Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation

Abstract

In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional n-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media.

1. Introduction

For the development of nonlinear mathematical physics in the past few decades, it has to be mentioned that various analytical methods of nonlinear evolution equations (NLEEs) have made significant progress, such as IST [1,2,3], the Hirota bilinear method [4], Darboux transformation [5], etc. The IST [1,2,3] proposed by Gardner et al. in 1967 has its own advantages in solving certain NLEEs, especially soliton equations [3,6,7,8,9] such as KdV, modified KdV, and nonlinear Schrödinger (NLS) equations. These soliton equations have many applications in physics, oceanography, engineering, and other fields. When an NLEE can be solved by IST, we usually refer to it as inverse scattering integrable. The basic idea of IST [1,2,3] is to transform the considered NLEE into a solvable system of linear partial differential equations (PDEs), and then obtain the exact solution of the original NLEE by solving these linear PDEs. The ability of IST to convert nonlinear problems into linear ones has the benefit of simplifying the problem-solving process. Solving n-soliton solutions of NLEEs with zero/non-zero boundary conditions is a highly efficient use of IST.

Fractional calculus [10], as an extension of integer-order calculus, is crucial for some existing physical models with anomalous diffusion related to multiscale media, such as Bose–Einstein condensates, quantum mechanics, optics, polymer science, nanofluids, geotechnical engineering, and other areas [11,12,13]. Fractional-order differential equations (fDEs) are frequently used to predict physical measurable values that follow power–law relationships, for example, $t^{\alpha }(\alpha >0)$ of mean square displacement versus time $t$ in anomalous diffusion [14,15,16,17]. Similarly, for integrable soliton equations, fractional-order generalizations predict the anomalous dispersion of local solitary waves, which is related to the power–law relationship between their velocity and amplitude [18]. Fractional-order nonlinear integrable equations (fNIEs) have been studied for about ten years, with over 200 definitions of fractional-order derivatives. However, due to the fact that most of the fractional-order derivatives do not have the general properties of integer-order derivatives, researchers are not optimistic about fNIEs.

In 2022, Ablowitz, Been, and Carr [18,19] proposed fNIEs with the Riesz fractional-order derivative $({\left|-{\partial }_x^2\right|}^{\alpha };\alpha \in (0,1))$ [20,21,22], including the fractional-order NLS (fNLS) equation, fKdV equation, fractional-order mKdV (fmKdV) equation, fractional-order sine-Gordon (fsine-G) equation, and fractional-order sinh-Gordon (fsinh-G) equation. Subsequently, based on Ablowitz et al.’s algorithm [18,19], Meng, Zhang, Zhang, and Yan [22] presented the integrable higher-order fNLS equations; Zhang, Meng, and Yan [23] obtained the integrable higher-order fmKdV hierarchy; Mou, Dai, and Wang [24] derived the integrable fractional-order n-component coupled NLS equations; and An, Ling, and Zhang [25] studied the integrable fractional coupled Hirota (fcH) equation. The commonality of these studies is that they link anomalous dispersion relations with fNIEs, resulting in fractional soliton solutions. However, different methods were used in the solution process. It is noted that the fractional soliton solutions of fNIEs can be found using both Riemann–Hilbert (RH) approach and IST. Ablowitz et al. [18,19] obtained the fractional soliton solutions of these fNIEs by solving the linear solvable Gel’fand–Levitan–Marchenko (GLM) integral equation [26] associated with IST, yet the teams [22,23,24,25] led by Yan, Ling, and Dai obtained fractional soliton solutions using the RH approach. Compared with the RH approach, the theoretical basis of IST is really more robust. In addition to this, IST mostly uses linear transformations and straightforward integral procedures; thus, it typically has low computational complexity. The RH approach, on the other hand, might require more intricate boundary value issues and integral equations, which would increase processing complexity. In the aforementioned references, some fractional soliton solutions are found, and the physical meaning of hyperdispersive transport as an equation is also demonstrated. Due to their close relationship with non-Gaussian statistics [27], fDEs with Riesz fractional-order derivatives are useful tools for explaining the behavior of complex systems and can be used to describe the flow of water in porous media, temperature transfer in fluid dynamics, and power–law decay in materials [28,29,30,31,32,33].

The vcfKdV equation to be derived and solved in this article will be presented later, and we will first trace its integer-order prototype with inverse scattering integrability. For this purpose, we introduce the vcKdV equation:

$$u_t+\sigma (t)u_x+\gamma (t)u_{xxx}+\beta (t)u{u}_x=0,$$

(1)

where the wave function $u$ is a differentiable of the spatial variable $x$ and the temporal variable $t$, the subscript represents the partial derivative of $u$ with respect to the referred variable, and $\sigma (t)$, $\beta (t)$, and $\gamma (t)$ are assumed to be integrable real-valued functions of $t$, representing the dispersion coefficient, dissipation coefficient, and convection coefficient, respectively. vcKdV-type Equation (1), as a generalized form of the classical KdV equation $u_t+6u{u}_x+u_{xxx}=0$, is an important physical model that takes into account the non-uniformity of the background medium. Its potential applications may include describing nonlinear waves in fluids, optical pulse transmission in optical fibers, etc. If $\beta (t)=6\gamma (t)$, Equation (1) becomes $u_t+\sigma (t)u_x+\gamma (t)u_{xxx}+6\gamma (t)u{u}_x=0$, which has Lax integrability and can be reduced from the coupled system of NLEEs with the time-dependent coefficients $a_0(t)$, $a_1(t)$, $a_2(t)$, and $a_3(t)$ [34]:

$$\left\{\begin{array}{l}{u}_t=a_3(t)(u_{xxx}-6u{u}_{x}v)-a_2(t)(u_{xx}-2u^{2}v)+a_1(t)u_x-a_0(t)u\\ v_t=a_3(t)(v_{xxx}-6uv{v}_x)+a_2(t)(v_{xx}-2v^{2}u)+a_1(t)v_x+a_0(t)v\end{array}\right.,$$

(2)

This is achieved by setting $a_3(t)=\gamma (t)$, $a_2(t)=0$, $a_1(t)=\sigma (t)$, $a_0(t)=0$, and $v=-1$, or can be obtained from the special case of $u_t+2\beta (t)u+[\alpha (t)+\beta (t)x]u_x-3c\gamma (t)u{u}_x+\gamma (t)u_{xxx}=0$ [35] with the time-dependent coefficients $\beta (t)=0$, $\alpha (t)=\delta (t)$, and the constant coefficient $c=-2$. To the best of our knowledge, the nonlinear fNIEs derived and solved by Ablowitz et al.’s algorithm [18,19] are all constant-coefficient equations, but a case with variable coefficients has not been reported yet.

