideration Eq. (63), we select

$$ g^{(3)} = E_{\alpha } \bigl(\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + \eta _{1}^{\alpha } + \theta _{12}^{\alpha } + \theta _{13}^{\alpha } + \theta _{23}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + \eta _{2}^{\alpha } + \theta _{12}^{\alpha } + \theta _{14}^{\alpha } + \theta _{24}^{\alpha } \bigr), $$
(85)

where \(\theta _{12}^{\alpha } \) is a constant. We then obtain from Eq. (63)

$$\begin{aligned}& E_{\alpha } \bigl(\theta _{12}^{\alpha } \bigr) = - (k_{1} - k_{2})^{2}. \end{aligned}$$
(86)

In view of Eqs. (64) and (65), we set

$$\begin{aligned}& h^{(3)} = E_{\alpha } \bigl(\xi _{1}^{\alpha } + \eta _{1}^{\alpha } + \eta _{2}^{\alpha } + \theta _{13}^{\alpha } + \theta _{14}^{\alpha } + \theta _{34}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{2}^{\alpha } + \eta _{1}^{\alpha } + \eta _{2}^{\alpha } + \theta _{23}^{\alpha } + \theta _{24}^{\alpha } + \theta _{34}^{\alpha } \bigr), \end{aligned}$$
(87)
$$\begin{aligned}& f^{(4)} = E_{\alpha } \bigl(\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + \eta _{1}^{\alpha } + \eta _{2}^{\alpha } + \theta _{12}^{\alpha } + \theta _{13}^{\alpha } + \theta _{14}^{\alpha } + \theta _{23}^{\alpha } + \theta _{24}^{\alpha } + \theta _{34}^{\alpha } \bigr), \end{aligned}$$
(88)

and then obtain

$$ E_{\alpha } \bigl(\theta _{34}^{\alpha } \bigr) = - (p_{1} - p_{2})^{2}. $$
(89)

Substituting Eqs. (86)–(89) into Eq. (66), we can see that if

$$ g^{(5)} = h^{(5)} = f^{(6)} = \cdots = 0, $$
(90)

then Eq. (67) and those equations behind all hold. In this case, we obtain two-fractal solutions of the stfisAKNS hierarchy (34):

$$ u = \frac{g_{2}}{f_{2}}, \qquad v = \frac{h_{2}}{f_{2}}, $$
(91)

where

$$\begin{aligned}& g_{2} = E_{\alpha } \bigl(\xi _{1}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{2}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + \eta _{1}^{\alpha } + \theta _{12}^{\alpha } + \theta _{13}^{\alpha } + \theta _{23}^{\alpha } \bigr) \\& \hphantom{g_{2} =}{}+ E_{\alpha } \bigl(\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + \eta _{2}^{\alpha } + \theta _{12}^{\alpha } + \theta _{14}^{\alpha } + \theta _{24}^{\alpha } \bigr), \end{aligned}$$
(92)
$$\begin{aligned}& h_{2} = E_{\alpha } \bigl(\xi _{1}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{2}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{1}^{\alpha } + \eta _{1}^{\alpha } + \eta _{2}^{\alpha } + \theta _{13}^{\alpha } + \theta _{14}^{\alpha } + \theta _{34}^{\alpha } \bigr) \\& \hphantom{h_{2} =}{} + E_{\alpha } \bigl(\xi _{2}^{\alpha } + \eta _{1}^{\alpha } + \eta _{2}^{\alpha } + \theta _{23}^{\alpha } + \theta _{24}^{\alpha } + \theta _{34}^{\alpha } \bigr), \end{aligned}$$
(93)
$$\begin{aligned}& f_{2} = 1 + E_{\alpha } \bigl(\xi _{1}^{\alpha } + \eta _{1}^{\alpha } + \theta _{13}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{1}^{\alpha } + \eta _{2}^{\alpha } + \theta _{14}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{2}^{\alpha } + \eta _{1}^{\alpha } + \theta _{23}^{\alpha } \bigr) \\& \hphantom{f_{2} =}{} + E_{\alpha } \bigl(\xi _{2}^{\alpha } + \eta _{2}^{\alpha } + \theta _{24}^{\alpha } \bigr) \\& \hphantom{f_{2} =}{}+ E_{\alpha } \bigl(\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + \eta _{1}^{\alpha } + \eta _{2}^{\alpha } + \theta _{12}^{\alpha } + \theta _{13}^{\alpha } + \theta _{14}^{\alpha } + \theta _{23}^{\alpha } + \theta _{24}^{\alpha } + \theta _{34}^{\alpha } \bigr), \end{aligned}$$
(94)
$$\begin{aligned}& \begin{aligned}&E_{\alpha } \bigl(\theta _{12}^{\alpha } \bigr) = - (k_{1} - k_{2})^{2},\qquad E_{\alpha } \bigl(\theta _{34}^{\alpha } \bigr) = - (p_{1} - p_{2})^{2}, \\ &E_{\alpha } \bigl[\theta _{i(j + 2)}^{\alpha } \bigr] = - \frac{1}{(k_{i} + p_{j})^{2}}\quad (i,j = 1,2). \end{aligned} \end{aligned}$$
(95)

In Figs. 46, the two-fractal solutions (91) restricted to the Cantor set are shown, where \(k_{1} = \mathrm{i}^{\alpha }\), \(k_{2} = 3\mathrm{i}^{\alpha }\), \(p_{1} = 0.01\), \(p_{2} = 3\), and \(\alpha = \ln 2/\ln 3\).

Figure 4
figure4

Space-time structures of two-fractal solutions (91) with \(\alpha = \ln 2/\ln 3\).

Figure 5
figure5

Profiles along the \(t_{2}\) axis of one-fractal solutions (91) with \(\alpha = \ln 2/\ln 3\).

Figure 6
figure6

Profiles along the x axis of one-fractal solutions (91) with \(\alpha = \ln 2/\ln 3\).

Proceeding with a similar manipulation, we can obtain three-fractal solutions and induce the uniform formulas of the N-fractal solutions of the stfisAKNS hierarchy (34):

$$ u = \frac{g_{N}}{f_{N}}, \qquad v = \frac{h_{N}}{f_{N}}, $$
(96)

with

$$\begin{aligned}& g_{N} = \sum_{\mu = 0,1} Z_{2}(\mu )E_{\alpha } \Biggl( \sum _{i = 1}^{2N} \mu _{i}\xi _{i} + \sum_{1 \le i < j}^{2N} \mu _{i}\mu _{j}\theta _{ij} \Biggr), \end{aligned}$$
(97)
$$\begin{aligned}& h_{N} = \sum_{\mu = 0,1} Z_{3}(\mu )E_{\alpha } \Biggl( \sum _{i = 1}^{2N} \mu _{i}\xi _{i} + \sum_{1 \le i < j}^{2N} \mu _{i}\mu _{j}\theta _{ij} \Biggr), \end{aligned}$$
(98)
$$\begin{aligned}& f_{N} = \sum_{\mu = 0,1} Z_{1}(\mu )E_{\alpha } \Biggl( \sum _{i = 1}^{2N} \mu _{i}\xi _{i} + \sum_{1 \le i < j}^{2N} \mu _{i}\mu _{j}\theta _{ij} \Biggr), \end{aligned}$$
(99)
$$\begin{aligned}& \xi _{i} = k_{i}x + \sum _{m = 1}^{\infty } w_{i,m + 1}t_{m + 1} + \xi _{i}^{(0)},\qquad w_{i,m + 1} = - k_{i}w_{i,m},\qquad w_{i,1} = k_{i}, \end{aligned}$$
(100)
$$\begin{aligned}& \begin{aligned}&\eta _{i} = p_{i}x + \sum _{m = 1}^{\infty } q_{i,m + 1}t_{m + 1} + \eta _{i}^{(0)},\qquad q_{i,m + 1} = p_{i}q_{i,m}, \\ & q_{i,1} = p_{i},\qquad \xi _{N + i}^{\alpha } = \eta _{i}^{\alpha } \quad (i = 1,2, \ldots ,N), \end{aligned} \end{aligned}$$
(101)
$$\begin{aligned}& E_{\alpha } (\theta _{ij}) = - (k_{i} - k_{j})^{2}, \qquad E_{\alpha } [\theta _{(i + N)(j + N)}] = - (p_{i} - p_{j})^{2} \quad (i < j = 2,3, \ldots ,N), \end{aligned}$$
(102)
$$\begin{aligned}& E_{\alpha } [\theta _{i(j + N)}] = - \frac{1}{(k_{i} + p_{j})^{2}} \quad (i,j = 1,2, \ldots ,N), \end{aligned}$$
(103)

