Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation

Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the N-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger (NLS)-type equation, are obtained by extending the Riemann–Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales.

The LFD is also called fractal derivative, which was first proposed by Kolwankar and Gangal [20] in 1996. Specifically, if the limit exists for a given function f (x) : [0, 1] → R , then the LFD of order q(0 < q < 1) at the point x = x 0 denoted by D q f (x 0 ) exists. For the function u(x) defined on a fractal set, Yang et al. [21] extended Kolwankar and Gangal's LFD as below: where Γ(·) is the Gamma function. The LFD (2) has many applications and developments [5,15,21,[27][28][29][30] due to its graceful properties (see the monograph [5] and the references therein). In 2017, Yang et al. [28] presented a family of special functions defined on the fractal sets and obtained non-differentiable exact solutions of some nonlinear local fractional ordinary differential equations.
In nonlinear mathematical physics, the inverse scattering transform (IST) [31,32] is a systematic method for solving boundary value problems of nonlinear partial differential equations (PDEs). One of the modern developments of IST [31,32] is the Riemann-Hilbert (RH) approach [33]. In comparison, the RH approach [33] has more advantages than the IST method [31,32]. For example, the RH approach [33] can give insights into the asymptotic behavior of the obtained solution when the time variable t goes to infinity. For recent research on the RH approach [33], readers can refer to [34][35][36][37][38][39][40][41][42][43][44][45]. As pointed out by Liu and Kengne [46], in media with weak nonlinearity, the celebrated NLS equation describing the wave-packet behavior has wide applications in many fields of applied science and engineering, and this model can be used to represent not only the evolution of light waves in nonlinear optical fibers but also the wave-packet envelope of ocean waves in the deep sea, dynamical systems in biology, and option prices in economics. In this article, we shall extend the RH approach [33] to the following local time-fractional focusing NLS-type equation: with u(x, 0) = u(x) as the initial condition. Here i 2 = −1 is the imaginary unit in the space variable x, D α t u is the LFD with respect to t in a fractal set, u xx /2 is the dispersion term, |u| 2 u is the self-focusing nonlinear term, and it is assumed that u and its partial derivatives with respect to x and partial derivatives of fractional order with respect to t quickly approach zeros when |x|→ ∞ . When α = 1, Equation (3) degenerates into the classical focusing NLS equation [32]. It is interesting to extend the NLS equation to fractional-order forms. In 2014, Fujioka et al. [10] used an extended NLS-type equation with fractional-and integer-order terms to describe fractional optical solitons. In 2020, Ain et al., [47] presented a thorough study of a time-dependent NLS-type equation with a time-fractional derivative by using the fractional time complex transformation and the homotopy perturbation method. In 2021, Xu et al. [14] first proposed fractional rogue waves and constructed one-and two-order fractional rogue wave solutions of a conformable space-time fractional NLS-type equation. It should be noted that Equation (3) can be reduced from the local time-fractional isospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy [15]. In 2021, Xu et al. [15] constructed N-fractal solutions of AKNS hierarchy by the IST method. Recently, Ghanbari [29] obtained exact fractional solutions to a generalized NLS-type equation with an LFD defined on Cantor sets via the generalized version of the exponential rational function method. Though the classical focusing NLS equation [32] and its generalized forms [42][43][44][45] were solved by the RH approach [33], to our knowledge, the local time-fractional NLS-type Equation (3) has not been studied by such a method. In the literature, there is no RH approach [33] for fractional-order models except our work on the associated IST for the local time-fractional KdV equation [48] and the local time-fractional AKNS hierarchy [15]. However, the RH approach [33] is the development and improvement of the IST [31,32]. This enables the RH approach [33] to obtain some meaningful results that the IST [31,32] cannot get, for example, the long-time asymptotic solutions mentioned above. In this paper, referring to the idea of the RH approach [33], we shall transform the solution of Equation (3) into the solution of a related RH problem, and then derive the long-time asymptotic solution and N-fractal-soliton solution of Equation (3) by determining the time-dependence of scattering data contained in the jump matrix of the RH problem. At the same time, we employ the Lax pair of Equation (3) to construct infinitely many conservation laws of Equation (3).
The second to sixth sections of this paper are organized as follows. In Section 2, we establish the relation between the solution of Equation (3) and a related RH problem. In Section 3, we solve the RH problem and derive the time evolution of scattering data in the related RH problem. In Section 4, we obtain the long-time asymptotic solution and N-fractal-soliton solution of Equation (3). In Section 5, we focus on the construction of infinitely many conservation laws of Equation (3). In Section 6, we conclude this paper.