In this paper, we shall derive and solve the vcfKdV-type equation with Riesz fractional-order derivative via IST. Because fractional-order media are “coarse” or multiscale media that are neither regular nor random, the combination of the Riesz fractional derivative and IST can be beneficial for solving nonlinear fNIEs in the context of fractional-order media. In Section 2, the considered vcfKdV-type equation is derived utilizing an anomalous dispersion relation by relating a class of fNIEs to the linear scattering problem. Section 3 lists some conclusions for the direct scattering problem and briefly details related research on spectral problems. Furthermore, a law for the transformation of scattering data with time is derived. In Section 4, we use the completeness of eigenfunctions to determine the explicit form of the vcfKdV-type equation. In Section 5, the corresponding GLM integrable equation is solved to create the n-soliton solution for the vcfKdV-type equation. In particular, the fractional one-soliton solution and two-soliton solution of the vcfKdV-type equation are computed in the absence of reflection potential, and the spatial structures of these one- and two-soliton solutions are analyzed. The obtained fractional soliton solutions could potentially aid in the explanation of hyperdispersive transport linked to nonlinear wave phenomena in mediums that are fractionally nonlinear.

2. The vcfKdV-Type Equation and Anomalous Dispersion Relation

Firstly, we consider the following Schrödinger spectral equation:

$${\vartheta }_{xx}(x,k)+u(x)\vartheta (x,k)=\lambda \vartheta (x,k),\hspace{0.33em}\left|x\right|<\infty ,$$

(3)

where $u(x)$ act as the potential and $\lambda$ is the spectral parameter, assumed to be $\lambda =-k^2$. Incorporate Equation (3) into the matrix spectral problems:

$${\vartheta }_x=M\vartheta , M=\left(\begin{array}{cc}0& 1\\ \lambda -u& 0\end{array}\right),$$

(4)

$${\vartheta }_t=N\vartheta , N=\left(\begin{array}{cc}Ћ& Ж\\ Й& -Ћ\end{array}\right),$$

(5)

where $Ћ$, $Ж$, and $Й$ are three functions of potential $\lambda$ and $u$ determined by the compatibility condition of matrices $M$ and $N$, i.e., the zero-curvature equation (ZCE) $M_t-N_x+[M,N]=0$ with $[M,N]=MN-NM$. As the above introduction of $-k^2$ provides a convention $\lambda =-k^2$ for $\lambda$, we note that in a case where $\lambda =-k^2>0$ is a real number, the corresponding solution $u$ of the vcfKdV equation to be derived later is a stable soliton solution. According to the ZCE combined with Equations (4) and (5), and Ablowitz et al.’s algorithm [18,19], we can relate the spectral problems (4) and (5) to a class of integrable NLEEs:

$$u_t+\theta (\varpi )u_x=0, \varpi \hspace{0.33em}=-{\displaystyle \frac{1}{4}}{\partial }_x^2-u-{\displaystyle \frac{1}{2}}{u}_x{\displaystyle {\int }_x^{\infty }dy},$$

(6)

where ${\displaystyle {\int }_x^{\infty }dy}$ operates on the function to which $\varpi$ is applied by integrating it.

Specifically, if $\theta (\varpi )$ is a rational function of $\varpi$, then we can use IST to solve Equation (6) through Equation (3). For the case $\theta (\varpi )={\varpi }^n$, given an arbitrary concrete value of $n(n=1,2,\cdots )$, we are able to obtain integer-order integrable KdV-type or higher-order KdV-type equations with variable coefficients, but we have to adjust $\theta (\varpi )$ for the derivation of the corresponding vcfKdV-type equation series. When assuming $\theta (\varpi )=[\sigma (t)-4\gamma (t)\varpi ]{\left|4\varpi \right|}^{\alpha }$, $\alpha \in [0,1)$; then, we can derive the vcfKdV-type equation:

$$u_t+{\left|4\varpi \right|}^{\alpha }[\sigma (t)u_x+\gamma (t)u_{xxx}+\beta (t)u{u}_x]=0,$$

(7)

where $\beta (t)=6\gamma (t)$. In fact, since $\theta (\varpi )$ is strongly related to the anomalous dispersion relation of the linearization part of Equation (6), then we substitute the formal solution $u=e^{ikx-i{\displaystyle {\int }_0^t\omega (k)dm}}$(here $\omega (k)$ ignores the variable $m$, and its derivative with respective to $t$ will also ignore $t$) in the linearization part of Equation (7):

$$u_t+{\left|-{\partial }_x^2\right|}^{\alpha }[\sigma (t)u_x+\gamma (t)u_{xxx}]=0,$$

(8)

where ${\left|-{\partial }_x^2\right|}^{\alpha }$ is the Riesz fractional derivative defined by [19]:

$${\left|-{\partial }_x^2\right|}^{\alpha }u(x,t)={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }\widehat{u}(x,t){\left|k^2\right|}^{\alpha }}{e}^{ikx}dk,$$

(9)

$$\widehat{u}(x,t)={\displaystyle {\int }_{-\infty }^{\infty }u(x,t)e^{-ikx}}dx.$$

(10)

Then, the dispersion relation $\omega (k)$ corresponding to Equation (8) yields

$$\omega (k)=\sigma (t)k{\left|k^2\right|}^{\alpha }-\gamma (t)k^3{\left|k^2\right|}^{\alpha }$$

(11)