where the summation \(\sum _{\mu = 0,1}\) refers to all possible combinations of each \(\mu _{i} = 0,1\) for \(i = 1,2, \ldots ,2N\), \(Z_{1}(\mu )\), \(Z_{2}(\mu )\), and \(Z_{3}(\mu )\) denote that when we select all the possible combinations \(\mu _{i} = 0,1\) (\(i = 1,2, \ldots ,2N\)) the following conditions hold, respectively:

$$ \sum_{i = 1}^{N} \mu _{i} = \sum_{i = 1}^{N} \mu _{N + i}, \qquad \sum_{i = 1}^{N} \mu _{i} = \sum_{i = 1}^{N} \mu _{N + i} + 1, \qquad \sum_{i = 1}^{N} \mu _{i} + 1 = \sum_{i = 1}^{N} \mu _{N + i}. $$
(104)

Second reduced bilinear forms and N-fractal solutions

Without loss of generality, we suppose

$$ H_{x}^{2\alpha } f \cdot f + 2gh = a^{2}f^{2}, $$
(105)

then the fractional order bilinear forms (45) and (46) become

$$\begin{aligned}& \bigl(H_{t_{m + 1}}^{\alpha } + H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)g \cdot f = cfg, \end{aligned}$$
(106)
$$\begin{aligned}& \bigl(H_{t_{m + 1}}^{\alpha } - H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)h \cdot f = - cfh, \end{aligned}$$
(107)

where \(c=a^{2}\) is a non-zero constant. Further letting

$$ f = \tilde{f}, \qquad g = E_{\alpha } \bigl(ct_{m + 1}^{\alpha } \bigr)\tilde{g},\qquad h = E_{\alpha } \bigl( - ct_{m + 1}^{\alpha } \bigr)\tilde{h}, $$
(108)

we transform Eqs. (105)–(107) into

$$\begin{aligned}& H_{x}^{2}\tilde{f} \cdot \tilde{f} = - 2\tilde{g} \tilde{h} + a^{2}\tilde{f}^{2}, \end{aligned}$$
(109)
$$\begin{aligned}& \bigl(H_{t_{m + 1}}^{\alpha } + H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)\tilde{g} \cdot \tilde{f} = 0, \end{aligned}$$
(110)
$$\begin{aligned}& \bigl(H_{t_{m + 1}}^{\alpha } - H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)\tilde{h} \cdot \tilde{f} = 0. \end{aligned}$$
(111)

In what follows, we employ Eqs. (109)–(111) to construct N-fractal solutions of the stfisAKNS hierarchy (34). To construct one-fractal solutions, we suppose that

$$\begin{aligned}& \tilde{f} = 1 + \varepsilon \tilde{f}^{(1)} + \varepsilon ^{2}\tilde{f}^{(2)} + \cdots + \varepsilon ^{j}\tilde{f}^{(j)} + \cdots , \end{aligned}$$
(112)
$$\begin{aligned}& \tilde{g} = \tilde{g}^{(0)} + \varepsilon \tilde{g}^{(1)} + \varepsilon ^{2}\tilde{g}^{(2)} + \cdots + \varepsilon ^{j}\tilde{g}^{(j)} + \cdots , \end{aligned}$$
(113)
$$\begin{aligned}& \tilde{h} = \tilde{h}^{(0)} + \varepsilon \tilde{h}^{(1)} + \varepsilon ^{2}\tilde{h}^{(2)} + \cdots + \varepsilon ^{j}\tilde{h}^{(j)} + \cdots . \end{aligned}$$
(114)

Substituting Eqs. (112)–(114) into Eqs. (109)–(110) and collecting the coefficients of the same order of ε yield a system of fractional differential equations, the first several equations of which are as follows:

$$\begin{aligned}& 2\tilde{g}^{(0)}\tilde{h}^{(0)} = a^{2}, \end{aligned}$$
(115)
$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } \tilde{g}^{(0)} + D_{x,t_{m}}^{2\alpha } \tilde{g}^{(0)} = 0, \end{aligned}$$
(116)
$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } \tilde{h}^{(0)} + D_{x,t_{m}}^{2\alpha } \tilde{h}^{(0)} = 0, \end{aligned}$$
(117)
$$\begin{aligned}& D_{x}^{2\alpha } f^{(1)} = - \tilde{g}^{(0)} \tilde{h}^{(1)} - \tilde{g}^{(1)}\tilde{h}^{(0)} + a^{2}\tilde{f}^{(1)}, \end{aligned}$$
(118)
$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } \tilde{g}^{(1)} + D_{x,t_{m}}^{2\alpha } \tilde{g}^{(1)} = - \bigl(H_{t_{m + 1}}^{\alpha } + H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)\tilde{g}^{(0)} \cdot \tilde{f}^{(1)}, \end{aligned}$$
(119)
$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } \tilde{h}^{(1)} - D_{x,t_{m}}^{2\alpha } \tilde{h}^{(1)} = - \bigl(H_{t_{m + 1}}^{\alpha } - H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr)\tilde{h}^{(0)} \cdot \tilde{f}^{(1)}, \end{aligned}$$
(120)
$$\begin{aligned}& D_{x}^{2\alpha } \tilde{f}^{(2)} = - \frac{1}{2}H_{x}^{2\alpha } \tilde{f}^{(1)} \cdot \tilde{f}^{(1)} - \tilde{g}^{(0)} \tilde{h}^{(2)} - \tilde{g}^{(1)}\tilde{h}^{(1)} - \tilde{g}^{(2)}\tilde{h}^{(0)} + a^{2} \biggl[ \tilde{f}^{(2)} + \frac{1}{2}\bigl( \tilde{f}^{(1)}\bigr)^{2} \biggr], \end{aligned}$$
(121)
$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } \tilde{g}^{(2)} + D_{x,t_{m}}^{2\alpha } \tilde{g}^{(2)} = - \bigl(H_{t_{m + 1}}^{\alpha } + H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr) \bigl(\tilde{g}^{(0)} \cdot \tilde{f}^{(2)} + \tilde{g}^{(1)} \cdot \tilde{f}^{(1)}\bigr), \end{aligned}$$
(122)
$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } \tilde{h}^{(2)} + D_{x,t_{m}}^{2\alpha } \tilde{h}^{(2)} = - \bigl(H_{t_{m + 1}}^{\alpha } - H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr) \bigl(\tilde{h}^{(0)} \cdot \tilde{f}^{(2)} + \tilde{h}^{(1)} \cdot \tilde{f}^{(1)}\bigr), \end{aligned}$$
(123)
$$\begin{aligned}& D_{x}^{\alpha } \tilde{f}^{(3)} = - H_{x}^{2\alpha } \tilde{f}^{(1)} \cdot \tilde{f}^{(2)} - \tilde{g}^{(0)}\tilde{h}^{(3)} - \tilde{g}^{(1)}\tilde{h}^{(2)} - \tilde{g}^{(2)} \tilde{h}^{(1)} - \tilde{g}^{(3)}\tilde{h}^{(0)} + a^{2}\bigl(\tilde{f}^{(3)} + \tilde{f}^{(1)} \tilde{f}^{(2)}\bigr), \end{aligned}$$
(124)
$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } \tilde{g}^{(3)} + D_{x,t_{m}}^{2\alpha } \tilde{g}^{(3)} = - \bigl(H_{t_{m + 1}}^{\alpha } + H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr) \bigl(\tilde{g}^{(0)} \cdot \tilde{f}^{(3)} + \tilde{g}^{(1)} \cdot \tilde{f}^{(2)} + \tilde{g}^{(2)} \cdot \tilde{f}^{(1)}\bigr), \end{aligned}$$
(125)
$$\begin{aligned}& D_{t_{m + 1}}^{\alpha } \tilde{h}^{(3)} + D_{x,t_{m}}^{2\alpha } \tilde{h}^{(3)} = - \bigl(H_{t_{m + 1}}^{\alpha } - H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr) \bigl(\tilde{h}^{(0)} \cdot \tilde{f}^{(3)} + \tilde{h}^{(1)} \cdot \tilde{f}^{(2)} + \tilde{h}^{(2)} \cdot \tilde{f}^{(1)}\bigr). \end{aligned}$$
(126)