Lax Pair and the Related RH Problem
It is easy to see that the local time-fractional focusing NLS-type Equation (3) has the following Lax pair in matrix form: where the isospectral parameter λ is complex, φ = φ(x, t, λ) is the eigenfunction, and where * denotes the complex conjugate. The Jost solutions with the following asymptotic property of the Lax pair in Equations (4) and (5) can be easily obtained: where the Mittag-Leffler function E α (t α ) is defined on a fractal set [5]: Taking the following transformation: We can convert the Lax pair in Equations (4) and (5) into: Obviously, the eigenfunction ϕ in Equations (10) and (11) has the following asymptotic behavior: where I is the second-order identity matrix. Note that solutions exist for Equation (4) [33]: Symmetry 2021, 13, 1593 where S(λ) is the scattering matrix: Since detϕ ± = 1 and detS(λ) = 1 [33], from Equation (15), we can easily calculate the inverse matrix S −1 (λ) = (ŝ ij (λ)) 2×2 : Because [33], here H denotes Hermitian conjugate, Equation (15) hints at the symmetry of S H (λ * ) = S −1 (λ). Thus, we have We T , and then construct where (ϕ ± ) s and (ϕ −1 ± ) s represent the s-th row vector and the s-th column vector of ϕ ± , respectively, and Λ 1 and Λ 2 are the diagonal matrices: Then we can verify that K + and K − satisfy Equation (10) and its adjoint equation: Expanding K ± in the Taylor series [33], and substituting K + and K − into Equations (19) and (20), then comparing the same power . We therefore build the following connection between solution u of Equation (3) and where (λK ± 1 ) 12 denotes the element at the intersection of the first row and the second column of the matrix λK ± 1 . With the above preparations, we can see from [33] that the following matrix RH problem can be established by using the K ± constructed by Equations (19) and (20): Symmetry 2021, 13, 1593 where the corresponding jump matrix reads:

Solutions of the Related RH Problem and Time-Dependence of Scattering Data
Whether the RH problem (27) is regular or not, it is always solvable [33]. In view of Equations (15), (19) and (20) together with the symmetry S H (λ * ) = S −1 (λ), we have For the regular case, namely detK ± (λ) = 0, with the help of the Plemelj formula [49], we can verify that a unique solution exists for the RH problem (27) [33]: When detK ± (λ) = 0, the symmetry S H (λ * ) = S −1 (λ) tells us that detK + (λ) = 0 and detK − (λ) = 0 has the same number of conjugate zeros [33], which are respectively denoted as λ j ∈ C + and λ = λ * j ∈ C − for j = 1, 2, · · · , N. In this irregular case, we construct the following two systems of homogeneous linear equations: where w j (λ j ) and w j (λ j ) are the row vector solution and column vector solution of Equations (32) and (33), respectively. Taking the Hermitian conjugate of Equation (32) and using the symmetry (K + ) It is easy to see from Equations (33) and (34) that w j (λ j ) = w H j (λ j ). By using Zakharov and Shabat's theorem [50], we can transform the irregular RH problem with detK ± (λ) = 0 into a regular one, and then indirectly prove that there exists a unique solution to the regular RH problem. Thus, we can determine the K + 1 in Equation (24) as following where The time evolution rules of the scattering data associated with Equation (3) are given by the following Theorem 1.