Furthermore, the operator $\theta (\varpi )$ can be directly related to the dispersion relation of the linearization part of Equation (6). As $\varpi \to -{\partial }_x^2/4$ states that $\theta (\varpi )\to \theta (-{\partial }_x^2/4)$, this linearization is

$$u_t+\theta (-{\displaystyle \frac{1}{4}}{\partial }_x^2)u_x=0$$

(12)

By substituting $u=e^{ikx-i{\displaystyle {\int }_0^t\omega (k)dm}}$ into Equation (12), we have

$$\theta (k^2)={\displaystyle \frac{\omega (2k)}{2k}}$$

(13)

Further comparing Equation (11) with Equation (13), we arrive at

$$\theta (k^2)=[\sigma (t)-4\gamma (t)k^2]{\left|4k^2\right|}^{\alpha }$$

(14)

As a result, with the dispersion relation $\omega (k)=\sigma (t)k{\left|k^2\right|}^{\alpha }-\gamma (t)k^3{\left|k^2\right|}^{\alpha }$ and linear Equation (12), we have $\theta (\varpi )=[\sigma (t)-4\gamma (t)\varpi ]{\left|4\varpi \right|}^{\alpha }$. Thus, the derivation of vcfKdV-type Equation (7) is feasible.

3. IST for the vcfKdV-Type Equation

3.1. The Direct Scattering Problem

For ease of calculation, we replace $\lambda$ with $-k^2$; then, Equation (3) becomes

$${\vartheta }_{xx}(x,k)+k^2\vartheta (x,k)=-u(x)\vartheta (x,k)$$

(15)

Let $u(x)\in Z_1\cap Z_2$, where

$$Z_i=\{u(x):{\displaystyle {\int }_{-\infty }^{\infty }(1+{\left|x\right|}^i)\left|u(x)\right|}dx<+\infty \}, i=1,2$$

(16)

This means that $u(x)$ is defined along the entire real axis with necessary derivatives, and at infinity, $u(x)$ and all its derivatives rapidly approach zero. Then, for any $k$ whose imaginary part is non-negative, spectral Equation (15) contains a pair of fundamental solutions, also known as the Jost solutions ${\vartheta }_1(x,k)$ and ${\vartheta }_2(x,k)$, which are analytic in the upper half-plane of $k$ and continuous up to the real axis, satisfying the asymptotic conditions [36]:

$${\vartheta }_1(x,k)\approx e^{ikx}, x\to +\infty$$

(17)

$${\vartheta }_2(x,k)\approx e^{-ikx}, x\to -\infty$$

(18)

From the boundary value condition, we have

$$W({\vartheta }_1(x,-k),{\vartheta }_1(x,k))=2ik,$$

(19)

$$W({\vartheta }_2(x,-k),{\vartheta }_2(x,k))=-2ik,$$

(20)

where $W({\vartheta }_1(x,k),{\vartheta }_2(x,k))={\vartheta }_1(x,k){\vartheta }_{2,x}(x,k)-{\vartheta }_{1,x}(x,k){\vartheta }_2(x,k)$ represents the Wronskian of ${\vartheta }_1(x,k)$ and ${\vartheta }_2(x,k)$. When $k\ne 0$, we can see that the sets of fundamental solutions $\{{\vartheta }_1(x,-k),{\vartheta }_1(x,k)\}$ and $\{{\vartheta }_2(x,-k),{\vartheta }_2(x,k)\}$ are linearly independent; then, any solution of Equation (15) can be represented by a linear combination of them; thus, we obtain the following two expressions:

$${\vartheta }_2(x,k)=\mu (k){\vartheta }_1(x,-k)+\nu (k){\vartheta }_1(x,k)$$

(21)

$${\vartheta }_2(x,-k)=\nu (k){\vartheta }_1(x,-k)+\mu (-k){\vartheta }_1(x,k)$$

(22)

From Equations (21) and (22), we can express $\mu (k)$ and $\nu (k)$ as follows:

$$\mu (k)={\displaystyle \frac{W({\vartheta }_2(x,k),{\vartheta }_1(x,k))}{2ik}}$$

(23)

$$\nu (k)={\displaystyle \frac{W({\vartheta }_1(x,-k),{\vartheta }_2(x,k))}{2ik}}$$

(24)

We let

$$\stackrel{-}{E}(x,k)={\vartheta }_1(x,k)e^{ikx}, \stackrel{.}{E}={\vartheta }_1(x,-k)e^{ikx}$$

(25)

$$\stackrel{-}{I}(x,k)={\vartheta }_2(x,k)e^{ikx}, \stackrel{.}{I}={\vartheta }_2(x,-k)e^{ikx}$$

(26)

Then

$$\stackrel{-}{E}(x,k)\approx 1, \stackrel{.}{E}(x,k)\approx e^{2ikx}, x\to -\infty$$

(27)

$$\stackrel{-}{I}(x,k)\approx e^{2ikx}, \stackrel{.}{I}(x,k)\approx 1, x\to +\infty$$

(28)

Rewrite Equations (21) and (22) as

$${\displaystyle \frac{\stackrel{-}{E}(x,k)}{\mu (k)}}=\stackrel{.}{I}(x,k)+\rho (k)\stackrel{-}{I}(x,k)$$

(29)

$${\displaystyle \frac{\stackrel{.}{E}(x,-k)}{\mu (-k)}}=-\stackrel{-}{I}(x,k)+\overline{\rho }(k)\stackrel{.}{I}(x,k)$$

(30)

where $\rho (k)=\nu (k)/\mu (k)$ and $\overline{\rho }(k)=\nu (-k)/\mu (-k)$. When we let $\tau \left(k\right)=1/\mu (k)$, then $\rho (k)$ and $\tau (k)$ are the reflection and transmission coefficients, respectively. Moreover, there is the relationship:

$${\left|\rho (k)\right|}^2+{\left|\tau (k)\right|}^2=1$$

(31)