Setting

$$ \tilde{g}^{(0)} = \tilde{h}^{(0)} = \frac{a}{\sqrt{2}} $$
(127)

and then substituting Eq. (127) into Eqs. (118)–(120), we have

$$\begin{aligned}& \tilde{f}^{(1)} = 2E_{\alpha } \bigl(\xi _{1}^{\alpha } \bigr), \qquad \tilde{g}^{(1)} = \sqrt{2} aE_{\alpha } \bigl(\xi _{1}^{\alpha } + 2\vartheta _{1}^{\alpha } \bigr),\qquad \tilde{h}^{(1)} = \sqrt{2} aE_{\alpha } \bigl(\xi _{1}^{\alpha } - 2\vartheta _{1}^{\alpha } \bigr), \end{aligned}$$
(128)
$$\begin{aligned}& \begin{aligned}&\xi _{1} = k_{1}x + \sum _{m = 1}^{\infty } w_{1,m + 1}t_{m + 1} + \xi _{1}^{(0)}, \\ & w_{1,1}^{2\alpha } = k_{1}^{2\alpha } = - 2a^{2}\sinh _{\alpha }^{2}\vartheta _{1}^{\alpha }, \qquad w_{1,m + 1}^{2\alpha } = - 2a^{2}w_{1,m}^{2\alpha } \cosh _{\alpha }^{2} \vartheta _{1}^{\alpha }, \end{aligned} \end{aligned}$$
(129)

where the hyperbolic functions \(\sinh _{\alpha } \mu ^{\alpha } \) and \(\cosh _{\alpha } \mu ^{\alpha } \) are defined on the fractal set Ω [16]:

$$ \sinh _{\alpha } \mu ^{\alpha } = \frac{E_{\alpha } (\mu ^{\alpha } ) - E_{\alpha } ( - \mu ^{\alpha } )}{2}, \qquad \cosh _{\alpha } \mu ^{\alpha } = \frac{E_{\alpha } (\mu ^{\alpha } ) + E_{\alpha } ( - \mu ^{\alpha } )}{2}. $$
(130)

If we set \(\tilde{g}^{(2)} = \tilde{g}^{(3)} = \tilde{h}^{(2)} = \tilde{h}^{(3)} = \tilde{f}^{(2)} = \tilde{f}^{(3)} = \cdots = 0\), we can see that Eqs. (128) and (129) satisfy Eqs. (118)–(126) and the other equations not explicitly written in the system of fractional differential equations. In this case, we write

$$ \begin{aligned}&\tilde{f}_{1} = 1 + 2E_{\alpha } \bigl(\xi _{1}^{\alpha } \bigr), \qquad \tilde{g}_{1} = \frac{a[1 + 2E_{\alpha } (\xi _{1}^{\alpha } + 2\vartheta _{1}^{\alpha } )]}{\sqrt{2}}, \\ & \tilde{h}_{1} = \frac{a[1 + 2E_{\alpha } (\xi _{1}^{\alpha } - 2\vartheta _{1}^{\alpha } )]}{\sqrt{2}}, \end{aligned} $$
(131)

and then we obtain one-fractal solutions of the stfisAKNS hierarchy (34):

$$ u = \frac{aE_{\alpha } (ct_{m + 1}^{\alpha } )[1 + 2E_{\alpha } (\xi _{1}^{\alpha } + 2\vartheta _{1}^{\alpha } )]}{\sqrt{2} (1 + 2E_{\alpha } (\xi _{1}^{\alpha } ))}, \qquad v = \frac{aE_{\alpha } ( - ct_{m + 1}^{\alpha } )[1 + 2E_{\alpha } (\xi _{1}^{\alpha } - 2\vartheta _{1}^{\alpha } )]}{\sqrt{2} (1 + 2E_{\alpha } (\xi _{1}^{\alpha } ))}, $$
(132)

by using Eqs. (108), (112)–(114), (128), (129), and (131).

In order to construct two-fractal solutions, we let

$$\begin{aligned}& \tilde{f}^{(1)} = 2\bigl(E_{\alpha } \bigl(\xi _{1}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{2}^{\alpha } \bigr)\bigr), \end{aligned}$$
(133)
$$\begin{aligned}& \tilde{g}^{(1)} = \sqrt{2} a\bigl[E_{\alpha } \bigl(\xi _{1}^{\alpha } + 2\vartheta _{1}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{2}^{\alpha } + 2 \vartheta _{2}^{\alpha } \bigr)\bigr], \end{aligned}$$
(134)
$$\begin{aligned}& \tilde{h}^{(1)} = \sqrt{2} a\bigl[E_{\alpha } \bigl(\xi _{1}^{\alpha } - 2\vartheta _{1}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{2}^{\alpha } - 2 \vartheta _{2}^{\alpha } \bigr)\bigr], \end{aligned}$$
(135)

where

$$ \begin{aligned}&\xi _{i} = k_{i}x + \sum _{m = 1}^{\infty } w_{i,m + 1}t_{m + 1} + \xi _{i}^{(0)},\qquad w_{i,1}^{2\alpha } = k_{i}^{2\alpha } = - 2a^{2}\sinh _{\alpha }^{2}\vartheta _{i}^{\alpha }, \\ & w_{i,m + 1}^{2\alpha } = - 2a^{2}w_{i,m}^{2\alpha } \cosh _{\alpha }^{2}\vartheta _{i}^{\alpha } \quad (i = 1,2), \end{aligned} $$
(136)

and suppose

$$\begin{aligned}& \bigl(H_{t_{m + 1}}^{\alpha } + H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr) \bigl(\tilde{g}^{(1)} \cdot \tilde{f}^{(2)} + \tilde{g}^{(2)} \cdot \tilde{f}^{(1)}\bigr) = 0, \end{aligned}$$
(137)
$$\begin{aligned}& \bigl(H_{t_{m + 1}}^{\alpha } - H_{x}^{\alpha } H_{t_{m}}^{\alpha } \bigr) \bigl(\tilde{h}^{(1)} \cdot \tilde{f}^{(2)} + \tilde{h}^{(2)} \cdot \tilde{f}^{(1)}\bigr) = 0. \end{aligned}$$
(138)
$$\begin{aligned}& H_{x}^{2\alpha } \tilde{f}^{(1)} \cdot \tilde{f}^{(2)} + \tilde{g}^{(1)}\tilde{h}^{(2)} + \tilde{g}^{(2)}\tilde{h}^{(1)} - a^{2} \tilde{f}^{(1)}\tilde{f}^{(2)} = 0. \end{aligned}$$
(139)