Theorem 1.
Suppose that u(x, t) develops according to the local time-fractional focusing NLS-type Equation (3), then the scattering data determined by the irregular RH problem (27): have the following time evolution rules: Proof of Theorem 1. We rewrite Equation (15) as Taking the LFD of the left-hand side of Equation (46) with respect to t and using Equation (15), we can obtain This shows that the left part of Equation (46) solves Equation (11). Hence, the right part of Equation (46) also solves Equation (11). By substituting the right part of Equation (46) into Equation (11), let x → +∞ , and then considering the boundary condition in Equation (12), we get Since ϕ + e −iλσ 3 namely ϕ − e −iλσ 3 S −1 (t, λ) satisfies Equation (11), we substitute ϕ − e −iλσ 3 S −1 (λ) into Equation (11), then set x → −∞ , and consider the boundary condition in Equation (12). The following equation is derived: Substituting the scattering matrix S(t, λ) and its inverse matric S −1 (t, λ) in Equations (16) and (17) into Equations (48) and (49), respectively, and then comparing the elements of the obtained matrices, we gain We easily see from Equations (29) and (52) that λ j (t) and λ j (t) are all independent from t and then Equation (43) holds. Directly solving Equations (50) and (51) yields Equations (41) and (42).

Long-Time Asymptotic Solution and N-Fractal-Soliton Solution
In view of Equations (41) and (42), we can determine the time-dependence of the Jump matrix: Generally, whenĜ(x, t, λ) = 0 it is difficult or impossible to work out the integral in Equation (35). Even so, we can consider the asymptotic property of the corresponding non-explicit solution (25) when the time variable t tends to infinity. For example, we set λ = λt α/2 (t α > 0), the integral in Equation (35) tends to zero at a rate of t −α/2 . In this case, we can obtain a long-time asymptotic solution of the local time-fractional focusing NLS-type Equation (3): where M and w k are determined by Equations (39) and (44), while w k can also be determined by using the symmetry w j = w H j . We next consider the N-fractal-soliton solution in the reflectionless case. For this purpose, we letŝ 12 (0, λ) = 0 and s 21 (0, λ) = 0 so thatĜ(x, t, λ) = 0. Then Equation (35) is simplified as In order to further determine M −1 in Equation (60), we select w j (0, 0, λ j ) = (c j , 1). Here, c j is a complex number. We then obtain from Equation (44)  and from this further gain Finally, from Equations (25) and (60)-(62), we obtain the N-fractal-soliton solution of the local time-fractional focusing NLS-type Equation (3): where Solution (63) can also be expressed by the following simple from where Letting N = 1, λ 1 = ξ + iη (ξ, η > 0 ∈ R), and c 1 = e −ηδ 0 +iρ 0 (δ 0 , ρ 0 ∈ R), from Equations (64)-(66), we can obtain the one-fractal-soliton solution of the local time-fractional focusing NLS-type Equation (3): When the time variable t is limited to the Cantor set, we show in Figures 1-5 the one-fractal-soliton solution (67) by selecting the parameters ξ = 0.5, η = 1, k 0 = 1.5, and δ 0 = 1. Figure 1 shows the spatio-temporal structure of solution (67), in which the temporal variation has fractal characteristics while the spatial variation possesses soliton structure. Figures 2-4 show the time-varying law of the continuous but non-differentiable amplitude with fractal characteristics of solution (67) at three different positions x = −1, x = 1.5 and x = 2. Figure 5 shows the dynamic evolution of the continuous and differentiable amplitude with soliton characteristics of solution (67) propagating along the x-axis at two different times t = 0.1 and t = 0.9.
Symmetry 2021, 13, 1593 9 of 13 0 temporal variation has fractal characteristics while the spatial variation possesses soliton structure. Figures 2-4 show the time-varying law of the continuous but non-differentiable amplitude with fractal characteristics of solution (67) at three different positions 1 x = − , 1.5 x = and 2 x = . Figure 5 shows the dynamic evolution of the continuous and differentiable amplitude with soliton characteristics of solution (67) propagating along the x-axis at two different times 0.1 t = and 0.9 t = . Figure 1. Spatio-temporal structure of one-fractal-soliton solution (67).