Here $\stackrel{-}{E}(x,k)$ and $\mu (k)$ can be extended analytically to the upper half-plane of $k$, and they tend to $1$ when $\left|k\right|\to \infty$. Meanwhile $\stackrel{.}{I}(x,k)$ can be extended analytically to the lower half-plane of $k$, which likewise tends to $1$ when $\left|k\right|\to \infty$. Instead, $\nu (k)$ is defined only on the real axis of the $k$-plane because that $\stackrel{.}{I}(x,k)$ is analytic in the lower half-plane of $k$. Thus, they are written separately as

$$\stackrel{-}{E}(x,k)=1+{\displaystyle {\int }_{-\infty }^x\frac{1}{2ik}}\{1-e^{2ik(x-ђ)}\}u(ђ)\stackrel{-}{E}(ђ,k)dђ$$

(32)

$$\stackrel{.}{I}(x,k)=1-{\displaystyle {\int }_x^{\infty }\frac{1}{2ik}}\{1-e^{2ik(x-ђ)}\}u(ђ)\stackrel{.}{I}(ђ,k)dђ$$

(33)

$$\mu (k)=1+{\displaystyle \frac{1}{2ik}}{\displaystyle {\int }_{-\infty }^{\infty }u(ђ)}\stackrel{-}{E}(ђ,k)dђ$$

(34)

$$\nu (k)=-{\displaystyle \frac{1}{2ik}}{\displaystyle {\int }_{-\infty }^{\infty }u(ђ)}\stackrel{-}{E}(ђ,k)e^{-2kђ}dђ$$

(35)

Moreover, $\mu (k)\to 0$ when $\left|k\right|\to \infty$ and $\mathrm{Im}k>0$. In the upper half-plane of $k$, the zeros $k_j(j=1,2,\cdots ,n)$ of the analytic function $\mu (k)$ are the discrete spectrum of the spectral Equation (15). Because of the Wronskian of ${\vartheta }_1(x,k_j)$ and ${\vartheta }_2(x,k_j)$ is zero, it shows that these two functions are linearly dependent:

$${\vartheta }_2(x,k_j)={Ъ}_j{\vartheta }_1(x,k_j)$$

(36)

where ${Ъ}_j$ is a constant. The analytic function $\mu (k)$ has a finite number of zeros at $k_1,k_2,\cdots ,k_n$ on the imaginary axis of the upper half-plane of $k$, that is, $k_j=i{ฟ}_j$, ${ฟ}_j>0$, $j=1,2,\cdots ,n$. At these eigenvalues, whether $x$ goes to positive or negative infinity, the corresponding eigenfunctions decay exponentially, and they are bound states.

In this way, the scattering data $S$ of spectral Equation (15) can be defined as:

$$S=\{\mathrm{Im}k=0, \rho (k)={\displaystyle \frac{\nu (k)}{\mu (k)}}, i{ฟ}_j, {ฤ}_j; j=0,1,2,\cdots ,n\}$$

(37)

where ${ฤ}_j^2=-i{Ъ}_j/{\mu }_k(i{ฟ}_j)$, ${ฤ}_j$ is the normalization factor of the eigenfunction $\vartheta (x,i{ฟ}_j)$, and

$${\displaystyle {\int }_{-\infty }^{\infty }{ฤ}_j^2{\vartheta }^2(x,i{ฟ}_j)dx=1}$$

(38)

3.2. Time Dependences of the Scattering Data

In this subsection, we determine how the scattering data (37) vary with time $t$. In Equation (5), we let $\vartheta ={(\vartheta ,\hspace{0.33em}{\vartheta }_x)}^T$; then, the development of the eigenfunction $\vartheta (x,k)$ over time is transformed into the following form:

$${\vartheta }_t=Ћ\vartheta +Ж{\vartheta }_x$$

(39)

For the integer-order integrable case, $Ћ$ and $Ж$ can be expressed as functions with respect to $\lambda$ and $u$, while for a case of fNIEs with a Riesz fractional derivative, $Ћ$ and $Ж$ cannot usually be given explicitly. In such a case, we need to give the constraints as $\left|x\right|\to \infty$:

$$Ћ\to 0$$

(40)

$$Ж\to -\theta (k^2)$$

(41)

Theorem 1.

Assuming $u(x,t)$ solves the vcfKdV Equation (7), then the scattering data

$$S(t)=\{\mathrm{Im}k=0, \rho (k,t)={\displaystyle \frac{\nu (k,t)}{\mu (k,t)}}, i{ฟ}_j(t), {ฤ}_j(t); j=0,1,2,\cdots ,n\}$$

(42)

for the spectral Equation (15) possess the following the time dependences:

$${ฟ}_j(t)={ฟ}_j$$

(43)

$${ฤ}_j^2(t)={ฤ}_j^2(0)e^{{\displaystyle {\int }_0^{t}2\{{ฟ}_j[\sigma (m)+4{ฟ}_j^2\gamma (m)]{(4{ฟ}_j^2)}^{\alpha }\}dm}}$$

(44)

$$\mu (k,t)=\mu (k,0), \nu (k,t)=\nu (k,0)e^{{\displaystyle {\int }_0^t\{-2ik[\sigma (m)-4k^2\gamma (m)]|4k^2{|}^{\alpha }\}dm}}$$

(45)

$$\rho (k,t)=\rho (k,0)e^{{\displaystyle {\int }_0^t\{-2ik[\sigma (m)-4k^2\gamma (m)]|4k^2{|}^{\alpha }\}dm}}$$

(46)

Proof. 

We introduce a function $\Xi$ about $t$ that will be determined later, and modify Equation (39) to

$${\vartheta }_t=(Ћ+\Xi )\vartheta +Ж{\vartheta }_x$$

(47)

It is not difficult to verify that the compatibility condition ${\vartheta }_{txx}={\vartheta }_{xxt}$ of Equations (15) and (39) is the same as that of Equations (15) and (47). Therefore, the same compatibility condition can only be equivalent to the same fNIE, namely the vcKdV Equation (7). In fact, the purpose of introducing $\Xi$ is to utilize Equations (15) and (47), which are equivalent to Equations (15) and (39), resulting from its modification, in order to determine the corresponding scattering data.

