Integrable Riesz Fractional-Order Generalized NLS Equation with Variable Coefficients: Inverse Scattering Transform and Analytical Solutions

Abstract

Significant new progress has been made in nonlinear integrable systems with Riesz fractional-order derivative, and it is impressive that such nonlocal fractional-order integrable systems exhibit inverse scattering integrability. The focus of this article is on extending this progress to nonlocal fractional-order Schrödinger-type equations with variable coefficients. Specifically, based on the analysis of anomalous dispersion relation (ADR), a novel variable-coefficient Riesz fractional-order generalized NLS (vcRfgNLS) equation is derived. By utilizing the relevant matrix spectral problems (MSPs), the vcRfgNLS equation is solved through the inverse scattering transform (IST), and analytical solutions including n-soliton solution as a special case are obtained. In addition, an explicit form of the vcRfgNLS equation depending on the completeness of squared eigenfunctions (SEFs) is presented. In particular, the 1-soliton solution and 2-soliton solution are taken as examples to simulate their spatial structures and analyze their structural properties by selecting different variable coefficients and fractional orders. It turns out that both the variable coefficients and fractional order can influence the velocity of soliton propagation, but there is no energy dissipation throughout the entire motion process. Such soliton solutions may not only have important value for studying the super-dispersion transport of nonlinear waves in non-uniform media, but also for realizing a new generation of ultra-high-speed optical communication engineering.

1. Introduction

Nonlinear problems often involve multiple disciplines, including mathematics, physics, chemistry, biology, engineering, computer science, etc. With the significant development of mathematical physics, more and more researchers have begun to pay attention to nonlinear integrable systems, which in turn has promoted the continuous deepening of nonlinear science. The classical nonlinear Schrödinger (NLS) equation, as a celebrated nonlinear integrable model, plays a crucial role in describing the evolution of wave functions in nonlinear media and has been widely applied in physics [1,2,3,4]. It is worth mentioning that a solitary wave solution of the NLS equation with particle-like properties [5], also known as soliton solution, is expected to become an important carrier for realizing the next generation of ultra-high-speed optical communication engineering.

Fractional calculus [6] originated from the extension of traditional calculus and has independently developed into an effective mathematical tool for simulating many physical processes existing in multi-scale media [7,8], demonstrating its superiority in dealing with non-Gaussian statistics or power-law behavior [9,10,11]. This indicates that studying differential equations with fractional derivatives has practical significance. As a result, various fractional-order ordinary/partial differential equations have emerged. The field of solitons has also been affected, and different forms of fractional-order NLS equations have been explored [12,13,14,15]. However, it is worth noting that based on IST solvability, all these fractional-order NLS (fNLS) equations are non-integrable.

In 2021, significant progress was made in fractional-order inverse scattering integrable systems. That is, Ablowitz et al. [16,17] proposed several new nonlinear integrable models that not only have Riesz fractional-order derivative [18,19] but also possess IST solvability, such as the fractional-order KdV (fKdV) equation and fNLS equation, and obtained soliton solutions by solving the transformed integral equations (TIEs) [2]. Generally speaking, Ablowitz et al.’s method [16,17] involves three key elements, namely IST integrability, ADR, and CSEs. The TIEs and Riemann–Hilbert problem (RHP) are two important alternative approaches for solving nonlinear integral systems through IST. Afterwards, based on the feasible method [16,17], different fractional-order equations related to the NLS type equation were reported. In 2022, Weng et al. [20] studied a higher order fNLS equation and obtained its multi-soliton solution based on the IST with RHP. In 2023, Mou et al. [21] derived the integrable n-component coupled fNLS equation and derived its soliton solutions by using the IST with RHP as well. In addition, within the framework of IST theory, Ablowitz et al. [22] presented the first integrable discrete fNLS equation and obtained its soliton solutions.

In the real world, there are many complex phenomena and evolution laws that can be described by mathematical physics equations. Compared with the case of constant-coefficient equations, the variable-coefficient models can sometimes describe these phenomena and evolution laws more accurately, especially in the physical background of inhomogeneous media. However, solving nonlinear models with variable coefficients is more challenging than solving the ones with constant coefficients. This is because the influence of the coefficient functions on the existing analytical methods in the solution process needs to be taken into account. All types of fNLS equations mentioned above are of constant coefficients and, as far as we know, the variable-coefficient Riesz fractional-order NLS-type equations with IST integrability in the sense of Ablowitz et al.’s method [16,17] have not yet been reported in the literature. Based on this research gap, this article aims to explore the soliton structure characteristics of IST integrable models under the dual influence of Riesz fractional derivative and variable coefficients, to adapt to the physical background environment of irregular and complex media.

The work of this paper is organized as follows. In Section 2, we derive a novel vcRfgNLS equation by choosing the appropriate matrix operator in a class of integrable Riesz fractional-order nonlinear evolution equations associated with the $2\times 2$ MSPs and use the linearization of the derived equation to obtain its ADR. In Section 3, we recall some necessary conclusions about the direct scattering problem and look for the variation in its scattering data with time according to the ADR, and finally, construct the corresponding TIEs to construct n-soliton solutions of the vcRfgNLS equation. In Section 4, the explicit form of the vcRfgNLS equation is presented by using the SEFs, adjoint SEFs and their completeness. In Section 5, we analyze the spatial structural properties of 1-soliton solution and 2-soliton solution. In the last section, some conclusions of our findings are given.

2. ADR and the Derivation of vcRfgNLS Equation

In this section, we first introduce the variable-coefficient generalized NLS (vcgNLS) equation [23]:

$$i{\psi }_z={\displaystyle \frac{a(z)}{2}}{\psi }_{\tau \tau }-b(z){\left|\psi \right|}^2\psi +i{\displaystyle \frac{c(z)}{2}}\psi ,$$

(1)

with gain form proposed in nonlinear fiber optics, where $a(z)$, $b(z)$ and $c(z)$ depend on the propagation distance variable $z$, which, respectively, describe the parameters of group velocity dispersion, nonlinearity, and distributed gain/absorption inhomogeneities. Here, we assume that $\alpha (t)$ and $\gamma (t)$ are integrable real functions with respect to $t$ and $q$ and all their partial derivatives with respect to $x$ decay rapidly enough to zero at infinity. If we let $\psi (z,\tau )=q(x,t)$, $a(z)=-\alpha (t)$, $b(z)=\alpha (t)$, and $c$, then Equation (1) becomes

$$i{q}_t+{\displaystyle \frac{1}{2}}\alpha (t)q_{xx}+\alpha (t){\left|q\right|}^{2}q=i\gamma (t)q,$$

(2)

which is equivalent to the one [24] with the Lax and IST integrabilities and can be derived from the following more generalized NLS equation [25],

$$i{\displaystyle \frac{\partial \Psi }{\partial Z}}+{\displaystyle \frac{1}{2}}D(Z){\displaystyle \frac{{\partial }^2\Psi }{\partial T^2}}+R(Z){\left|\Psi \right|}^2\Psi =i\Gamma (Z)\Psi ,$$

(3)

by setting $\Psi (T,Z)=q(x,t)$, $D(Z)=\alpha (t)$, $R(Z)=\alpha (t)$, and $\Gamma (Z)=\gamma (t)$. In Equation (3), the Wronskian of $D(Z)$ and $R(Z)$ is originally allowed to be non-zero, in which case it follows that $D(Z)$ and $R(Z)$ are linearly independent. As a generalization of the classical NLS equation $i{q}_t+q_{xx}+2{\left|q\right|}^{2}q=0$, the vcgNLS Equation (2) or (3) is important in many areas of physics, especially optical communication engineering. For example, the management principles of optical solitons dispersion can be found in [25] and the references there.

