Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

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 nonisospectral 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 with the fractional order operator and the tfnisAKNS hierarchy with the following operator: and then solve the fractional order AKNS hierarchies (1) and (3) by extending HBM [15] and IST [1], respectively. Here, D α x and I α x,a represent the local fractional order partial derivative operator [16] D α x φ(x, t) = (1 + α) lim and the local fractional integral operator [16] I α x,a ϕ(x, t) =

(+ α)
x a ϕ(ω, t)(dω) α (0 < α ≤ 1) (6) for any nondifferentiable functions φ(x, t) and ϕ(x, t) defined on a fractal set , 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 [18][19][20][21][22][23][24][25]. 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
where i α is the necessary formal imaginary number unit [16] to connect the relationships between Mittag-Leffler functions and trigonometric functions defined in a fractal set . Then the stfisAKNS hierarchy (1) can be derived from the fractional order zero curvature equation, i.e., the fractional order compatibility condition of the following fractional order linear spectral problem: and its associated time-fractional order evolution equation 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, i α 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: On the other hand, the local fractional derivative of Eq. (10) with respect to x gives 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 α [25]. Thus, Eqs. (9)-(12) lead to 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 In view of Eq. (7), we rewrite Eqs. (14) and (15) as Further supposing and substituting it into Eq. (16), then comparing the coefficients of the same powers of 2i α k in Eq. (17), we have 2i α k m : Equations (19) and (20) give 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 and suppose then the time-fractional order zero curvature equation i.e., the time-fractional order compatibility condition of the following linear spectral problem: 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 which can be written as and substituting it into Eq. (28), then comparing the coefficients of the same powers of 2ik in Eq. (28) yield Then we have 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: by introducing the variables t 1 = x, t 2 , t 3 , . . . 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 the representative of which is the flnNLS equation The second one is the frtNLS hierarchy one representative of which is the frtNLS equation And the last one is the fmKdV hierarchy one representative of which is the fmKdV equation

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

Fractional bilinear forms
then the stfiAKNS hierarchy (34) can be bilinearized as where H α t m+1 , H α t m , and H α x are local fractional versions [27] of the Hirota bilinear operator [15].
Proof We write the component forms of Eq. (34) as In view of Eq. (44), we have Using Eqs. (49)-(51), we convert the left-and right-hand sides of Eq. (47) into With the help of Eqs. (52) and (53) [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 2α x f ·f +2gh = 0, and the other weakens this constraint by letting H 2α x f · f + 2gh = a 2 f 2 , here a is a constant.

First reduced bilinear forms and N-fractal solutions
We set then the fractional order bilinear forms (45) and (46) become For the fractional one-fractal solutions, we assume that 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: D 2α 2D 2α 2D 2α . . . and so forth. Letting from Eqs. (60) and (61) we have In view of Eq. (62), we suppose that where θ α 13 is a constant determined later and the Mittag-Leffler function is defined on a fractal set [16]: Then Eqs. (69), (70), and (73) hint Substituting Eqs. (69)-(74) into Eqs. (63)-(65) and setting we can see that Eq. (61) and the equations behind all hold. In this case, we write and hence we obtain the fractional one-fractal solutions of the stfisAKNS hierarchy (34): . (78) The one-fractal solutions (78) restricted to the Cantor set are shown in Figs. 1-3, where the parameters k 1 = i and p 1 = 0.5.
Similarly, we can obtain three-fractal and then determine the uniform formulas (96) for the N -fractal solutions of the stfisAKNS hierarchy (34) by

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).
is a solution of Eq. (25). So, P(x, k) can be expressed by F(x, k) andF(x, k), which is another solution of Eq. (25) but independent of F(x, k), i.e., there exist two functions μ(t, k) and τ (t, k) such that We next consider the first case, i.e., discrete spectral k = κ j (Im κ j > 0). It is easy to see that F(x, κ j ) decays exponentially, whileF(x, κ j ) increases as x → +∞. Thus, we have τ (t, k) = 0 and simplify Eq. (162) as The left-multiplication on Eq. (163) by the inner product (F 2 (x, κ j ), F 1 (x, κ j )) yields Since F(x, κ j ) is a normalized eigenfunction and we can rewrite Eq. (164) as Making use of Eq. (25), we have The integration of Eq. (167) with respect to x from -∞ to +∞ gives At the same time, we rewrite Eq. (29) as Then with the help of the conjugation operatorL [3] and the results we gain from Eq. (167) Thus, Eq. (163) becomes Noting that when x → +∞, from Eq. (174) we then have Similarly, the following fractional order equations can be obtained: We further consider the second case, i.e., real continuous spectral k. We select a solution G(x, k) of Eq. (25), then solves Eq. (25). Therefore, there exists a pair of linearly independent fundamental solutions G(x, k) andḠ(x, k) and two determined functions ω(t, k) and ϑ(t, k) such that In view of the asymptotic properties as x → -∞, from Eqs. (180) and (181) we have The substitution of the Jost relationship G(x, k) = γ (t, k)F(x, k) + δ(t, k)F(x, k) into Eq. (180) yields Letting x → +∞ and using then Eq. (183) hints Similarly, we have
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.
Proof Considering the reflectionless case, i.e., R(t, k) =R(t, k) = 0, from Eq. (189) we have 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 h(x, t) = h 1 (x, t), h 2 (x, t), . . . , hN (x, t) T ,   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 α = 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).