By utilizing the asymptotic property $\vartheta (x,k)\approx e^{-ikx}$ when $x\to -\infty$ and those in Equations (40) and (41), from Equation (47), we have

$$\Xi =-ik\theta (k^2)$$

(48)

where $\theta (k^2)=[\sigma (t)-4k^2\gamma (t)]{\left|4k^2\right|}^{\alpha }$ corresponds to the vcfKdV Equation (7). It is easy to see from Equations (17) and (18) that ${\vartheta }_1(x,k)$ and ${\vartheta }_1(x,-k)$ are linearly independent. We insert the linear combination $\vartheta (x,k)=\mu (k,t){\vartheta }_1(x,k)+\nu (k,t){\vartheta }_1(x,-k)$, together with two undetermined functions $\mu (k,t)$ and $\nu (k,t)$, into Equation (47) and let $x\to +\infty$; then, we can obtain

$${\mu }_t(k,t)e^{-ikx}+{\nu }_t(k,t)e^{ikx}=\Xi [\mu (k,t)e^{-ikx}+\nu (k,t)e^{ikx}]+Ж[-ik\mu (k,t)e^{-ikx}+ik\nu (k,t)e^{ikx}]$$

(49)

Thus, we have

$${\mu }_t(k,t)=0, {\nu }_t(k,t)=-2ik[\sigma (t)-4k^2\gamma (t)]{\left|4k^2\right|}^{\alpha }\nu (k,t)$$

(50)

Since ${\mu }_t(k,t)=0$, all zeros $k_j=i{ฟ}_j$, ${ฟ}_j>0$, $j=1,2,\cdots ,n$ of $\mu (k,t)$ are independent of $t$, that is,

$${ฟ}_{j,t}=0$$

(51)

Considering the composition of the normalization factor ${ฤ}_j^2=-i{Ъ}_j/{\mu }_k(i{ฟ}_j)$, we have to determine the variation in ${Б}_j$ with time $t$. When $k=i{ฟ}_j$, we have $\vartheta (x,k)\approx {Б}_j{e}^{-{ฟ}_{j}x}$ as $x\to +\infty$. Then, from Equation (47), we obtain

$${Б}_{j,t}=(\Xi -{ฟ}_jЖ){Б}_j$$

(52)

As a result, we have

$${ฤ}_{j,t}(t)={ฤ}_j(t)\{{ฟ}_j[\sigma (t)+4{ฟ}_j^2\gamma (t)]{(4{ฟ}_j^2)}^{\alpha }\}$$

(53)

Directly solving Equations (50)–(53), we reach Equations (43)–(45). Finally, the reflection coefficient $\rho (k,t)$ can be determined by

$$\rho (k,t)={\displaystyle \frac{\nu (k,t)}{\mu (k,t)}}={\displaystyle \frac{\nu (k,0)}{\mu (k,0)}}{e}^{{\displaystyle {\int }_0^t(-2ik(\sigma (m)-4k^2\gamma (m))|4k^2{|}^{\alpha })dm}}=\rho (k,0)e^{{\displaystyle {\int }_0^t(-2ik(\sigma (m)-4k^2\gamma (m))|4k^2{|}^{\alpha })dm}}$$

(54)

We finish the proof. □

3.3. Reconstruction of Potential

Based on the time dependences (42)–(46) of the scattering data (41), we can obtain the solution of vcfKdV-type Equation (7) by solving the GLM integral equation.

Theorem 2.

Based on the scattering data (42), we can construct the GLM integrable equation:

$$M(x,y)+N(x+y)+{\displaystyle {\int }_x^{\infty }M(x,s)N(s+y)}ds=0$$

(55)

$$N(x)={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }\rho (t,k)e^{ikx}dk+}{\displaystyle \sum _{j=1}^n{ฤ}_j^2(t)e^{-{ฟ}_{j}x}}$$

(56)

where  $n$  is the number of zeros of the function $\mu (k,t)$; then, we have the formal solution of vcfKdV-type Equation (7):

$$u(x,t)=2{\displaystyle \frac{\partial }{\partial x}}M(x,x,t)$$

(57)

where $M(x,x,t)=M(x,y,t){|}_{y=x}$ can be calculated from Equation (55) , and ${ฤ}_j(t)$  is determined by Equation (44).

Proof. 

The derivation of Equations (56) and (57) is the same as that [36] of the integer-order case, and we will not repeat it here, only determining $M(x,x,t)$ in Equation (57). In order to give the explicit form of $M(x,x,t)$, we set $\nu (k,0)=0$ such that the reflection coefficient $\rho (k,t)=0$; then, Equation (55) can be rewritten as:

$$M(t,x,y)+{\displaystyle \sum _{j=1}^n{ฤ}_j^2}(t)e^{-{ฟ}_j(x+y)}+{\displaystyle \sum _{j=1}^n{ฤ}_j^2}(t)e^{-{ฟ}_{j}y}{\displaystyle {\int }_x^{\infty }M(t,x,s)}{e}^{-{ฟ}_{j}s}ds=0$$

(58)

Supposing that

$$M(t,x,y)={\displaystyle \sum _{m=1}^n{ฤ}_m}(t)g_m(x)e^{-{ฟ}_{m}y}$$

(59)

and substituting Equation (59) into Equation (58), we can transform the resulting equation into a system of linear algebraic equations about $g_1(x)$, $g_2(x)$, …, and $g_n(x)$. Then, we further eliminate the common factors $e^{-{ฟ}_{1}y}$, $e^{-{ฟ}_{2}y}$, …, and $e^{-{ฟ}_{n}y}$ from these $n$ equations, respectively; then, we can obtain

$$M(x,x,t)={\displaystyle \frac{\partial }{\partial x}}\ln \hspace{0.33em}\det H(t,x)$$

(60)

with

$$H(t,x)={(h_{ij}(t,x))}_{i\times j}, h_{ij}(t,x)={\delta }_{ij}+{\displaystyle \frac{1}{{ฟ}_i+{ฟ}_j}}{ฤ}_i(t){ฤ}_j(t)e^{-({ฟ}_i+{ฟ}_j)x}$$