In this paper, the vcRfgNLS equation we are going to derive and solve by Ablowitz et al.’s method [16,17] is

$$i{\left[\begin{array}{c}q\\ -q^{\ast }\end{array}\right]}_t+\sigma {\left|4{({\Omega }^a)}^2\right|}^{\epsilon }\left[\begin{array}{c}-{\displaystyle \frac{1}{2}}\alpha (t)q_{xx}-\alpha (t){\left|q\right|}^{2}q+i\gamma (t)q\\ {\displaystyle \frac{1}{2}}\alpha (t)q_{xx}^{\ast }+\alpha (t){\left|q\right|}^2{q}^{\ast }-i\gamma (t)q^{\ast }\end{array}\right]=0,\epsilon \in (0,1),$$

(4)

where

$$\sigma =\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right],{\Omega }^a={\displaystyle \frac{1}{2i}}\left[\begin{array}{cc}-\partial +2q{\partial }_x^{-1}r& -2q{\partial }_x^{-1}q\\ 2r{\partial }_x^{-1}r& \partial -2r{\partial }_x^{-1}q\end{array}\right],\partial ={\displaystyle \frac{\partial }{{\partial }_x}},{\partial }_x^{-1}={\displaystyle {\int }_{-\infty }^{x}dy}.$$

(5)

Note that the use of the operator ${\partial }_x^{-1}$, as, for instance ${\partial }_x^{-1}q$ acting on a function, must be integrated from $-\infty$ to $x$ over the product of $q(y,t)$ and the function acted upon. Obviously, Equation (4) is a fractional-order extended form of the vcgNLS Equation (2), and the limit of the first component in Equation (4) is the vcgNLS Equation (2) when $\epsilon \to 0$, while the second component is the negative conjugate of the first component. Here, Equation (4) is written in matrix form for the purpose of facilitating our subsequent derivation and solution. As for the explicit form of Equation (4), we will derive it separately later.

Firstly, we introduce the following $2\times 2$ MSPs [2]:

$${\phi }_x=A(x,k,t)\phi ,A(x,k,t)=ik\sigma +Q,Q=\left[\begin{array}{cc}0& q(x,t)\\ r(x,t)& 0\end{array}\right],$$

(6)

$${\phi }_t=B(x,k,t)\phi ,B(x,k,t)=\left[\begin{array}{cc}{B}_1(x,k,t)& B_2(x,k,t)\\ B_3(x,k,t)& -B_1(x,k,t)\end{array}\right],$$

(7)

where $\phi ={({\phi }_1(x,k,t),{\phi }_2(x,k,t))}^T$ is the matrix eigenfunction, $k\in C$ is the complex spectral parameter, $q(x,t)$ and $r(x,t)$ are a pair of smooth potentials, $B_1(x,k,t)$, $B_2(x,k,t)$, and $B_3(x,k,t)$ are three undetermined functions of the variables $t$ and $x$, the potentials $q(x,t)$ and $r(x,t)$, and the spectral parameter $k$. When Equations (6) and (7) hold simultaneously, the matrices $A(x,k,t)$ and $B(x,k,t)$ satisfy the matrix equation:

$$A_t(x,k,t)-B_x(x,k,t)+A(x,k,t)B(x,k,t)-B(x,k,t)A(x,k,t)=0.$$

(8)

With the help of Equation (8), a hierarchy of nonlinear equations can be derived, which has the following form:

$$u_t+\sigma \Lambda ({\Omega }^a)u=0,$$

(9)

where $u={(q(x,t),r(x,t))}^T$, $\Lambda ({\Omega }^a)={({\Omega }^a)}^n$, $n=1,2,\cdots$.

When considering a special case where $\Lambda ({\Omega }^a)$ is a rational form of ${\Omega }^a$, the relevant integer-order and/or higher-order NLS equations with IST solvability can be obtained from Equation (9) by providing the selected value of $n$$(n=1,2,\cdots )$ and the appropriate relation between $q(x,t)$ and $r(x,t)$. Of course, if we begin to derive Riesz fractional-order NLS-type equations with variable coefficients, we need to make appropriate adjustments for $\Lambda ({\Omega }^a)$, which means not only a fractional power of ${\Omega }^a$ but also some variable-coefficient functions have to be embedded into $\Lambda ({\Omega }^a)$. Following the ideas of [1,16,17,20,21,22], we take

$$\Lambda ({\Omega }^a)=[\gamma (t)-2i\alpha (t){({\Omega }^a)}^2]{\left|4{({\Omega }^a)}^2\right|}^{\epsilon },$$

(10)

and then obtain

$$i{\left[\begin{array}{c}q\\ r\end{array}\right]}_t+\sigma {\left|4{({\Omega }^a)}^2\right|}^{\epsilon }\left[\begin{array}{c}-{\displaystyle \frac{1}{2}}\alpha (t)q_{xx}+\alpha (t)q^{2}r+i\gamma (t)q\\ -{\displaystyle \frac{1}{2}}\alpha (t)r_{xx}+\alpha (t)r^{2}q+i\gamma (t)r\end{array}\right]=0.$$

(11)

Without loss of generality, we further set $r(x,t)=-q^{\ast }(x,t)$, where the superscript $\ast$ represents taking a complex conjugate on the object being imposed, the vcRfgNLS Equation (4) can be obtained from Equation (11).

It is worth noting that $\Lambda ({\Omega }^a)$ is related to the ADR $\omega (k,t)$ of the linear part of Equation (4):

$$i{q}_t+{\left|-{\partial }_x^2\right|}^{\epsilon }\left[{\displaystyle \frac{\alpha (t)}{2}}{q}_{xx}-i\gamma (t)q\right]=0,$$

(12)

where ${\left|-{\partial }_x^2\right|}^{\epsilon }$ is used to denote the Riesz fractional-order derivative [18,19], see [16] for its explicit expression. To determine the ADR $\omega (k,t)$ for Equation (12), we substitute

$$q=e^{ikx-i{\displaystyle {\int }_0^t\omega (k,m)dm}}$$

(13)

into Equation (12). Then, the ADR can be determined as

$$\omega (k,t)=\left[{\displaystyle \frac{1}{2}}\alpha (t)k^2+i\gamma (t)\right]{\left|k^2\right|}^{\epsilon }.$$

(14)

On the other hand, if we consider the zero-boundary situation, where both $q(x,t)$ and $r(x,t)$ decay to zero quickly enough as $\left|x\right|\to \infty$, then we can associate $\Lambda ({\Omega }^a)$ with the ADR of the linear part of Equation (9). In this limit case, it is easy to see that ${\Omega }^a$ is a diagonal differential operator:

$${\Omega }^a\to {\displaystyle \frac{1}{2i}}{\displaystyle \frac{\partial }{{\partial }_x}}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right].$$

(15)

Thus, Equation (9) is transformed into a pair of equations:

$$q_t-\Lambda \left(-{\displaystyle \frac{1}{2i}}\partial \right)q=0,$$

(16)

$$r_t+\Lambda \left({\displaystyle \frac{1}{2i}}\partial \right)r=0.$$

(17)

Substituting Equation (13) into Equation (16) yields

$$\omega (k,t)=i\Lambda \left(-{\displaystyle \frac{k}{2}}\right).$$

(18)

Further comparing Equations (14) and (18), we then have

$$\Lambda (k)=-i\omega (-2k,t)=[\gamma (t)-2i\alpha (t)k^2]{\left|4k^2\right|}^{\epsilon },$$

(19)

which gives the expression for $\Lambda ({\Omega }^a)$ in Equation (10).