We then obtain from Eqs. (121)–(123) and Eqs. (136)–(138)

$$\begin{aligned}& \tilde{f}^{(2)} = 4E_{\alpha } \bigl(\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + A_{12}^{\alpha } \bigr), \end{aligned}$$
(140)
$$\begin{aligned}& \tilde{g}^{(2)} = 2\sqrt{2} aE_{\alpha } \bigl(\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + 2 \vartheta _{1}^{\alpha } + 2\vartheta _{2}^{\alpha } + A_{12}^{\alpha } \bigr), \end{aligned}$$
(141)
$$\begin{aligned}& \tilde{h}^{(2)} = 2\sqrt{2} aE_{\alpha } \bigl(\xi _{1}^{\alpha } + \xi _{2}^{\alpha } - 2 \vartheta _{1}^{\alpha } - 2\vartheta _{2}^{\alpha } + A_{12}^{\alpha } \bigr). \end{aligned}$$
(142)

Substituting Eqs. (133)–(135) and Eqs. (140)–(142) into Eqs. (137)–(139) yields

$$ \tilde{f}^{(3)} = \tilde{g}^{(3)} = \tilde{h}^{(3)} = \tilde{f}^{(4)} = \tilde{g}^{(4)} = \tilde{h}^{(4)} = \cdots = 0. $$
(143)

Thus, from Eqs. (105)–(107) we have

$$\begin{aligned}& f_{2} = 1 + 2\bigl(E_{\alpha } \bigl(\xi _{1}^{\alpha } \bigr) + E_{\alpha } \bigl(\xi _{2}^{\alpha } \bigr) \bigr) + 4E_{\alpha } \bigl(\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + A_{12}^{\alpha } \bigr), \end{aligned}$$
(144)
$$\begin{aligned}& g_{2} = \frac{aE_{\alpha } (ct_{m + 1}^{\alpha } )\{ 1 + 2[E_{\alpha } (\xi _{1}^{\alpha } + 2\vartheta _{1}^{\alpha } ) + E_{\alpha } (\xi _{2}^{\alpha } + 2\vartheta _{2}^{\alpha } )] + 4E_{\alpha } (\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + 2\vartheta _{1}^{\alpha } + 2\vartheta _{2}^{\alpha } + A_{12}^{\alpha } )\}}{\sqrt{2}}, \end{aligned}$$
(145)
$$\begin{aligned}& h_{2} = \frac{aE_{\alpha } ( - ct_{m + 1}^{\alpha } )\{ 1 + 2[E_{\alpha } (\xi _{1}^{\alpha } - 2\vartheta _{1}^{\alpha } ) + E_{\alpha } (\xi _{2}^{\alpha } - 2\vartheta _{2}^{\alpha } )] + 4E_{\alpha } (\xi _{1}^{\alpha } + \xi _{2}^{\alpha } - 2\vartheta _{1}^{\alpha } - 2\vartheta _{2}^{\alpha } + A_{12}^{\alpha } )\}}{\sqrt{2}}, \end{aligned}$$
(146)

and hence we obtain two-fractal solutions of the stfisAKNS hierarchy (34):

$$\begin{aligned}& u = \frac{aE_{\alpha } (ct_{m + 1}^{\alpha } )\{ 1 + 2[E_{\alpha } (\xi _{1}^{\alpha } + 2\vartheta _{1}^{\alpha } ) + E_{\alpha } (\xi _{2}^{\alpha } + 2\vartheta _{2}^{\alpha } )] + 4E_{\alpha } (\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + 2\vartheta _{1}^{\alpha } + 2\vartheta _{2}^{\alpha } + A_{12}^{\alpha } )\}}{\sqrt{2} [1 + 2(E_{\alpha } (\xi _{1}^{\alpha } ) + E_{\alpha } (\xi _{2}^{\alpha } )) + 4E_{\alpha } (\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + A_{12}^{\alpha } )]}, \end{aligned}$$
(147)
$$\begin{aligned}& v = \frac{aE_{\alpha } ( - ct_{m + 1}^{\alpha } )\{ 1 + 2[E_{\alpha } (\xi _{1}^{\alpha } - 2\vartheta _{1}^{\alpha } ) + E_{\alpha } (\xi _{2}^{\alpha } - 2\vartheta _{2}^{\alpha } )] + 4E_{\alpha } (\xi _{1}^{\alpha } + \xi _{2}^{\alpha } - 2\vartheta _{1}^{\alpha } - 2\vartheta _{2}^{\alpha } + A_{12}^{\alpha } )\}}{\sqrt{2} [1 + 2(E_{\alpha } (\xi _{1}^{\alpha } ) + E_{\alpha } (\xi _{2}^{\alpha } )) + 4E_{\alpha } (\xi _{1}^{\alpha } + \xi _{2}^{\alpha } + A_{12}^{\alpha } )]}, \end{aligned}$$
(148)

where

$$ E_{\alpha } \bigl(A_{12}^{\alpha } \bigr) = \frac{\sinh _{\alpha }^{2}\frac{\vartheta _{1}^{\alpha } - \vartheta _{2}^{\alpha }}{2}}{\sinh _{\alpha }^{2}\frac{\vartheta _{1}^{\alpha } + \vartheta _{2}^{\alpha }}{2}}. $$
(149)

Selecting \(c = 2.5\), \(k_{1} = 1\), \(k_{2} = - 4\), \(\vartheta _{1} = 1\), \(\vartheta _{2} = 2\), and \(\alpha = \ln 2/\ln 3\), we show the two-fractal solutions (147) and (148) in Figs. 79.

Figure 7
figure7

Space-time structures of two-fractal solutions (147) and (148) with \(\alpha = \ln 2/\ln 3\).

Figure 8
figure8

Profiles along the \(t_{2}\) axis of two-fractal solutions (147) and (148) with \(\alpha = \ln 2/\ln 3\).

Figure 9
figure9

Profiles along the x axis of two-fractal solutions (147) and (148) with \(\alpha = \ln 2/\ln 3\).

Similarly, we can obtain three-fractal and then determine the uniform formulas (96) for the N-fractal solutions of the stfisAKNS hierarchy (34) by

$$\begin{aligned}& f_{N} = \sum_{\mu = 0,1} E_{\alpha } \Biggl[ \sum_{i = 1}^{N} \mu _{i}\bigl(\xi _{i}^{\alpha } + \ln _{\alpha } 2\bigr) + \sum_{1 \le i < l}^{N} \mu _{i}\mu _{l}A_{il}^{\alpha } \Biggr], \end{aligned}$$
(150)
$$\begin{aligned}& g_{N} = \frac{aE_{\alpha } (ct_{m + 1}^{\alpha } )\sum_{\mu = 0,1} E_{\alpha } [ \sum_{i = 1}^{N} \mu _{i}(\xi _{i}^{\alpha } + 2\vartheta _{i}^{\alpha } + \ln _{\alpha } 2) + \sum_{1 \le i < l}^{N} \mu _{i}\mu _{l}A_{il}^{\alpha } ]}{\sqrt{2}}, \end{aligned}$$
(151)
$$\begin{aligned}& h_{N} = \frac{aE_{\alpha } ( - ct_{m + 1}^{\alpha } )\sum_{\mu = 0,1} E_{\alpha } [ \sum_{i = 1}^{N} \mu _{i}(\xi _{i}^{\alpha } - 2\vartheta _{i}^{\alpha } + \ln _{\alpha } 2) + \sum_{1 \le i < l}^{N} \mu _{i}\mu _{l}A_{il}^{\alpha } ]}{\sqrt{2}}, \end{aligned}$$
(152)
$$\begin{aligned}& \xi _{i} = k_{i}x + \sum _{m = 1}^{\infty } w_{i,m + 1}t_{m + 1} + \xi _{i}^{(0)}, \end{aligned}$$
(153)
$$\begin{aligned}& w_{i,1}^{2\alpha } = k_{i}^{2\alpha } = - 2a^{2}\sinh _{\alpha }^{2}\vartheta _{i}^{\alpha },\qquad w_{i,m + 1}^{2\alpha } = - 2a^{2}w_{i,m}^{2\alpha } \cosh _{\alpha }^{2}\vartheta _{i}^{\alpha } \quad (i = 1,2, \ldots ,N), \end{aligned}$$
(154)
$$\begin{aligned}& \mathrm{e}^{A_{il}^{\alpha }} = \frac{\sinh ^{2}\frac{\theta _{i}^{\alpha } - \theta _{l}^{\alpha }}{2}}{\sinh ^{2}\frac{\theta _{i}^{\alpha } + \theta _{l}^{\alpha }}{2}} \quad (1 \le i < l \le N). \end{aligned}$$
(155)