(61)

where ${\delta }_{ij}=1$ when $i=j$; otherwise, ${\delta }_{ij}=0$ for $i\ne j$. Thus, with the help of Equations (57) and (60), we can obtain the n-soliton solution of vcfKdV-type Equation (7). The specific one-soliton solution and two-soliton solution will be derived in Section 5. □

4. Explicit Form of the vcfKdV-Type Equation

Following the steps in [2], we introduce the squared eigenfunctions of the $\varpi$ operator, which are any of the three functions $\{{\partial }_x{\vartheta }_1^2,{\partial }_x{\vartheta }_2^2,{\partial }_x({\vartheta }_1{\vartheta }_2)\}$, generally represented by $\mathcal{№}(x,k,t)$, where ${\vartheta }_1$ and ${\vartheta }_2$ solve Equation (15) under asymptotic conditions (17) and (18) as $x\to \pm \infty$. Then, we have

$$\theta (\varpi )\mathcal{№}(x,k,t)=\theta (k^2)\mathcal{№}(x,k,t)$$

(62)

In order to apply $\theta (\varpi )=[\sigma (t)-4\gamma (t)\varpi ]{\left|4\varpi \right|}^{\alpha }$ to any function, we need to represent this function in terms of $\mathcal{№}(x,k,t)$. As pointed out in [2], this means that we need a completeness relation for this set of eigenfunctions. It is known [37] that the eigenfunctions $\mathcal{№}(x,k,t)$ are complete; then, a sufficiently regular function $\Theta (x)$ can be expanded into the form expressed by the eigenfunctions $\mathcal{№}(x,k,t)$:

$$\Theta (x)={\displaystyle {\int }_{{\Gamma }_{\infty }}dk\frac{{\tau }^2(k)}{4\pi ik}{\displaystyle {\int }_{-\infty }^{\infty }dyЦ(x,y,k)}}\Theta (y)$$

(63)

where

$$Ц(x,y,k)={\partial }_x[{\vartheta }_2^2(x,k){\vartheta }_1^2(y,k)-{\vartheta }_1^2(x,k){\vartheta }_2^2(y,k)], {\Gamma }_{\infty }=\underset{R\to \infty }{\lim }{\Gamma }_R$$

(64)

${\Gamma }_R$ is the semicircular contour integral in the upper half-plane from $k=-R$ to $k=R$, and $\tau \left(k\right)=1/\mu (k)$ is the transmission coefficient introduced earlier. Then, we let the operator $\theta (\varpi )$ act on the function $\Theta (x)$ in Equation (63), while taking into account Equation (62), to obtain:

$$\theta (\varpi )\Theta (x)={\displaystyle {\int }_{{\Gamma }_{\infty }}dk\theta (k^2)\frac{{\tau }^2(k)}{4\pi ik}{\displaystyle {\int }_{-\infty }^{\infty }dyЦ(x,y,k)}}\Theta (y)$$

(65)

Thus, combining Equations (7) and (65), we use the operator $\theta (\varpi )$ to act on the function $u_x$, and then obtain:

$$[\sigma (t)-4\gamma (t)\varpi ]{\left|4\varpi \right|}^{\alpha }{u}_x={\displaystyle {\int }_{{\Gamma }_{\infty }}dk{\left|4k^2\right|}^{\alpha }\frac{{\tau }^2(k)}{4\pi ik}{\displaystyle {\int }_{-\infty }^{\infty }dyЦ(x,y,k)}}[\sigma (t)u_y+\gamma (t)u_{yyy}+\beta (t)u{u}_y]$$

(66)

where $\beta (t)=6\gamma (t)$. Then, we obtain the explicit form of vcfKdV-type Equation (7):

$$u_t+{\displaystyle {\int }_{{\Gamma }_{\infty }}dk{\left|4k^2\right|}^{\alpha }\frac{{\tau }^2(k)}{4\pi ik}{\displaystyle {\int }_{-\infty }^{\infty }dyЦ(x,y,k)}}[\sigma (t)u_y+\gamma (t)u_{yyy}+6\gamma (t)u{u}_y]=0$$

(67)

In particular, as $\alpha =0$, Equation (67) reduces to the integer-order vcKdV-type equation:

$$u_t+\sigma (t)u_x+\gamma (t)u_{xxx}+6\gamma (t)u{u}_x=0$$

(68)

On this basis, Equation (67) reduces to the classical KdV equation by setting $\sigma (t)=0$ and $\gamma (t)=1$:

$$u_t+u_{xxx}+6u{u}_x=0$$

(69)

In addition, as $\alpha \ne 0$, $\sigma (t)=0$, and $\gamma (t)=1$, we obtain the explicit form of the fKdV version [18] of Equation (69):

$$u_t+{\displaystyle {\int }_{{\Gamma }_{\infty }}dk{\left|4k^2\right|}^{\alpha }\frac{{\tau }^2(k)}{4\pi ik}{\displaystyle {\int }_{-\infty }^{\infty }dyЦ(x,y,k)}}(u_{yyy}+6u{u}_y)=0$$

(70)

5. Fractional Soliton Solutions of the vcfKdV-Type Equation

In this section, we explore fractional one- and two-soliton solutions for the vcfKdV-type equation and analyze the structural features of the soliton solutions.