3. IST and Analytical Solutions for the vcRfgNLS Equation

In this section, we mainly apply the IST to the vcRfgNLS Equation (4) based on Ablowitz et al.’s method [16,17]. This includes a necessary review of the known results [26,27], as well as a specific discussion of the temporal variations in scattering data.

Firstly, we rewrite Equation (6) as a component form:

$${\phi }_{1x}(x,k)=-ik{\phi }_1(x,k)+q(x){\phi }_2(x,k),$$

(20)

$${\phi }_{2x}(x,k)=ik{\phi }_2(x,k)+r(x){\phi }_1(x,k).$$

(21)

It is assumed that the smooth potentials $q(x)$ and $r(x)$ defined along the entire x-axis have the required derivatives and decay quickly enough to zero as $x$ approaches infinity, such that the following integrals

$${\displaystyle {\int }_{-\infty }^{\infty }\left|x^{j}q(x)\right|dx},{\displaystyle {\int }_{-\infty }^{\infty }\left|x^{j}r(x)\right|dx},(j=0,1),$$

(22)

are finite. Then, there exists a set of fundamental solutions $\varphi (x,k)$, $\overline{\varphi }(x,k)$, $\phi (x,k)$, and $\overline{\phi }(x,k)$ of Equations (20) and (21) satisfying the asymptotic conditions [24,25]:

$$\varphi (x,k)\approx \left[\begin{array}{c}1\\ 0\end{array}\right]e^{-ikx},\overline{\varphi }(x,k)\approx \left[\begin{array}{c}0\\ -1\end{array}\right]e^{ikx},x\to -\infty ,$$

(23)

$$\phi (x,k)\approx \left[\begin{array}{c}0\\ 1\end{array}\right]e^{ikx},\overline{\phi }(x,k)\approx \left[\begin{array}{c}1\\ 0\end{array}\right]e^{-ikx},x\to \infty .$$

(24)

where $\phi (x,k)$ and $\overline{\phi }(x,k)$ for every real number $k$ are the fundamental set of solutions of Equations (20) and (21). Therefore, the pair of solutions $\varphi (x,k)$ and $\overline{\varphi }(x,k)$ can be expressed as the linear combinations of $\phi (x,k)$ and $\overline{\phi }(x,k)$, and the coefficients of these linear combinations depend on $k$, such that

$$\varphi (x,k)=g(k)\overline{\phi }(x,k)+h(k)\phi (x,k),$$

(25)

$$\overline{\varphi }(x,k)=-\overline{g}(k)\phi (x,k)+\overline{h}(k)\overline{\phi }(x,k),$$

(26)

with $g(k)$, $h(k)$, $\overline{g}(k)$, and $\overline{h}(k)$ determined by

$$g(k)=W(\varphi ,\phi ),h(k)=-W(\varphi ,\overline{\phi }),$$

(27)

$$\overline{g}(k)=W(\overline{\varphi },\overline{\phi }),\overline{h}(k)=W(\overline{\varphi },\phi ),$$

(28)

where $W(\varphi ,\phi )={\varphi }_1{\phi }_2-{\varphi }_2{\phi }_1$ is the Wronskian of $\varphi$ and $\phi$.

Next, we recall the case when $g(k)$ and $\overline{g}(k)$ on the upper and lower half-planes of $k$, respectively, which have only a finite number of zeros. Let ${\theta }_j(j=1,2,\cdots ,n)$ be the zeros of $g(k)$, namely $g({\theta }_j)=0$, $j=1,2,\cdots ,n$. That is to say, ${\theta }_j(j=1,2,\cdots ,n)$ are the discrete spectra of Equations (20) and (21). Since the linear relation between $\varphi (x,{\theta }_j)$ and $\phi (x,{\theta }_j)$ can be indicated by their Wronskian, there exist $\varphi (x,{\theta }_j)={\delta }_j\phi (x,{\theta }_j)$ for $j=1,2,\cdots ,n$. Similarly, we have $\overline{\varphi }(x,{\overline{\theta }}_j)={\overline{\delta }}_j\overline{\phi }(x,{\overline{\theta }}_j)$ for $j=1,2,\cdots ,\overline{n}$, when ${\overline{\theta }}_j(j=1,2,\cdots ,\overline{n})$ are the zeros of $\overline{g}(k)$, where ${\delta }_j$ and ${\overline{\delta }}_j$ are constants. If ${\theta }_j$ and ${\overline{\theta }}_j$ are the simple zeros of $g(k)$ and $\overline{g}(k)$, respectively, there must exist constants $c_j$ and ${\overline{c}}_j$ such that

$$c_j^2={\left(2{\displaystyle {\int }_{-\infty }^{\infty }{\phi }_1(x,{\theta }_j)}{\phi }_2(x,{\theta }_j)dx\right)}^{-1}=-{\displaystyle \frac{i{\delta }_j}{g_k({\theta }_j)}},$$

(29)

$${\overline{c}}_j^2={\left(2{\displaystyle {\int }_{-\infty }^{\infty }{\overline{\phi }}_1(x,{\overline{\theta }}_j)}{\overline{\phi }}_2(x,{\overline{\theta }}_j)dx\right)}^{-1}=-{\displaystyle \frac{i{\overline{\delta }}_j}{{\overline{g}}_k({\overline{\theta }}_j)}}.$$

(30)

In addition to the above discrete spectra, the entire real axis of $k$-plane is filled with the continuous spectra for the non-normalized eigenfunctions. In this case, Equations (25) and (26) still hold since the solutions satisfying Equations (20), (21), (23), and (24) are bounded for any real number $k$. Thus, they can also be expressed as follows:

$$\tau (k)\varphi (x,k)=\overline{\phi }(x,k)+\rho (k)\phi (x,k),$$

(31)

$$\overline{\tau }(k)\overline{\varphi }(x,k)=-\phi (x,k)+\overline{\rho }(k)\overline{\phi }(x,k),$$

(32)

where $\tau (k)=1/g(k)$, $\overline{\tau }(k)=1/\overline{g}(k)$, $\rho (k)=h(k)/g(k)$, and $\overline{\rho }(k)=\overline{h}(k)/\overline{g}(k)$.

In this way, the following zero-time scattering data $S(k)$ and $\overline{S}(k)$ of the MSP (6) can be defined as

$$S(k)=\{\mathrm{Im}k=0,\rho (k)={\displaystyle \frac{h(k)}{g(k)}},{\theta }_j,c_j^2;j=1,2,\cdots ,n\},$$

(33)

$$\overline{S}(k)=\{\mathrm{Im}k=0,\overline{\rho }(k)={\displaystyle \frac{\overline{h}(k)}{\overline{g}(k)}},{\overline{\theta }}_j,{\overline{c}}_j^2;j=1,2,\cdots ,\overline{n}\},$$

(34)

the temporal variations of which are determined below.

We know that the matrix $B(x,k,t)$ in the corresponding integer-order integrable NLS equations is a function associated with $k$, $q(x,t)$, and $r(x,t)$. However, for the Riesz fractional-order case, it is not possible to provide a direct expression for it. Therefore, the following constraints need to be given for the vcRfgNLS Equation (4) as $\left|x\right|\to \infty$:

$$B_1(x,k,t)\to {\displaystyle \frac{1}{2}}\Lambda (k),$$

(35)

$$B_2(x,k,t)\to 0,B_3(x,k,t)\to 0,$$

(36)

$$B_4(x,k,t)\to -{\displaystyle \frac{1}{2}}\Lambda (k),$$

(37)

where $\Lambda (k)$ is given by Equation (19).

Theorem 1.