IST for N-fractal solutions of the tfnisAKNS hierarchy

Since the space derivative of the linear spectral problem (25) is integer order, all the existing results [2, 3] about the spectral problem (25) are valid for the tfnisAKNS hierarchy (3). The main objective of this section is to determine the time-dependence of scattering data by using the associated time-fractional order evolution Eq. (10).

Time-dependence of scattering data

Theorem 4

Suppose that the potentials u and v of the linear spectral problem (25) equipped with the non-isospectral parameter ik satisfying Eq. (22) develop according to the tfnisAKNS hierarchy (3), then the scattering data

$$\begin{aligned}& \biggl\{ \kappa _{j}(t), c_{j}(t), R(t,k) = \frac{\gamma (t,k)}{\delta (t,k)}, j = 1,2, \ldots ,N \biggr\} , \end{aligned}$$
(156)
$$\begin{aligned}& \biggl\{ \bar{\kappa }_{l}(t), \bar{c}_{l}(t), \bar{R}(t,k) = \frac{\bar{\gamma } (t,k)}{\bar{\delta } (t,k)}, l = 1,2, \ldots ,\bar{N} \biggr\} \end{aligned}$$
(157)

have the following time-dependence:

$$\begin{aligned}& D_{t}^{\alpha } \kappa _{j} = - \frac{\mathrm{i}}{2}(2\mathrm{i}\kappa _{j})^{m},\qquad c_{j}^{2}(t) = c_{j}^{2}(0)E_{\alpha } \bigl(mI_{0,t}^{\alpha } (2\mathrm{i}\kappa _{j})^{m - 1}\bigr), \end{aligned}$$
(158)
$$\begin{aligned}& \delta (t,k) = \delta (0,k), \qquad \gamma (t,k) = \gamma (0,k), \end{aligned}$$
(159)
$$\begin{aligned}& D_{t}^{\alpha } \bar{\kappa }_{l} = - \frac{\mathrm{i}}{2}(2\mathrm{i}\bar{\kappa }_{l})^{m}, \qquad \bar{c}_{l}^{2}(t) = \bar{c}_{l}^{2}(0)E_{\alpha } \bigl( - mI_{0,t}^{\alpha } (2\mathrm{i}\bar{\kappa }_{l})^{m - 1}\bigr), \end{aligned}$$
(160)
$$\begin{aligned}& \bar{\delta } (t,k) = \bar{\delta } (0,k),\qquad \bar{\gamma } (t,k) = \bar{ \gamma } (0,k), \end{aligned}$$
(161)

where \(c_{j}^{2}(0)\), \(\bar{c}_{l}^{2}(0)\), \(R(0,k) = \beta (0,k)/\alpha (0,k)\), and \(\bar{R}(0,k) = \bar{\beta } (0,k)/\bar{\alpha } (0,k)\) are the corresponding scattering data of the linear spectral problem (25) in the case of \((u(x,0),v(x,0))^{T}\).

Proof

Since \(F(x,k)\) solves Eq. (25), \(P(x,k) = D_{t}^{\alpha } F(x,k) - NF(x,k)\) is a solution of Eq. (25). So, \(P(x,k)\) can be expressed by \(F(x,k)\) and \(\tilde{F}(x,k)\), which is another solution of Eq. (25) but independent of \(F(x,k)\), i.e., there exist two functions \(\mu (t,k)\) and \(\tau (t,k)\) such that

$$ D_{t}^{\alpha } F(x,k) - V\phi (x,k) = \mu (t,k)F(x,k) + \tau (t,k)\tilde{F}(x,k). $$
(162)

We next consider the first case, i.e., discrete spectral \(k = \kappa _{j}\) (\(\operatorname{Im}\kappa _{j} > 0\)). It is easy to see that \(F(x,\kappa _{j})\) decays exponentially, while \(\tilde{F}(x,\kappa _{j})\) increases as \(x \to + \infty \). Thus, we have \(\tau (t,k) = 0\) and simplify Eq. (162) as

$$ D_{t}^{\alpha } F(x,\kappa _{j}) - VF(x,\kappa _{j}) = \mu (t,\kappa _{j})F(x,\kappa _{j}). $$
(163)

The left-multiplication on Eq. (163) by the inner product (\(F_{2}(x,\kappa _{j})\), \(F_{1}(x,\kappa _{j})\)) yields

$$ D_{t}^{\alpha } F_{1}(x,\kappa _{j})F_{2}(x,\kappa _{j}) - \bigl[CF_{1}^{2}(x,\kappa _{j}) + BF_{2}^{2}(x,\kappa _{j})\bigr] = 2\mu (t, \kappa _{j})F_{1}(x,\kappa _{j})F_{2}(x, \kappa _{j}). $$
(164)

Since \(F(x,\kappa _{j})\) is a normalized eigenfunction and

$$ 2 \int _{ - \infty }^{\infty } c_{j}^{2}F_{1}(x, \kappa _{j})F_{2}(x,\kappa _{j})\, \mathrm{d}x = 1, $$
(165)

we can rewrite Eq. (164) as

$$ \mu (t,\kappa _{j}) = - c_{j}^{2}\bigl( \bigl(F_{2}^{2}(x,\kappa _{j}),F_{1}^{2}(x, \kappa _{j})\bigr)^{T},(B,C)^{T}\bigr). $$
(166)

Making use of Eq. (25), we have

$$ \bigl(F_{1}(x,\kappa _{j})F_{2}(x,\kappa _{j})\bigr)_{x} = u(x)F_{2}^{2}(x, \kappa _{j}) + v(x)F_{1}^{2}(x,\kappa _{j}). $$
(167)

The integration of Eq. (167) with respect to x from −∞ to +∞ gives

$$ \int _{ - \infty }^{\infty } \bigl(u(x)F_{2}^{2}(x, \kappa _{j}) + v(x)F_{1}^{2}(x,\kappa _{j})\bigr)\,\mathrm{d}x = \int _{ - \infty }^{\infty } \bigl(F_{1}(x, \kappa _{j})F_{2}(x,\kappa _{j}) \bigr)_{x}\,\mathrm{d}x = 0. $$
(168)

At the same time, we rewrite Eq. (29) as

( B C ) = l = 1 m ( 2 i k ) m l L ¯ l 1 ( x u x v ) , L ¯ =σ2 ( u v ) 1 (v,u).
(169)

Then with the help of the conjugation operator [3]