Infinitely Many Conservation Laws
The conservation laws of Equation (3) are given by the following Theorem 2. (3) has infinitely many conservation laws determined by:
Proof of Theorem 2. According to the steps in [33], we first set Then, from Equations (4) and (5), we have

Infinitely Many Conservation Laws
The conservation laws of Equation (3) are given by the following Theorem 2.

Theorem 2.
The local time-fractional focusing NLS-type Equation (3) has infinitely many conservation laws determined by: and conserved quantities: where Proof of Theorem 2. According to the steps in [33], we first set φ = ( f 1 , f 2 ) T and g = f 1 / f 2 . Then, from Equations (4) and (5), we have Differentiating Equations (72) and (73) with respect to t (local fractional order) and x, respectively, and using [33], we gain Secondly, we expand g as a power series of λ −1 : and substitute Equation (75) into Equation (74). Then comparing the same power of λ −1 yields infinitely many conservation laws determined by Equation (68) of the local time-fractional focusing NLS-type Equation (3), including the density u * g n and the flux u * g n+1 − u * x g n . Thirdly, in view of Equation (68) we can obtain infinitely many conserved quantities in Equation (69). From Equation (4) we easily verify that g satisfies the Riccati equation: Substituting Equation (75) into Equation (76), we finally obtain g n in Equations (70) and (71).

Conclusions
In summary, the Riemann-Hilbert approach [33] was extended for constructing Nfractal-soliton solution and long-time asymptotic solution of the local time-fractional NLStype Equation (3). Specifically, solutions (25) or (26) of Equation (3) are first transformed into the solution of the related RH problem (27); then the long-time asymptotic solution (59) and the N-fractal-soliton solution (63) or its simplified form (65) are obtained by determining the time-dependence of scattering data (40) contained in the jump matrix (58) of the RH problem (27). Besides, infinitely many conservation laws of Equation (3), determined by Equation (68), one end of which is the partial derivative of the local fractional order in time, and the other end is the partial derivative of integral order in space, are also obtained. In order to simulate the obtained one-fractal-soliton solution (67), we constrained the time variable to the Cantor set and then illustrated solution (67) in figures which show that solution (67) has different geometric characteristics on two scales. One scale with fractal dimension ln 2/ ln 3 ≈ 0.631 makes the time variable of solution (67) continuous but non-differentiable, generating a fractal-soliton. The other scale with integer dimension 1 keeps the space variable, possessing the continuity and differentiability characteristics of the classical soliton. These two different scales are the essence leading to the fractal-soliton feature. In view of the existing literature, the novelty and significance of the results of this paper is that by taking the time-fractional NLS-type Equation (3) as an example, the RH approach [33] was extended to the fractional-order models for the first time and meaningful results were obtained. This will be a useful supplement for expanding and enriching the research content of the RH approach [33] in the field of fractals. As a comparison between the obtained results of this paper and those in the literature, we are willing to draw a conclusion that because the time-variable derivative of Equation (3) is of fractional order and the space-variable derivative is of integer order, the corresponding results related to argument t are novel, such as (5), (7), (9), (11), (15), (41), (42), (45), (47) (74), while the results about argument x are known [33]. Some necessary known results [33] are listed in this paper because of the integrity and comparison of the RH approach for Equation (3). In recent years, the RH approach [33] has received further developments like those in [34][35][36][37][38][39][40][41][42][43][44][45]. Extending the RH approach [33] and its recent developments [34][35][36][37][38][39][40][41][42][43][44][45] to the space-time fractional models is worthy of study. In addition, it is feasible to extend the research on stability analysis [51] and nonlinear instability [52] to some local fractional models.

Data Availability Statement:
The data in the manuscript are available from the corresponding author upon request.