Assuming $q(x,t)$ solves the vcRfgNLS Equation (4), then the scattering data

$$S(k,t)=\{\mathrm{Im}k=0,\rho (k,t)={\displaystyle \frac{h(k,t)}{g(k,t)}},{\theta }_j(t),c_j^2(t);j=1,2,\cdots ,n\},$$

(38)

$$\overline{S}(k,t)=\{\mathrm{Im}k=0,\overline{\rho }(k,t)={\displaystyle \frac{\overline{h}(k,t)}{\overline{g}(k,t)}},{\overline{\theta }}_j(t),{\overline{c}}_j^2(t);j=1,2,\cdots ,\overline{n}\},$$

(39)

for the MSP (6) possess the following temporal variations:

$$g(k,t)=g(k,0),h(k,t)=h(k,0)e^{{\displaystyle {\int }_0^t[2i\alpha (m)k^2-\gamma (m)]{\left|4k^2\right|}^{\epsilon }dm}},$$

(40)

$$\overline{g}(k,t)=\overline{g}(k,0),\overline{h}(k,t)=\overline{h}(k,0)e^{{\displaystyle {\int }_0^t[\gamma (m)-2i\alpha (m)k^2]{\left|4k^2\right|}^{\epsilon }dm}},$$

(41)

$$c_j^2(t)=c_j^2(0)e^{{\displaystyle {\int }_0^t[2i\alpha (m){\theta }_j^2-\gamma (m)]{\left|4{\theta }_j^2\right|}^{\epsilon }dm}},{\overline{c}}_j^2(t)={\overline{c}}_j^2(0)e^{{\displaystyle {\int }_0^t[\gamma (m)-2i\alpha (m){\overline{\theta }}_j^2]{\left|4{\overline{\theta }}_j^2\right|}^{\epsilon }dm}},$$

(42)

$$\rho (k,t)=\rho (k,0)e^{{\displaystyle {\int }_0^t[2i\alpha (m)k^2-\gamma (m)]{\left|4k^2\right|}^{\epsilon }dm}},\overline{\rho }(k,t)=\overline{\rho }(k,0)e^{{\displaystyle {\int }_0^t[\gamma (m)-2i\alpha (m)k^2]{\left|4k^2\right|}^{\epsilon }dm}},$$

(43)

 where ${\theta }_j$ and ${\overline{\theta }}_j$ are the complex constants in upper and lower half-planes of $k$, respectively.

Proof. 

Firstly, we introduce the matrix parameter $\upsilon I$, where $\upsilon$ is a function with respect to $t$, and $I$ is the $2\times 2$ unit matrix. Then, Equation (7) can be adjusted into

$${\phi }_t=\left[\begin{array}{cc}{B}_1& B_2\\ B_3& -B_1\end{array}\right]\phi -\upsilon I\phi .$$

(44)

After introducing the above matrix parameter $\upsilon I$, we can easily confirm that Equations (6) and (7) have the same compatibility condition ${\phi }_{xxt}={\phi }_{txx}$ as Equations (6) and (44). Therefore, Equations (6) and (44) can also generate Equation (4). The reason why we embed the parameter $\upsilon$ is that it can assist in deriving the corresponding scattering data.

When $x\to -\infty$, the eigenfunction $\varphi (x,k)\approx {(1,0)}^T{e}^{-ikx}$, where the superscript $T$ denotes the transpose of matrix. According to Equation (44) and using Equations (35)–(37), we have the following equation:

$$0=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}\Lambda (k)& 0\\ 0& -{\displaystyle \frac{1}{2}}\Lambda (k)\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]e^{-ikx}-\left[\begin{array}{cc}\upsilon & 0\\ 0& \upsilon \end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]e^{-ikx}.$$

(45)

Since $\phi (x,k)$ and $\overline{\phi }(x,k)$ are linearly independent, we know that the eigenfunction $\varphi (x,k)$ can be expressed as a linear combination of $\phi (x,k)$ and, i.e.,:

$$\varphi (x,k)=g(k,t)\overline{\phi }(x,k)+h(k,t)\phi (x,k).$$

(46)

Then, when $x\to \infty$, we have

$$\left[\begin{array}{c}{g}_t{e}^{-ikx}\\ h_t{e}^{ikx}\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}\Lambda (k)-\upsilon & 0\\ 0& -{\displaystyle \frac{1}{2}}\Lambda (k)-\upsilon \end{array}\right]\left(g\left[\begin{array}{c}1\\ 0\end{array}\right]e^{-ikx}+h\left[\begin{array}{c}0\\ 1\end{array}\right]e^{ikx}\right).$$

(47)

Considering the coefficients of $e^{-ikx}$ and $e^{ikx}$ in Equations (45) and (47), we derive

$$\upsilon ={\displaystyle \frac{1}{2}}\Lambda (k),g_t=0,h_t=-\Lambda (k)h.$$

(48)

Similarly, when $\overline{\varphi }(x,k)\approx {(0,-1)}^T{e}^{ikx}$ as $x\to -\infty$, we obtain

$$\upsilon =-{\displaystyle \frac{1}{2}}\Lambda (k),{\overline{g}}_t=0,{\overline{h}}_t=\Lambda (k)\overline{h}.$$

(49)

It is important to note that the parameters $\upsilon$ corresponding to different eigenfunctions are different. We can conclude from $g_t=0$ and ${\overline{g}}_t=0$ that their zeros are independent of $t$, that is to say, ${\theta }_{j,t}=0$ and ${\overline{\theta }}_{j,t}=0$. Then, Equations (29) and (30) are only relevant for ${\delta }_j$ and ${\overline{\delta }}_j$, respectively. When $x\to \infty$, $\varphi (x,{\theta }_j)$ has the asymptotic property $\varphi (x,{\theta }_j)\approx {\delta }_j{(0,1)}^T{e}^{i{\theta }_{j}x}$. Similarly, there exists $\overline{\varphi }(x,{\overline{\theta }}_j)\approx {\overline{\delta }}_j{(1,0)}^T{e}^{-i{\overline{\theta }}_{j}x}$. Then, we have

$${\delta }_{j,t}=-\Lambda ({\theta }_j){\delta }_j,{\overline{\delta }}_{j,t}=\Lambda ({\overline{\theta }}_j){\overline{\delta }}_j.$$

(50)

The combination of Equations (29), (30) and (50) gives the following results:

$${(c_j^2)}_t=-\Lambda ({\theta }_j)c_j^2,{({\overline{c}}_j^2)}_t=\Lambda ({\overline{\theta }}_j){\overline{c}}_j^2.$$

(51)

Integrating Equations (48), (50) and (51) yields Equations (40)–(42), while $\rho (k,t)$ and $\overline{\rho }(k,t)$ in Equation (43) can be determined from the equalities $\rho (k,t)=h(k,t)/g(k,t)$ and $\overline{\rho }(k,t)=\overline{h}(k,t)/\overline{g}(k,t)$ by using Equations (40) and (41), and the assumptions $\rho (k,0)=h(k,0)/g(k,0)$ and $\overline{\rho }(k,0)=\overline{h}(k,0)/\overline{g}(k,0)$. Thus, we have proven the variations in scattering data $S(k,t)$ and $\overline{S}(k,t)$ with time $t$.

Finally, using the temporal variations (40)–(42) of scattering data (38) and (39), we derive the analytical solutions of vcRfgNLS Equation (4) with the help of corresponding TIEs associated with IST. □

Theorem 2.