L ¯ =σ+2 ( v u ) 1 (u,v)
(170)

and the results

( L ¯ l 1 ( F 2 2 ( x , κ j ) , F 1 2 ( x , κ j ) ) T , ( x u x v ) ) = ( 2 i κ j ( t ) ) l 1 ( ( F 2 2 ( x , κ j ) , F 1 2 ( x , κ j ) ) T , ( x u x v ) ) ,
(171)
( ( F 2 2 ( x , κ j ) , F 1 2 ( x , κ j ) ) T , ( x u x v ) ) = x [ F 1 ( x , κ j ) F 2 ( x , κ j ) ] x dx= 1 2 c j 2 ,
(172)

we gain from Eq. (167)

μ ( t , κ j ) = c j 2 ( ( F 2 2 ( x , κ j ) , F 1 2 ( x , κ j ) ) T , ( B , C ) T ) μ ( t , κ j ) = c j 2 ( ( F 2 2 ( x , κ j ) , F 1 2 ( x , κ j ) ) T , l = 1 m ( 2 i k ) m l L ¯ l 1 ( x u x v ) ) μ ( t , κ j ) = c j 2 l = 1 m ( 2 i κ j ) m l ( ( F 2 2 ( x , κ j ) , F 1 2 ( x , κ j ) ) T , L ¯ l 1 ( x u x v ) ) μ ( t , κ j ) = 1 2 m ( 2 i κ j ) m 1 .
(173)

Thus, Eq. (163) becomes

$$ D_{t}^{\alpha } F(x,\kappa _{j}) - VF(x,\kappa _{j}) = \frac{1}{2}m(2\mathrm{i}\kappa _{j})^{m - 1}F(x,\kappa _{j}). $$
(174)

Noting that when \(x \to + \infty \),

V ( 1 2 ( 2 i κ j ) m x 0 0 1 2 ( 2 i κ j ) m x ) ,
(175)
F ( x , κ j ) c j ( 0 1 ) e i κ j x , D t α F ( x , κ j ) ( D t α c j ) ( 0 1 ) e i κ j x + i ( D t α κ j ) x c j ( 0 1 ) e i κ j x ,
(176)

from Eq. (174) we then have

$$ \mathrm{i}D_{t}^{\alpha } \kappa _{j} - \frac{1}{2}(2\mathrm{i}\kappa _{j})^{m} = 0, \qquad D_{t}^{\alpha } c_{j} - \frac{1}{2}m(2\mathrm{i}\kappa _{j})^{m - 1}c_{j} = 0. $$
(177)

Similarly, the following fractional order equations can be obtained:

$$ \mathrm{i}D_{t}^{\alpha } \bar{\kappa }_{l} - \frac{1}{2}(2\mathrm{i}\bar{\kappa }_{l})^{m} = 0, \qquad D_{t}^{\alpha } \bar{c}_{l} + \frac{1}{2}m(2\mathrm{i}\bar{\kappa }_{l})^{m - 1} \bar{c}_{l} = 0. $$
(178)

We further consider the second case, i.e., real continuous spectral k. We select a solution \(G(x,k)\) of Eq. (25), then

$$ Q(x,k) = D_{t}^{\alpha } G(x,k) - NG(x,k) $$
(179)

solves Eq. (25). Therefore, there exists a pair of linearly independent fundamental solutions \(G(x,k)\) and \(\bar{G}(x,k)\) and two determined functions \(\omega (t,k)\) and \(\vartheta (t,k)\) such that

$$ G_{t}(x,k) - VG(x,k) = \omega (t,k)G(x,k) + \vartheta (t,k) \bar{G}(x,k). $$
(180)

In view of the asymptotic properties

D t α G ( x , k ) i ( D t α k ) x ( 1 0 ) e i k x , G ( x , k ) ( 1 0 ) e i k x , G ¯ ( x , k ) ( 0 1 ) e i k x ,
(181)

as \(x \to - \infty \), from Eqs. (180) and (181) we have

$$ \vartheta (t,k) = 0, \qquad \omega (t,k) = 0. $$
(182)

The substitution of the Jost relationship \(G(x,k) = \gamma (t,k)\bar{F}(x,k) + \delta (t,k)F(x,k)\) into Eq. (180) yields

$$ D_{t}^{\alpha } \bigl(\gamma (t,k)\bar{F}(x,k) + \delta (t,k)F(x,k)\bigr) - V\bigl(\gamma (t,k)\bar{F}(x,k) + \delta (t,k)F(x,k)\bigr) = 0. $$
(183)

Letting \(x \to + \infty \) and using

F(x,k) ( 0 1 ) e i k x , F ¯ (x,k) ( 1 0 ) e i k x ,
(184)

then Eq. (183) hints

$$ D_{t}^{\alpha } \gamma (t,k) = 0,\qquad D_{t}^{\alpha } \delta (t,k) = 0. $$
(185)

Similarly, we have

$$ D_{t}^{\alpha } \bar{\gamma } (t,k) = 0,\qquad D_{t}^{\alpha } \bar{\delta } (t,k) = 0. $$
(186)

Finally, from Eqs. (177), (178), (185), and (186) we can obtain Eqs. (158)–(161). We finish the proof. □

Exact solutions

Theorem 5

The tfnisAKNS hierarchy (3) has the following exact solutions:

$$ u = - 2K_{1}(x,x,t), \qquad v = \frac{K_{2x}(x,x,t)}{K_{1}(x,x,t)}, $$
(187)

where \(K(x,y,t) = (K_{1}(x,y,t),K_{2}(x,y,t))^{T}\) satisfies the Gel’fand–Levitan–Marchenko (GLM) integral equation

K ( x , y , t ) ( 1 0 ) F ¯ ( x + y , t ) + ( 0 1 ) x F ( z + x , t ) F ¯ ( z + y , t ) d z + x K ( x , s , t ) x F ( z + s , t ) F ¯ ( z + y , t ) d z d s = 0 ,
(188)

while

$$\begin{aligned}& F(x,t) = \frac{1}{2\pi } \int _{ - \infty }^{\infty } R(t,k)\mathrm{e}^{\mathrm{i}kx} \,\mathrm{d}k + \sum_{j = 1}^{N} c_{j}^{2} \mathrm{e}^{\mathrm{i}\kappa _{j}x}, \end{aligned}$$
(189)
$$\begin{aligned}& \bar{F}(x,t) = \frac{1}{2\pi } \int _{ - \infty }^{\infty } \bar{R}(t,k) \mathrm{e}^{ - \mathrm{i}kx}\,\mathrm{d}k - \sum_{j = 1}^{\bar{N}} \bar{c}_{j}^{2} \mathrm{e}^{ - \mathrm{i}\bar{\kappa }_{j}x} \end{aligned}$$
(190)

are determined by the scattering data (158)(160).

Proof

The proof is similar to the integer case [3], and the essential difference is that Theorem 5 has the different scattering data (158)–(160). We omit the proof here for simplification. □

N-fractal solutions

Theorem 6

Suppose that

$$\begin{aligned}& M(x,t) = I + P(x,t)P^{T}(x,t), \qquad P(x,t) = \biggl( \frac{c_{j}(t)\bar{c}_{m}(t)}{\kappa _{j} - \bar{\kappa }_{m}}\mathrm{e}^{\mathrm{i}(\kappa _{j} - \bar{\kappa }_{m})x} \biggr)_{\bar{N} \times N}, \end{aligned}$$
(191)
$$\begin{aligned}& \bar{z} = \bigl(\bar{c}_{1}(t)\mathrm{e}^{ - \mathrm{i}\bar{\kappa }_{1}x}, \bar{c}_{2}(t)\mathrm{e}^{ - \mathrm{i}\bar{\kappa }_{2}x}, \ldots , \bar{c}_{n}(t)\mathrm{e}^{ - \mathrm{i}\bar{\kappa }_{N}x}\bigr)^{T} \end{aligned}$$
(192)

are determined by the scattering data (158)(160) and \(M^{ - 1}(x,t)\) exists, while I is an \(\bar{N} \times \bar{N}\) unit matrix, then the tfnisAKNS hierarchy (3) has N-fractal solutions

$$\begin{aligned}& u = 2\operatorname{tr}\bigl(M^{ - 1}(x,t)\bar{z}(x,t) \bar{z}^{T}(x,t)\bigr), \end{aligned}$$
(193)
$$\begin{aligned}& v = - \frac{\frac{\partial }{\partial x}\operatorname{tr} ( M^{ - 1}(x,t)P(x,t)\frac{\partial }{\partial x}P^{T}(x,t) )}{\operatorname{tr}(M^{ - 1}(x,t)\bar{z}(x,t)\bar{z}^{T}(x,t))}, \end{aligned}$$
(194)

where \(\operatorname{tr}( \cdot )\) represents the trace of a given matrix.