5.1. One-Soliton Solution of the vcfKdV-Type Equation

Considering Equations (57) and (60), we have

$$u(x,t)=2{\displaystyle \frac{{\partial }^2}{\partial x^2}}\ln \det H(t,x)$$

(71)

If we let $n=1$ and substitute Equations (43), (44), and (61) into Equation (71), we can obtain the fractional one-soliton solution to the vcfKdV-type Equation (7):

$$u(x,t)=2{ฟ}_1^2{\mathrm{sech}}^2{ฟ}_1\{x-x_0-{\displaystyle {\int }_0^t[{(4{ฟ}_1^2)}^{\alpha }\sigma (m)+{(4{ฟ}_1^2)}^{1+\alpha }\gamma (m)]dm}\}$$

(72)

Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 simulate the spatial structures and wave propagation of the fractional one-soliton solution (72) with fixed ${ฟ}_1=1.5$ and $x_0=0$, different variable coefficients $\sigma (t)$ and $\gamma (t)$, and fractional-order values of $\alpha$. In Figure 1, we select $\sigma (t)=0.5$ and $\gamma (t)=0.1$ to simulate the three-dimensional structure of the fractional one-soliton solution (72), with $\alpha$ taken as $0$, $0.35$, and $0.7$. We can clearly see that any of these different values of $\alpha$ do not affect the amplitude of the fractional one-soliton, but they do affect the width of the linear trajectory on the coordinate plane and the speed of the fractional one-soliton. More specifically, as $\alpha$ increases, both the width of the linear trajectory and the speed of fractional soliton increase. Figure 2 simulates the wave propagation of the fractional one-soliton solution at different times, $t=-1$ and $t=1$, where the same parameters as in Figure 1 are chosen. In Figure 2, it can be seen that the direction of fractional soliton motion is from the negative $x$-axis through the origin to the positive $x$-axis, with the amplitude and width unchanged, indicating that there is no diffusible dissipation during the motion process.

In Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, the different variable coefficients $\sigma (t)$ and $\gamma (t)$ are set to $\sigma (t)=0.5+0.5t$ and $\gamma (t)=0.1+0.1t\hspace{0.33em}$ in Figure 3 and Figure 4; to $\sigma (t)=\sin t$ and $\gamma (t)=\cos t\hspace{0.33em}$ in Figure 5 and Figure 6; and to $\sigma (t)=e^t$ and $\gamma (t)=\cos t\hspace{0.33em}$ in Figure 7 and Figure 8, respectively. We can easily see that when the variable coefficients $\sigma (t)$ and $\gamma (t)$ are taken as functions of $t\hspace{0.33em}$, the fractional solitons traveling back and forth along the $x$-axis represent periodic changes in the velocity direction. Therefore, the trajectories formed by the fractional solitons on the coordinate plane are not on straight lines. In Figure 3, it is shown that the trajectory is parabolic in shape. As the fractional value $\alpha$ increases, the amplitude remains constant, the opening of the parabolic trajectory gradually narrows, the width decreases, and the overall symmetric structure is maintained. In Figure 5, the trajectory exhibits a sine/cosine shape, and the amplitude remains constant with the increase in $\alpha$. It can be observed that the trajectory appears to have a cyclic posture and the period of motion does not change with $\alpha$. However, the distance from the wave peaks to troughs increases with the increase in $\alpha$. In Figure 7, it can be clearly seen that the soliton has a local period in the interval $t<0$. Similar to other cases, the amplitude still does not change with the increase in $\alpha$. When $t>0$, the trajectory on the coordinate plane becomes longer with the increase in $\alpha$. In Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, a common feature is that the trajectory formed by the fractional soliton has a clustering process at the turning point, which is the brighter part of the top-view image. This may be the reason for the change in the direction of motion of the fractional soliton moving in a straight line.

From a mathematical perspective, by analyzing expression (72), it is not difficult to see that the amplitude of the one-soliton solution determined by Equation (72) is a constant value $2{ฟ}_1^2$, and the velocity $v$ propagating along the $x$-axis is:

$$v=4^{\alpha }{({{ฟ}_1}^2)}^{\alpha }\sigma (t)+4^{1+\alpha }{({{ฟ}_1}^2)}^{1+\alpha }\gamma (t)$$

(73)

This also indicates that the velocity of the one-soliton solution in Figure 1 and Figure 2 is constant, where we set $\sigma (t)=0.5$ and $\gamma (t)=0.1$. In Figure 9, the velocity (73) corresponding to the one-soliton solution in Figure 5 and Figure 6 is shown. We can see that the three time-varying velocities depicted in Figure 9 are completely consistent with the propagation of the solitons in Figure 5 and Figure 6. It is evident that there is a significant numerical difference between the velocities of the integer-order soliton and that of fractional soliton. However, for small-fractional order $\alpha$, such numerical differences are also small in the interval $t\in [1,5]$, as shown in

Table 1. Numerical difference in the velocities of integer-order and fractional-order solitons.

Time $\mathit{t}$	$\mathbf{Value} \mathbf{of} \mathit{v}{|}_{\mathit{\alpha }=0.0001}-\mathit{v}{|}_{\mathit{\alpha }=0}$	$\mathbf{Value} \mathbf{of} \mathit{v}{|}_{\mathit{\alpha }=0.002}-\mathit{v}{|}_{\mathit{\alpha }=0}$
1	7.25164 × 10–4	1.45336 × 10–2
2	1.14268 × 10–3	2.29014 × 10–2
3	1.5602 × 10–3	3.12692 × 10–2
4	1.97772 × 10–3	3.69371 × 10–2
5	2.39254 × 10–3	4.80049 × 10–2

.

5.2. Two-Soliton Solutions of the vcfKdV-Type Equation

If we let $n=2$, we obtain the two-soliton solution to vcfKdV-type Equation (7):

$$u(x,t)={\displaystyle \frac{8{ฟ}_1^2{e}^{\upsilon }+8{ฟ}_2^2{e}^{\varsigma }+8{ฟ}_1^2фe^{\upsilon +2\varsigma }+8{ฟ}_2^2фe^{2\upsilon +\varsigma }+16{({ฟ}_1-{ฟ}_2)}^2{e}^{\upsilon +\varsigma }}{{(1+e^{\upsilon }+e^{\varsigma }+фe^{\upsilon +\varsigma })}^2}}$$

(74)

where

$$\upsilon =2{ฟ}_1{\displaystyle {\int }_0^t[{(4{ฟ}_1^2)}^{\alpha }\sigma (m)+{(4{ฟ}_1^2)}^{1+\alpha }\gamma (m)]dm}-2{ฟ}_{1}x$$

(75)

$$\varsigma =2{ฟ}_2{\displaystyle {\int }_0^t[{(4{ฟ}_2^2)}^{\alpha }\sigma (m)+{(4{ฟ}_2^2)}^{1+\alpha }\gamma (m)]dm}-2{ฟ}_{2}x$$

(76)

$$ф={\left({\displaystyle \frac{{ฟ}_1-{ฟ}_2}{{ฟ}_1+{ฟ}_2}}\right)}^2$$

(77)

Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 simulate the spatial structures and wave propagation of the fractional two-soliton solution (74) with fixed ${ฟ}_1=1.5$ and ${ฟ}_2=1$, different variable coefficients $\sigma (t)$ and $\gamma (t)$, and fractional-order values of $\alpha$. In Figure 10, we select the parameters $\sigma (t)=0.1$ and $\gamma (t)=0.1$ to simulate the three-dimensional structure of the fractional two-soliton solution (74), with $\alpha$ taken as $0$, $0.35$, and $0.6$.