The equation hierarchy (9) has the following analytical solutions:

$$q(x,t)=-2Z_1(x,x,t),r(x,t)={\displaystyle \frac{Z_{2x}(x,x,t)}{Z_1(x,x,t)}},$$

(52)

 with $Z_i(x,x,t)=Z_i(x,y,t){|}_{y=x}$,$i=1,2$, satisfying the TIEs:

$$\overline{Z}(x,y)+\left[\begin{array}{c}0\\ 1\end{array}\right]V(x+y)+{\displaystyle {\int }_x^{\infty }Z(x,s)V(s+y)ds=0},$$

(53)

$$\overline{Z}(x,y)+\left[\begin{array}{c}0\\ 1\end{array}\right]V(x+y)+{\displaystyle {\int }_x^{\infty }Z(x,s)V(s+y)ds=0},$$

(54)

 where  $Z(x,y)={(Z_1(x,y),Z_2(x,y))}^T$, and

$$V(x)={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }\rho (k)e^{ikx}dk+{\displaystyle \sum _{j=1}^n{c}_j^2}}{e}^{i{\theta }_{j}x},$$

(55)

$$\overline{V}(x)={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }\overline{\rho }(k)e^{-ikx}dk-{\displaystyle \sum _{j=1}^{\overline{n}}{\overline{c}}_j^2}}{e}^{-i{\overline{\theta }}_{j}x},$$

(56)

 while $c_j^2(t)$ and ${\overline{c}}_j^2(t)$ can be traced back to Equation (42), $n$ and $\overline{n}$ represent the numbers of zeros of the functions $g(k,t)$ and $\overline{g}(k,t)$.

Proof. 

The derivation of Equation (52) is similar to the integer-order case in [24,25], we omit it here for simplification. □

To obtain the explicit forms of $Z_1(x,x,t)$ and $Z_2(x,x,t)$ in Equation (52), we let $\rho (k,t)$ and $\overline{\rho }(k,t)$ in Equations (38) and (39) be zero. In fact, we can achieve this as long as $h(k,0)$ and $\overline{h}(k,0)$ in Equations (40) and (41) are all zero. In this case, the potential $q(x,t)$ to be obtained solves Equation (4). For Equation (53), eliminating $\overline{Z}(x,y)$ yields

$$Z(x,y)-\left[\begin{array}{c}1\\ 0\end{array}\right]\overline{V}(x+y)+\left[\begin{array}{c}0\\ 1\end{array}\right]{\displaystyle {\int }_x^{\infty }V(x+s)\overline{V}(s+y)ds}+{\displaystyle {\int }_x^{\infty }Z(x,z)({\displaystyle {\int }_x^{\infty }V(z+s)}\overline{V}(s+y)ds)dz=0}.$$

(57)

By direct calculation, we obtain

$${\displaystyle {\int }_x^{\infty }V(z+s)}\overline{V}(s+y)ds=-{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{v=1}^{\overline{n}}\frac{i{c}_j^2(t){\overline{c}}_v^2(t)}{{\theta }_j-{\overline{\theta }}_v}}}{e}^{i{\theta }_j(x+z)-i{\overline{\theta }}_v(x+y)}.$$

(58)

Supposing that

$$Z_1(t,x,y)={\displaystyle \sum _{p=1}^{\overline{n}}{\overline{c}}_p(t){г}_p(t,x)e^{-i{\overline{\theta }}_{p}y}},$$

(59)

$$Z_2(t,x,y)={\displaystyle \sum _{p=1}^{\overline{n}}{\overline{c}}_p(t){б}_p}(t,x)e^{-i{\overline{\theta }}_{p}y},$$

(60)

we transform Equation (57) into the following system of equations for $v=1,2,\cdots ,\overline{n}$:

$${г}_v(x,t)+{\overline{c}}_v(t)e^{-i{\overline{\theta }}_{v}x}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{p=1}^{\overline{n}}\frac{c_j^2(t){\overline{c}}_v(t){\overline{c}}_p(t)}{({\theta }_j-{\overline{\theta }}_v)({\theta }_j-{\overline{\theta }}_p)}}}{e}^{i(2{\theta }_j-{\overline{\theta }}_v-{\overline{\theta }}_p)x}{г}_p(x,t)=0,$$

(61)

$${б}_v(x,t)-{\displaystyle \sum _{j=1}^n\frac{i{c}_j^2(t){\overline{c}}_v(t)}{{\theta }_j-{\overline{\theta }}_v}}{e}^{i(2{\theta }_j-{\overline{\theta }}_v)x}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{p=1}^{\overline{n}}\frac{c_j^2(t){\overline{c}}_v(t){\overline{c}}_p(t)}{({\theta }_j-{\overline{\theta }}_v)({\theta }_j-{\overline{\theta }}_p)}}}{e}^{i(2{\theta }_j-{\overline{\theta }}_v-{\overline{\theta }}_p)x}{б}_p(x,t)=0.$$

(62)

We simplify Equations (61) and (62) by writing them as matrix forms:

$$H(x,t)г(x,t)=-\overline{ш}(x,t),H(x,t)б(x,t)=iL(x,t)ш(x,t),$$

(63)

with

$$L(x,t)={\left[{\displaystyle \frac{c_j(t){\overline{c}}_v(t)}{{\theta }_j-{\overline{\theta }}_v}}{e}^{i({\theta }_j-{\overline{\theta }}_v)x}\right]}_{\overline{n}\times n},H(x,t)=I+L(x,t)L^T(x,t),$$

(64)

$$г(x,t)={({г}_1(x,t),{г}_2(x,t),\cdots ,{г}_{\overline{n}}(x,t))}^T,$$

(65)

$$б(x,t)={({б}_1(x,t),{б}_2(x,t),\cdots ,{б}_{\overline{n}}(x,t))}^T,$$

(66)

$$\overline{ш}(x,t)={({\overline{c}}_1(t)e^{-i{\overline{\theta }}_{1}x},{\overline{c}}_2(t)e^{-i{\overline{\theta }}_{2}x},\cdots ,{\overline{c}}_{\overline{n}}(t)e^{-i{\overline{\theta }}_{\overline{n}}x})}^T,$$

(67)

$$ш(x,t)={(c_1(t)e^{i{\theta }_{1}x},c_2(t)e^{i{\theta }_{2}x},\cdots ,c_n(t)e^{i{\theta }_{n}x})}^T,$$

(68)

where $I$ is the $\overline{n}\times \overline{n}$ unit matrix. Assuming that the inverse matrix of $H(t,x)$ exists, from Equation (63) we can obtain

$$г(x,t)=-H^{-1}(x,t)\overline{ш}(x,t),б(x,t)=i{H}^{-1}(x,t)L(x,t)ш(x,t).$$

(69)

Substituting Equation (69) into Equations (59) and (60) yields

$$Z_1(x,y,t)=-\mathrm{tr}(H^{-1}(x,t)\overline{ш}(x,t){\overline{ш}}^T(y,t)),$$

(70)

$$Z_2(x,y,t)=i\mathrm{tr}(H^{-1}(x,t)L(x,t)ш(x,t){\overline{ш}}^T(y,t)),$$

(71)

where $\mathrm{tr}(\cdot )$ denotes the trace of matrix. Then, from Equations (52), (70) and (71), we can determine the n-soliton solution of Equation (4):

$$q(x,t)=2\mathrm{tr}(H^{-1}(x,t)\overline{ш}(x,t){\overline{ш}}^T(x,t)),$$

(72)

$$r(x,t)=-{\displaystyle \frac{\frac{\partial }{{\partial }_x}\mathrm{tr}\left(H^{-1}(x,t)L(x,t)\frac{\partial }{{\partial }_x}{L}^T(x,t)\right)}{\mathrm{tr}(H^{-1}(x,t)\overline{ш}(x,t){\overline{ш}}^T(x,t))}}.$$

(73)