Proof

Considering the reflectionless case, i.e., \(R(t,k) = \bar{R}(t,k) = 0\), from Eq. (189) we have

$$ \int _{x}^{\infty } F_{d}(t,s + z) \bar{F}_{d}(t,z + y)\,\mathrm{d}z = - \sum _{j = 1}^{N} \sum_{m = 1}^{\bar{N}} \frac{\mathrm{i}c_{j}^{2}(t)\bar{c}_{m}^{2}(t)}{\kappa _{j} - \bar{\kappa }_{m}}\mathrm{e}^{\kappa _{j}(x + s) - \mathrm{i}\bar{\kappa }_{m}(x + y)}. $$
(195)

Further suppose that the components of \(K(x,y,t) = (K_{1}(x,y,t),K_{2}(x,y,t))^{T}\) can be expressed by

$$ K_{1}(x,y,t) = \sum_{p = 1}^{\bar{N}} \bar{c}_{p}(t) g_{p}(x,t)\mathrm{e}^{ - \mathrm{i}\bar{\kappa }_{p}y}, \qquad K_{2}(x,y,t) = \sum_{p = 1}^{\bar{N}} \bar{c}_{p}(t) h_{p}(x,t)\mathrm{e}^{ - \mathrm{i}\bar{\kappa }_{p}y}, $$
(196)

then we can convert Eq. (188) into a set of algebraic equations

$$\begin{aligned}& g_{m}(x,t) + \bar{c}_{m}(t)\mathrm{e}^{ - \mathrm{i}\bar{\kappa }_{m}x} + \sum_{j = 1}^{N} \sum _{p = 1}^{\bar{N}} \frac{c_{j}^{2}(t)\bar{c}_{m}(t)\bar{c}_{p}(t)}{(\kappa _{j} - \bar{\kappa }_{m})(\kappa _{j} - \bar{\kappa }_{p})} \mathrm{e}^{\mathrm{i}(2\kappa _{j} - \bar{\kappa }_{m} - \bar{\kappa }_{p})x}g_{p}(x,t) = 0, \end{aligned}$$
(197)
$$\begin{aligned}& h_{m}(x,t) - \sum_{j = 1}^{N} \frac{c_{j}^{2}(t)\bar{c}_{m}(t)}{\kappa _{j} - \bar{\kappa }_{m}} \mathrm{e}^{\mathrm{i}(2\kappa _{j} - \bar{\kappa }_{m})x} \\& \quad {} + \sum_{j = 1}^{N} \sum _{p = 1}^{\bar{N}} \frac{c_{j}^{2}(t)\bar{c}_{m}(t)\bar{c}_{p}(t)}{(\kappa _{j} - \bar{\kappa }_{m})(\kappa _{j} - \bar{\kappa }_{p})} \mathrm{e}^{\mathrm{i}(2\kappa _{j} - \bar{\kappa }_{m} - \bar{\kappa }_{p})x}h_{p}(x,t) = 0 \end{aligned}$$
(198)

for \(m = 1,2, \ldots ,\bar{N}\). Introducing vectors

$$\begin{aligned}& g(x,t) = \bigl(g_{1}(x,t),g_{2}(x,t), \ldots ,g_{\bar{N}}(x,t)\bigr)^{T}, \end{aligned}$$
(199)
$$\begin{aligned}& h(x,t) = \bigl(h_{1}(x,t),h_{2}(x,t), \ldots ,h_{\bar{N}}(x,t)\bigr)^{T}, \end{aligned}$$
(200)
$$\begin{aligned}& z = \bigl(c_{1}(t)\mathrm{e}^{\mathrm{i}\kappa _{1}x},c_{2}(t) \mathrm{e}^{\mathrm{i}\kappa _{2}x}, \ldots ,c_{n}(t)\mathrm{e}^{\mathrm{i}\kappa _{N}x} \bigr)^{T}, \end{aligned}$$
(201)

we can rewrite the above set of algebraic equations as

$$ M(x,t)g(x,t) = - \bar{N}(x,t),\qquad M(t,x)h(t,x) = \mathrm{i}P(t,x)z(t,x), $$
(202)

namely

$$ g(x,t) = - M^{ - 1}(x,t)z(x,t), \qquad h(x,t) = \mathrm{i}M^{ - 1}(x,t)P(x,t)z(x,t). $$
(203)

Substituting Eq. (203) into Eq. (196) yields

$$\begin{aligned}& K_{1}(x,y,t) = - \operatorname{tr}\bigl(M^{ - 1}(x,t) \bar{z}(x,t)\bar{z}^{T}(y,t)\bigr), \end{aligned}$$
(204)
$$\begin{aligned}& K_{2}(x,y,t) = \mathrm{i}\operatorname{tr}\bigl(M^{ - 1}(x,t)P(x,t)z(x,t) \bar{z}^{T}(y,t)\bigr). \end{aligned}$$
(205)

Then we reach Eqs. (193) and (194) by means of Eqs. (187), (204), and (205). The proof is complete. □

Particularly, when \(N = \bar{N} = 1\), Eqs. (193) and (194) give one-fractal solutions

$$\begin{aligned}& u = \frac{2\bar{c}_{1}^{2}(0)\mathrm{e}^{ - 2\mathrm{i}\bar{\kappa }_{1}x}E_{\alpha } ( - mI_{t,0}^{\alpha } (2\mathrm{i}\bar{\kappa }_{j})^{m - 1})}{1 + \frac{c_{1}^{2}(0)\bar{c}_{1}^{2}(0)}{(\kappa _{1} - \bar{\kappa }_{1})^{2}}\mathrm{e}^{2\mathrm{i}(\kappa _{1} - \bar{\kappa }_{1})x}E_{\alpha } (mI_{t,0}^{\alpha } ((2\mathrm{i}\kappa _{j})^{m - 1} - (2\mathrm{i}\bar{\kappa }_{j})^{m - 1}))}, \end{aligned}$$
(206)
$$\begin{aligned}& v = \frac{2c_{1}^{2}(0)\mathrm{e}^{2\mathrm{i}\kappa _{1}x}E_{\alpha } (mI_{t,0}^{\alpha } (2\mathrm{i}\kappa _{j})^{m - 1})}{1 + \frac{c_{1}^{2}(0)\bar{c}_{1}^{2}(0)}{(\kappa _{1} - \bar{\kappa }_{1})^{2}}\mathrm{e}^{2\mathrm{i}(\kappa _{1} - \bar{\kappa }_{1})x}E_{\alpha } (mI_{t,0}^{\alpha } ((2\mathrm{i}\kappa _{j})^{m - 1} - (2\mathrm{i}\bar{\kappa }_{j})^{m - 1}))}, \end{aligned}$$
(207)

where \(\kappa _{1}\) and \(\bar{\kappa }_{1}\) are determined by

$$ D_{t}^{\alpha } \kappa _{1} = - \frac{\mathrm{i}}{2}(2\mathrm{i}\kappa _{1})^{m}, \qquad D_{t}^{\alpha } \bar{\kappa }_{1} = - \frac{\mathrm{i}}{2}(2\mathrm{i}\bar{\kappa }_{1})^{m}. $$
(208)