From Figure 10 and Figure 11, we can see that the fractional two solitons pass through each other during the collision process, there is no dissipation or diffusion during the collision process, and as the fractional value $\alpha$ increases, the X-shaped trajectories on the coordinate plane formed by the two solitons rotate in a clockwise direction and the angle of the pinch becomes larger with the increase in $\alpha$. By using Figure 10 and Figure 11, we can analyze the direction of motion of the fractional two-soliton solution. As time $t$ increases, the two solitons travel from the negative $x$-axis to the positive $x$-axis and the amplitudes change during the collision process, but in the end, they all revert back to their pre-collision states. It can be seen in Figure 10 and Figure 11 that the direction of fractional soliton motion is from the negative $x$-axis to the positive $x$-axis, with the amplitude and width unchanged compared to before the collision, indicating that there is no diffusible dissipation during the motion process, including collision.

In Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17, the variable coefficients $\sigma (t)$ and $\gamma (t)$ are set to $\sigma (t)=0.5+t$ and $\gamma (t)=0.1+0.1t\hspace{0.33em}$ in Figure 12 and Figure 13; $\sigma (t)=\sin t$ and $\gamma (t)=\cos t\hspace{0.33em}$ in Figure 14 and Figure 15; and $\sigma (t)=e^t$ and $\gamma (t)=\cos t\hspace{0.33em}$ in Figure 16 and Figure 17, respectively. In Figure 12, the trajectory on the coordinate plane formed by the fractional two-soliton solution is in the form of two parabolic collisions which, instead of passing through each other, bounce back during the collision, with the size of the parabolic “opening” decreasing as $\alpha$ increases. In Figure 14, the sine/cosine trajectories on the coordinate plane formed by the fractional two-soliton solution have periodic collisions. In every collision, the fractional solitons will pass through each other. With the increase in $\alpha$, the distance from the wave crest to the trough of the sine/cosine trajectory increases, that is, the corresponding velocity of fractional soliton motion increases. In Figure 16, the fractional two-soliton solution locally assumes a periodic form, and similar to those in Figure 14, the increase in $\alpha$ is accompanied by a similar increase in the velocity of the fractional two-soliton solution’s motion.

In Figure 10, Figure 12, Figure 14 and Figure 16, we can see that choosing different parameters can control the characteristics of the trajectory on the coordinate plane formed by the fractional two-soliton solution. When the selection of the variable coefficients $\sigma (t)$ and $\gamma (t)$ are both periodic functions, the trajectories on the coordinate plane formed by the fractional two solitons are all fully periodic. When one of $\sigma (t)$ or $\gamma (t)$ is periodic, a part of the trajectory exhibits periodic characteristics. Similar to the fractional one-soliton solution, when the selection of the variable coefficients $\sigma (t)$ and $\gamma (t)$ is related to time $t$, the velocity of fractional two-soliton solution does not follow invariance, and both the direction and quantity of velocity are time-varying. Other than that, from Figure 11, Figure 13, Figure 15 and Figure 17, it is found that the amplitude of fractional two-soliton solution is not affected by the fractional order $\alpha$.

6. Conclusions

We have generalized Ablowitz et al.’s algorithm [18,19] for extending IST to solve vcfKdV-type Equation (7) defined by the Riesz fractional-order derivative. As for the results of this article, we would like to mention the following: (i) we determine the effective utilization of the dispersion relation $\omega (k)$ of one-dimensional integrable fNLEEs as well as the formal solution $u=e^{ikx-i{\displaystyle {\int }_0^t\omega (k)dm}}$ to derive the vcfKdV-type Equation (7); (ii) we derive the explicit form (67) for vcfKdV-type Equation (7) using the completeness of eigenfunctions; and (iii) fractional-order n-soliton expression (71) combined with Equations (60) and (61) for vcfKdV-type Equation (7) are obtained by IST, and the structural characteristics of its fractional one- and two-soliton solutions are investigated with different parameters chosen. As for our research findings, we highlight the following: (i) the fractional order $\alpha$ does not affect the amplitude of the fractional solitons; (ii) the overall shape of the trajectories on the coordinate plane formed by the fractional solitons is affected by the fractional order $\alpha$; (iii) the velocity of the fractional solitons is affected by not only the variable coefficients $\sigma (t)$ and $\gamma (t)$, but also the fractional order $\alpha$; (iv) for the different choices of other parameters, the effects of the fractional order $\alpha$ on the soliton structure are different (for example, the trajectories in Figure 1 and Figure 10 rotate clockwise with the increase in the fractional order $\alpha$, and the trajectories in Figure 3 and Figure 12 decrease the parabolic “openings” with the increase in the fractional order $\alpha$), but there is no dissipation and diffusion in the complete propagation process. The ideas presented in this paper can also be generalized to derive and solve other integrable fNLEEs with variable coefficients, opening up a new direction in the dynamical behavior of variable-coefficient fractional-order inverse scattering integrable systems, which may provide help for the prediction of hyperdispersive transmission of nonlinear waves in fractional-order nonlinear media. In addition to the results and findings obtained above, the main theoretical novelty of this article lies in the feasibility of using vcfKdV-type Equation (7) as an example to extend the algorithm [18,19] proposed by Ablowitz et al. of IST to fractional-order integrable systems with coefficient functions.