In this article, our goal is to solve the vcRfgNLS Equation (4). This requires conversion of Equation (9) to Equation (4). If we set $r(x,t)=-q^{\ast }(x,t)$ in Equation (9), then Equations (6) and (7) are transformed into the TIEs corresponding to Equation (4). Therefore, Equation (4) can also be reduced from Equation (9), which leads to

$${\overline{\phi }}_1(x,k)={\phi }_2^{\ast }(x,k^{\ast }),{\overline{\phi }}_2(x,k)=-{\phi }_1^{\ast }(x,k^{\ast }),$$

(74)

$${\overline{\varphi }}_1(x,k)={\varphi }_2^{\ast }(x,k^{\ast }),{\overline{\varphi }}_2(x,k)=-{\varphi }_1^{\ast }(x,k^{\ast }),$$

(75)

$$\overline{g}(k)=g^{\ast }(k^{\ast }),\overline{h}(k)=h^{\ast }(k^{\ast }),$$

(76)

$${\overline{\theta }}_j={\theta }_j^{\ast },{\overline{c}}_j^2=-c_j^{2\ast },\overline{\rho }(k)={\rho }^{\ast }(k^{\ast }),n=\overline{n},$$

(77)

$$\overline{V}(x)=V^{\ast }(x),{\overline{Z}}_1(x,y)=Z_2^{\ast }(x,y),{\overline{Z}}_2(x,y)=-Z_1^{\ast }(x,y).$$

(78)

In the above situation, Equation (54) is obviously the corresponding conjugate equation of Equation (53), the n-soliton solution of Equation (9) degenerate to the one of Equation (4), the matrices $L(x,t)$ and $H(x,t)$ in Equation (64) convert to the following forms:

$$L(x,t)={\left[{\displaystyle \frac{i{c}_j(t)c_v^{\ast }(t)}{{\theta }_j-{\theta }_v^{\ast }}}{e}^{i({\theta }_j-{\theta }_v^{\ast })x}\right]}_{n\times n},H(x,t)=I+L(x,t)L^{\ast }(x,t),$$

(79)

where $L^{\ast }(x,t)$ is the conjugate matrix of $L(x,t)$.

Finally, the formula of n-soliton solution of Equation (4) can be obtained:

$$q(x,t)=-2\mathrm{tr}(H^{-1}(x,t){ш}^{\ast }(x,t){ш}^{\ast T}(x,t)),$$

(80)

where ${ш}^{\ast }(x,t)=-i\overline{ш}(x,t)$. In the following Section 5, this article will provide the specific 1-soliton solution and 2-soliton solution for Equation (4).

4. Explicit Form of the vcRfgNLS Equation

Firstly, we introduce the operator:

$$\Omega ={\displaystyle \frac{1}{2i}}\left[\begin{array}{cc}\partial +2r{\partial }_x^{+}q& 2r{\partial }_x^{+}r\\ -2q{\partial }_x^{+}q& -\partial -2q{\partial }_x^{+}r\end{array}\right],{\partial }_x^{+}={\displaystyle {\int }_x^{+\infty }dy}.$$

(81)

It is obvious that the previously used operator ${\Omega }^a$ in Equation (5) is the adjoint of $\Omega$.

According to Equations (6) and (7), the SEFs $\theta$, $\overline{\theta }$, ${\theta }^a$, and ${\overline{\theta }}^a$, and the operators $\Omega$ and ${\Omega }^a$ satisfy the following relations [28,29]:

$$\theta (x,k,t)={({\phi }_2^2(x,k,t),{\phi }_1^2(x,k,t))}^T,$$

(82)

$$\overline{\theta }(x,k,t)={({\overline{\phi }}_2^2(x,k,t),{\overline{\phi }}_1^2(x,k,t))}^T,$$

(83)

$${\theta }^a(x,k,t)={(-{\varphi }_1^2(x,k,t),{\varphi }_2^2(x,k,t))}^T,$$

(84)

$${\overline{\theta }}^a(x,k,t)={(-{\overline{\varphi }}_1^2(x,k,t),{\overline{\varphi }}_2^2(x,k,t))}^T,$$

(85)

$$\Omega \theta =k\theta ,\Omega \overline{\theta }=k\overline{\theta },$$

(86)

$${\Omega }^a{\theta }^a=k{\theta }^a,{\Omega }^a{\overline{\theta }}^a=k{\overline{\theta }}^a,$$

(87)

where $\theta$ and $\overline{\theta }$ are the SEFs of the operator $\Omega$, while ${\theta }^a$ and ${\overline{\theta }}^a$ are the SEFs of the operator ${\Omega }^a$. According to the result [29], we know that $\theta$ and ${\theta }^a$ are complete.

Similarly, we have the relations:

$$\Lambda (\Omega )\theta =\Lambda (k)\theta ,\Lambda (\Omega )\overline{\theta }=\Lambda (k)\overline{\theta },$$

(88)

$$\Lambda ({\Omega }^a){\theta }^a=\Lambda (k){\theta }^a,\Lambda ({\Omega }^a){\overline{\theta }}^a=\Lambda (k){\overline{\theta }}^a.$$

(89)

If we want to write the explicit form of Equation (4), we should know what the operator $\Lambda ({\Omega }^a)$ acting on $u(x,t)$ in Equation (9) means. We can see that when $\Lambda ({\Omega }^a)$ acts on an eigenfunction, it provides the results given in Equation (89). To let $\Lambda ({\Omega }^a)$ act on any function, it is necessary to represent any vector-valued function, we note it as $b(x)={[b_1(x),b_2(x)]}^T$, in terms of the SEFs, which can be achieved by requiring the completeness of the SEFs. As mentioned earlier, the completeness of the SEFs holds, then $\Lambda ({\Omega }^a)$ can be used to act upon the vector function $b(x)$, which is sufficiently smooth and decays to zero at infinity, as follows:

$$\Lambda ({\Omega }^a)b(x)={\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{i=1}^2{\displaystyle {\int }_{\Gamma (\infty ,i)}dk}}\Lambda (k)f_i(k){\displaystyle {\int }_{-\infty }^{\infty }dy{J}_i}(x,y,k)b(y),$$

(90)

$$J_1(x,y,k)={\theta }^a(x,k)\theta {(y,k)}^T,f_1(k)=-{\tau }^2(k),$$

(91)

$$J_2(x,y,k)={\overline{\theta }}^a(x,k)\overline{\theta }{(y,k)}^T,f_2(k)={\overline{\tau }}^2(k),$$

(92)

where $\Gamma (\infty ,i)=\underset{R\to \infty }{\lim }\Gamma (R,i)$, $i=1,2$, $\Gamma (R,1)$ ($\Gamma (R,2)$) is the semicircular contour in the upper (lower) half-plane evaluated from $-R$ to $R$.