To be more specific, given \(m = 2\), the second member equations of the tfnisAKNS hierarchy (3) read

D t α ( u v ) = ( 2 u x x u x x + 2 u 1 ( u v ) + 2 x u 2 v 2 v x + x v x x 2 u 1 ( u v ) 2 x u v 2 ) ,
(209)

which have one-fractal solutions

$$\begin{aligned}& u = \frac{2\bar{c}_{1}^{2}(0)\mathrm{e}^{ - 2\mathrm{i}\bar{\kappa }_{1}x}E_{\alpha } ( - \frac{\mathrm{i}}{2}\ln [\Gamma (1 + \alpha ) - 2\mathrm{i}\bar{\kappa }_{1}(0)t^{\alpha } ] )}{1 + \frac{c_{1}^{2}(0)\bar{c}_{1}^{2}(0)}{(\kappa _{1} - \bar{\kappa }_{1})^{2}}\mathrm{e}^{2\mathrm{i}(\kappa _{1} - \bar{\kappa }_{1})x}E_{\alpha } ( \frac{\mathrm{i}}{2}\ln \frac{[\Gamma (1 + \alpha ) - 2\mathrm{i}\kappa _{1}(0)t^{\alpha } ]}{[\Gamma (1 + \alpha ) - 2\mathrm{i}\bar{\kappa }_{1}(0)t^{\alpha } ]} )}, \end{aligned}$$
(210)
$$\begin{aligned}& v = \frac{2c_{1}^{2}(0)\mathrm{e}^{2\mathrm{i}\kappa _{1}x}E_{\alpha } ( \frac{\mathrm{i}}{2}\ln [\Gamma (1 + \alpha ) - 2\mathrm{i}\kappa _{1}(0)t^{\alpha } ] )}{1 + \frac{c_{1}^{2}(0)\bar{c}_{1}^{2}(0)}{(\kappa _{1} - \bar{\kappa }_{1})^{2}}\mathrm{e}^{2\mathrm{i}(\kappa _{1} - \bar{\kappa }_{1})x}E_{\alpha } ( \frac{\mathrm{i}}{2}\ln \frac{[\Gamma (1 + \alpha ) - 2\mathrm{i}\kappa _{1}(0)t^{\alpha } ]}{[\Gamma (1 + \alpha ) - 2\mathrm{i}\bar{\kappa }_{1}(0)t^{\alpha } ]} )}, \end{aligned}$$
(211)

where

$$ \kappa _{1} = \frac{\mathrm{i}\kappa _{1}(0)\Gamma (1 + \alpha )}{2\kappa _{1}(0)t^{\alpha } + \mathrm{i}\Gamma (1 + \alpha )},\qquad \bar{\kappa }_{1} = \frac{\mathrm{i}\bar{\kappa }_{1}(0)\Gamma (1 + \alpha )}{2\bar{\kappa }_{1}(0)t^{\alpha } + \mathrm{i}\Gamma (1 + \alpha )}. $$
(212)

Figures 1012 show the spatial structures and profiles of one-fractal solutions (206) and (207) restricted to the Cantor set by selecting \(\kappa _{1}(0) = 1\), \(\bar{\kappa }_{1}(0) = 1.8\), \(c_{1}(0) = - 1\), \(\bar{c}_{1}(0) = 9\), and \(\alpha = \ln 2/\ln 3\).

Figure 10
figure10

Space time structures of one-fractal solutions (206) and (207) with \(\alpha = \ln 2/\ln 3\).

Figure 11
figure11

Profiles along the t axis of one-fractal solutions (206) and (207) with \(\alpha = \ln 2/\ln 3\).

Figure 12
figure12

Profiles along the t axis of one-fractal solutions (206) and (207) with \(\alpha = \ln 2/\ln 3\).

Conclusions

Though there are many results for both the important AKNS linear spectral and the nonlinear AKNS partial differential systems, the local fractional versions of them and N-fractal solutions of the AKNS systems are still blank. HBM and IST are two very important analytical methods for solving soliton equations in the field of nonlinear sciences, which are extended to the stfisAKNS and tfnisAKNS hierarchies by introducing the fractional order bilinear operators [27] and fractional order scattering data (see [24] for our preliminary work). Compared with the existing literature, the work of this paper is novel. For the comparisons, we would like to point out that when \(\alpha = 1\), both the derived stfisAKNS and tfnisAKNS hierarchies and the obtained N-fractal solutions become the known hierarchies [3, 7] and their corresponding exact solutions [3, 5, 7]. All the derived fractional AKNS hierarchies and the obtained N-fractal solutions benefit from the local fractional order partial derivatives with graceful properties [16] and the existing results [3] about the spectral problem (25) which are valid for the tfnisAKNS hierarchy (3).

As shown in Figs. 112, the obtained N-fractal solutions restricted to the Cantor set are continuous everywhere but nondifferentiable. This is essentially different from the existing classical soliton solutions. Though the obtained one-fractal solutions (78), (206), and (207) and two-fractal solutions (91), (147), and (148) are shown by figures, the features of one-fractal solutions and two-fractal solutions and the differences between them could not be concluded from them. On the one hand, these fractal solutions are limited to the Cantor set; on the other hand, they may be related to the properties of the fractional AKNS hierarchies and the parameters selected for plotting the figures.

This paper could extend IST to the tfnisAKNS hierarchy (3) but not the space-time fractional one. It is because of the fact that if the space derivative of the spectral problem (25) is fractional order, then the corresponding analysis is complex. At the same time, extending HBM to the tfnisAKNS hierarchy (3) is worth exploring. Integrable couplings [2832] include the original integrable systems as sub-systems. Extending HBM and IST to the integrable couplings and their fractional order generalizations is worthy of study. At the same time, the practical application of fractional calculus in medicine, for example the conformable fractional mathematical model [33], is worth further research and exploration. With the developments [3439] of fractional calculus, more and more theories and methods will be extended to fractional order nonlinear differential equations. Among them, two efficient methods called q-homotopy analysis transform method and fractional natural decomposition method used by Gao et al. [40] for numerical solution of fractional Benney–Lin equation deserve attention.

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Abbreviations

AKNS:

Ablowitz–Kaup–Newell–Segur

HBM:

Hirota bilinear method

IST:

Inverse scattering transform

isAKNS:

isospectral AKNS

nisAKNS:

non-isospectral AKNS

stfisAKNS:

space-time fractional order isospectral AKNS

tfnisAKNS:

time-fractional order non-isospectral AKNS

flnNLS:

fractional order local and nonlocal nonlinear Schrödinger

frtNLS:

fractional order reverse-t NLS

fmKdV:

fractional order modified KdV

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Acknowledgements

The authors sincerely express their thanks to the anonymous referees for their valuable comments. Dr. Bo Xu and Professor Sheng Zhang are grateful to Professor Xiao-Jun Yang for his helpful suggestions during their short visit at China University of Mining and Technology.

Funding

This research was supported by the National Science Foundation of China (11971475, 11547005), the Liaoning BaiQianWan Talents Program of China (2019 Year), the Natural Science Foundation of Education Department of Liaoning Province of China (LJ2020002), and the Natural Science Foundation of Xinjiang Autonomous Region of China (2020D01B01).

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Correspondence to Yufeng Zhang or Sheng Zhang.

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Xu, B., Zhang, Y. & Zhang, S. Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions. Adv Differ Equ 2021, 223 (2021). https://doi.org/10.1186/s13662-021-03374-0

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Keywords

  • 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
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