Next, letting $u={(q(x,t),-q^{*}(x,t))}^T$ and acting the operator $\Lambda ({\Omega }^a)$ in Equation (10) on it yields

$$[\gamma (t)-2i\alpha (t){({\Omega }^a)}^2]{\left|4{({\Omega }^a)}^2\right|}^{\epsilon }\left[\begin{array}{c}q\\ -q^{\ast }\end{array}\right]={\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{i=1}^2{\displaystyle {\int }_{\Gamma (\infty ,i)}dk}}{\left|4k^2\right|}^{\epsilon }{f}_i(k){\displaystyle {\int }_{-\infty }^{\infty }dy{J}_i}(x,y,k)\left[\begin{array}{c}{\displaystyle \frac{i}{2}}\alpha (t)q_{yy}+i\alpha (t){\left|q\right|}^{2}q+\gamma (t)q\\ -{\displaystyle \frac{i}{2}}\alpha (t)q_{yy}^{\ast }-i\alpha (t){\left|q\right|}^2{q}^{\ast }-\gamma (t)q^{\ast }\end{array}\right].$$

(93)

Therefore, we derive the explicit form of the vcRfgNLS Equation (4) by using Equations (11) and (93):

$$i{q}_t+{\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{i=1}^2{\displaystyle {\int }_{\Gamma (\infty ,i)}dk}}{\left|4k^2\right|}^{\epsilon }{f}_i(k){\displaystyle {\int }_{-\infty }^{\infty }dy{X}_i}(x,y,k)=0,$$

(94)

where

$$X_1(x,y,k)=-{\varphi }_1^2(x,k,t)[{\phi }_2^2(y,k,t)({\displaystyle \frac{1}{2}}\alpha (t)q_{yy}+\beta (t){\left|q\right|}^{2}q-i\gamma (t)q)+{\phi }_1^2(y,k,t)({\displaystyle \frac{1}{2}}\alpha (t)q_{yy}^{\ast }+\beta (t){\left|q\right|}^2{q}^{\ast }-i\gamma (t)q^{\ast })],$$

(95)

$$X_2(x,y,k)=-{\overline{\varphi }}_1^2(x,k,t)[{\overline{\phi }}_2^2(y,k,t)({\displaystyle \frac{1}{2}}\alpha (t)q_{yy}+\beta (t){\left|q\right|}^{2}q-i\gamma (t)q)+{\overline{\phi }}_1^2(y,k,t)({\displaystyle \frac{1}{2}}\alpha (t)q_{yy}^{\ast }+\beta (t){\left|q\right|}^2{q}^{\ast }-i\gamma (t)q^{\ast })].$$

(96)

In particularly, as $\epsilon =0$, Equation (94) reduces to the integer-order form of the vcgNLS Equation (4), the RHP of which has been established in [24]. As $\epsilon =0$, $\alpha (t)=2$, and $\gamma (t)=0$, Equation (94) reduces to the classical NLS equation [2]:

$$i{q}_t+q_{xx}+2{\left|q\right|}^{2}q=0.$$

(97)

As $\epsilon \ne 0$, $\alpha (t)=2$, and $\gamma (t)=0$, Equation (94) reduces to the explicit form of the Riesz fractional-order version [16] of Equation (97):

$$i{q}_t+{\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{i=1}^2{\displaystyle {\int }_{\Gamma (\infty ,i)}dk}}{\left|4k^2\right|}^{\epsilon }{f}_i(k){\displaystyle {\int }_{-\infty }^{\infty }dy{G}_i}(x,y,k)dy=0,$$

(98)

where

$$G_1(x,y,k)=-{\varphi }_1^2(x,k,t)[{\phi }_2^2(x,k,t)(q_{yy}+2{\left|q\right|}^{2}q)+{\phi }_1^2(x,k,t)(q_{yy}^{\ast }+2{\left|q\right|}^2{q}^{\ast })],$$

(99)

$$G_2(x,y,k)=-{\overline{\varphi }}_1^2(x,k,t)[{\overline{\phi }}_2^2(x,k,t)(q_{yy}+2{\left|q\right|}^{2}q)+{\overline{\phi }}_1^2(x,k,t)(q_{yy}^{\ast }+{\left|q\right|}^2{q}^{\ast })].$$

(100)

5. 1-Soliton Solution and 2-Soliton Solution of the vcRfgNLS Equation

In this section, we give the explicit 1-soliton solution and 2-soliton solution of the vcRfgNLS Equation (4) and analyze their structural characteristics.

For $\overline{n}=n=1$, letting $c_1^2(0)=e^{-2\eta {\zeta }_0+i{w}_0}$ and ${\theta }_1=\xi +i\eta$ yields ${\theta }_1^{\ast }=\xi -i\eta$, here $\eta >0$, and ${\zeta }_0$, $w_0$, and $\xi$ are all real numbers, then Equation (80) gives the expression for the 1-soliton solution of Equation (4):

$$q(x,t)=2\eta e^{\kappa }\mathrm{sech}\{2\eta [x+{\displaystyle {\int }_0^{t}2\xi \alpha (m){(4{\xi }^2+4{\eta }^2)}^{\epsilon }dm}-{\zeta }_0]\},$$

(101)

with

$$\kappa =-2i\xi x+{\displaystyle {\int }_0^t[\gamma (m)-2i\alpha (m)({\xi }^2-{\eta }^2)]{(4{\xi }^2+4{\eta }^2)}^{\epsilon }}dm+i{w}_0.$$

(102)

In Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, we show the spatial structures of 1-soliton solution (101) through default images, top-views, and evolutionary diagrams. They show that the choices of $\epsilon$ do not affect the linear shape of the bell 1-soliton, but, as the value of $\epsilon$ increases, the angle with the horizontal direction becomes smaller. This means that the speed of 1-soliton is faster, but its width projected onto the coordinate plane is smaller. Using the same parameters as in Figure 1, we use Figure 2 to simulate the evolution of the 1-soliton propagating from right to left. The amplitude and width of the 1-soliton shown in Figure 2 remain unchanged regardless of whether $\epsilon$ changes or not, indicating that no diffusion or dissipation occurs during the movement process. Furthermore, Figure 2 also provides good evidence for the conclusion we found in Figure 1 that the speed of the 1-soliton increases with the increase of $\epsilon$.

Table 1. The velocity and amplitude of 1-soliton with different fractional orders.

Time $\mathit{t}$	Velocity and Amplitude with $\mathit{\epsilon }=0$	Velocity and Amplitude with $\mathit{\epsilon }=0.4$	Velocity and Amplitude with $\mathit{\epsilon }=0.7$
−100	−1.48803 × 10−43; 1	−2.83269 × 10−43; 1	−4.59083 × 10−43; 1
−10	−1.816 × 10−4; 1	−3.45703 × 10−4; 1	−5.60266 × 10−4; 1
−1	−1.29611; 1	−2.46734; 1	−3.99871; 1
0	−2; 1	−3.80731; 1	−6.17034; 1
5	−2.69506 × 10−2; 1	−5.13045 × 10−2; 1	−8.31471 × 10−2; 1
200	−5.53559 × 10−87; 1	−1.05378 × 10−86; 1	−1.70782 × 10−86; 1
1000	−2.0030383559 × 10−424; 1	−3.8614765923 × 10−424; 1	−6.2640077051 × 10−424; 1

lists the velocity and amplitude of 1-soliton with different fractional orders corresponding to Figure 3 at certain moments. Table 1 supports the finding that the higher the speed of 1-soliton, the smaller its wave width along the $t$-axis, which results in a smaller width of the soliton projected onto the coordinate plane, which is clearly non parallel to the coordinate axis.

Table 2. The width of 1-soliton with different fractional orders at two horizontal heights.

Height $\left|\mathit{q}\right|$ and Time $\mathit{t}$	Width with $\mathit{\epsilon }=0$	Width with $\mathit{\epsilon }=0.4$	Width with $\mathit{\epsilon }=0.7$
0.1; −5	5.98642	5.98644	5.98643
0.1; 0	5.986443	5.986443	5.986443
0.1; 10	5.98644	5.986443	5.98644
0.2; −5	4.58486	4.58483	4.58484
0.2; 0	4.584862	4.584862	4.584862
0.2; 10	4.58486	4.58486	4.58487

shows that when different fractional orders are used, the width of 1-soliton along the $x$-axis in Figure 3 remains unchanged.

For $\overline{n}=n=2$, the expression for the 2-soliton solution of the vcRfgNLS Equation (4) is

$$q(x,t)=-2i{\displaystyle \frac{c_1^2(0)e^{{\varpi }_1-{\varpi }_2^{\ast }}{M}_{12}+c_2^2(0)e^{{\varpi }_2-{\varpi }_1^{\ast }}{M}_{21}-c_2^2(0)e^{{\varpi }_2-{\varpi }_2^{\ast }}{M}_{11}-c_1^2(0)e^{{\varpi }_1-{\varpi }_1^{\ast }}{M}_{22}}{M_{11}{M}_{22}-M_{12}{M}_{21}}},$$

(103)

with

$${\varpi }_j=-i{\theta }_{j}x-i{\displaystyle {\int }_0^t(\alpha (m){\theta }_j^2+\frac{i}{2}\gamma (m))}{\left|4{\theta }_j^2\right|}^{\epsilon }dm,j=1,2,$$

(104)

$$M_{11}={\displaystyle \frac{c_1^2(0){(c_1^2(0))}^{\ast }{e}^{{\varpi }_1^{\ast }+{\varpi }_1}+e^{-{\varpi }_1^{\ast }-{\varpi }_1}}{{\overline{\theta }}_1-{\theta }_1}},$$

(105)

$$M_{12}={\displaystyle \frac{c_2^2(0){(c_1^2(0))}^{\ast }{e}^{{\varpi }_1^{\ast }+{\varpi }_2}+e^{-{\varpi }_1^{\ast }-{\varpi }_2}}{{\overline{\theta }}_1-{\theta }_2}},$$

(106)

$$M_{21}={\displaystyle \frac{c_1^2(0){(c_2^2(0))}^{\ast }{e}^{{\varpi }_2^{\ast }+{\varpi }_1}+e^{-{\varpi }_2^{\ast }-{\varpi }_1}}{{\overline{\theta }}_2-{\theta }_1}},$$

(107)

$$M_{22}={\displaystyle \frac{c_2^2(0){(c_2^2(0))}^{\ast }{e}^{{\varpi }_2^{\ast }+{\varpi }_2}+e^{-{\varpi }_2^{\ast }-{\varpi }_2}}{{\overline{\theta }}_2-{\theta }_2}},$$

(108)

where ${\overline{\theta }}_1={\theta }_1^{\ast }$, ${\overline{\theta }}_2={\theta }_2^{\ast }$, $c_1^2(0)$, and $c_2^2(0)$ are constants.

In Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, we show the spatial structures of 2-soliton solution (103) through default images, top-views, and evolutionary diagrams. We can see from the top-views in Figure 9 that as $\epsilon$ increases, the soliton which forms smaller angle with $x$-axis moves faster, and conversely, the soliton forming larger angle with $x$-axis moves slower. It is shown in Figure 11 that two solitons with periodicity (except during collision) collide and completely penetrate each other. The velocity of soliton varies with $\epsilon$, the overall pattern of change is roughly the same as in Figure 9, but the periodicity remains unchanged. In Figure 13, two solitons resemble swallow shaped wave collision together, in which the solitons do not penetrate each other. As $\epsilon$ increases, the speed of the left soliton increases, but the speed of the right soliton decreases. In Figure 15, the solitons are similar to parabolas with opening to the left touch without penetrating each other. As $\epsilon$ increases, the opening of inner parabola decreases, which means that the speed of soliton is increasing. Conversely, the opening of outer parabola is increasing, which means that the speed of soliton is decreasing. In Figure 9, Figure 11, Figure 13 and Figure 15, we find that each soliton always has an upward bulge at the time of collision, which may be caused by energy aggregation during the collision process. In Figure 10, Figure 12, Figure 14 and Figure 16, we can observe a common characteristic that, regardless of the choices for $\alpha (t)$ and $\gamma (t)$, the amplitude of soliton is not affected by $\epsilon$. It is worth noting that the different observed times are due to the different degrees of collision of the dichotomous solitons when different $\alpha (t)$ and $\gamma (t)$ are chosen, which are observed after the collisions have stabilized. This suggests that diffuse dissipation does not occur throughout the motion process.

6. Conclusions

We have developed the fractional extension of the vcgNLS Equation (2) based on the Riesz fractional-order derivative and showed the explicit form of the vcRfgNLS Equation (4) based on the completeness of SEFs. We investigated the TIE in the context of IST to derive the n-soliton solutions of the vcRfgNLS Equation (4). In particular, we analyzed the 1-soliton solution (101) and 2-soliton solution (103). For the higher-order soliton solutions with $n\ge 3$, we can derive them by substituting Equations (42), (67), (68) and (79) into Equations (80). However, for the sake of convenience, we have omitted such higher-order soliton solutions. Regarding our conclusions, we highlight the following aspects: (1) the fractional order $\epsilon$ does not affect the overall shape of soliton, which is affected by $\alpha (t)$ and $\gamma (t)$; (2) the velocity of soliton is also affected by $\alpha (t)$ and $\gamma (t)$, as well as $\epsilon$. When $\alpha (t)$ and $\gamma (t)$ are fixed, for the 1-soliton, the speed of soliton increases as $\epsilon$ increases; for the 2-soliton, the speed of soliton shows an increase in 1-soliton and a decrease in the other soliton as $\epsilon$ increases; (3) neither the 1-soliton nor the 2-soliton, $\epsilon$ does not affect the amplitude, this means that its propagation does not undergo dissipation and diffusion; (4) for different choices of $\alpha (t)$ and $\gamma (t)$, the effect of $\epsilon$ on the soliton is also different (in the case of 1-soliton as shown in Figure 1, the angle between the formed trajectory and the horizontal direction decreases with the increase of $\epsilon$; in Figure 3, the horizontal distance between the two vertical portions of the formed trajectory increases with the increase of $\epsilon$; in Figure 5, the peak-to-trough distance of the resulting sine-cosine trajectory increases with increasing $\epsilon$; and in Figure 7, the opening of the resulting parabolic trajectory decreases as the fractional $\epsilon$ increases). These interesting phenomena greatly enrich the dynamic behavior of fractional-order nonlinear integrable systems and may play a significant role in predicting the hyper-dispersive transmission of nonlinear waves in inhomogeneous media. The computer algebra system used to generate the reproducible Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 is the Mathematica software 13 (Wolfram Research, Inc., Champaign City, IL, USA), which runs the commands Plot and Plot3D for 2D and 3D figures, respectively. Considering that the NLS-type equations, as pointed out in [30], and references therein, can not only be used to describe the soliton dynamics in nonlinear optics and Bose Einstein condensates, but also widely applied in fluid mechanics, quantum mechanics, and superconductivity, the vcgNLS Equation (4), due to its dual control mechanism of variable coefficients and fractional order, will have important applications and play a greater role in the above fields. This is because the vcNLS Equation (2) is more suitable for describing the soliton dynamics in non-uniform media, and the Riesz fractional-order version of Equation (2) can facilitate the use of fractional order to control the speed of soliton propagation under the same conditions. Furthermore, the ideas discussed in this study can be used to derive and solve other fractional-order models with variable coefficients. This article elaborates the feasibility of generalizing Ablowitz et al.’s method [16,17] of IST to variable-coefficient fractional-order integrable NLS-type systems, with the selection of the vcRfgNLS Equation (4) as a practical example. Comparison with some othere variable-coefficient fNLS models [31,32,33,34] and th research results [35,36,37,38], we would like to point out that the vcRfgNLS Equation (4) and its explicit expression in Equation (94) have not been reported before. More importantly, Equation (4) studied in this work is an inverse scattering integrable system, which is a beneficial supplement for variable-coefficient fNLS equations in the literature. However, we have not yet explored the stability of the soliton solutions with variable coefficients and fractional order obtained in this article. The methods mentioned and used in [30,39] may be helpful for studying the stability of such soliton solutions.