Research Article | Open Access
Sheng Zhang, Yuanyuan Wei, Bo Xu, "Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example", Complexity, vol. 2019, Article ID 7952871, 9 pages, 2019. https://doi.org/10.1155/2019/7952871
Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example
In this paper, the spectral transform with the reputation of nonlinear Fourier transform is extended for the first time to a local time-fractional Korteweg-de vries (tfKdV) equation. More specifically, a linear spectral problem associated with the KdV equation of integer order is first equipped with local time-fractional derivative. Based on the spectral problem with the equipped local time-fractional derivative, the local tfKdV equation with Lax integrability is then derived and solved by extending the spectral transform. As a result, a formula of exact solution with Mittag-Leffler functions is obtained. Finally, in the case of reflectionless potential the obtained exact solution is reduced to fractional n-soliton solution. In order to gain more insights into the fractional n-soliton dynamics, the dynamical evolutions of the reduced fractional one-, two-, and three-soliton solutions are simulated. It is shown that the velocities of the reduced fractional one-, two-, and three-soliton solutions change with the fractional order.
Since the increasing interest on fractional calculus and its applications, dynamical processes and dynamical systems of fractional orders have attracted much attention. In 2010, Fujioka et al.  investigated soliton propagation of an extended nonlinear Schröinger equation with fractional dispersion term and fractional nonlinearity term. In 2014, Yang et al.  used a local fractional KdV equation to model fractal waves on shallow water surfaces.
In the field of nonlinear mathematical physics, the spectral transform  with the reputation of nonlinear Fourier transform is a famous analytical method for constructing exact and explicit n-soliton solutions of nonlinear partial differential equations (PDEs). Since put forward by Gardner et al. in 1967, the spectral transform method has achieved considerable developments [4–26]. With the close attentions of fractional calculus and its applications [27–53], some of the natural questions are whether the existing methods like those in [54–70] in soliton theory can be extended to nonlinear PDEs of fractional orders and what about the fractional soliton dynamics and integrability of fractional PDEs. As far as we know there are no research reports on the spectral transform for nonlinear PDEs of fractional orders. This paper is motivated by the desire to extend the spectral transform to nonlinear fractional PDEs and then gain more insights into the fractional soliton dynamics of the obtained solutions. For such a purpose, we consider the following local tfKdV equation:Here we note that if then eq. (1) becomes the celebrated KdV equation . In eq. (1), the local time-fractional derivative at the point is defined as where ; some useful properties  of the local time-fractional derivative have been used in this paper.
The rest of this paper is organized as follows. In Section 2, we derive the local tfKdV eq. (1) by introducing a linear spectral problem equipped with local time-fractional derivative. In Section 3, we construct fractional n-soliton solution of the local tfKdV eq. (1) by extending the spectral transform method. In Section 4, we investigate the dynamical evolutions of the obtained fractional one-soliton solution, two-soliton solution, and three-soliton solution. In Section 5, we conclude this paper.
2. Derivation of the Local tfKdV Equation
Theorem 1. The local tfKdV eq. (1) is a Lax system, which can be derived from the linear spectral problem equipped with a local time-fractional evolution equation:where and are all differentiable functions with respect to and , the spectral parameter is independent of , and is an arbitrary constant.
Proof. Taking the time-fractional derivative of eq. (3) yieldsSubstituting eq. (4) into eq. (5), we haveTaking the derivative of eq. (4) with respect to twice givesWith the help of eq. (3), from eq. (7) we have On the other hand, at the aribitrary point we haveFinally, using eqs. (6), (8), and (9) we arrive at eq. (1). Thus, we finish the proof. The process of proof shows that eq. (1) is a Lax integrable system.
3. Local Fractional Spectral Transform
Since the local tfKdV eq. (1) is a local time-fractional system, the orders of the derivatives with respect to space variable are integers. So, all the existing results about the spectral problem (3), a part of the Lax pair for the classical KdV equation, can be translated to the local tfKdV eq. (1).
Lemma 2. If the real potential defined in the whole real axis and its various derivatives are differentiable functions which vanishes rapidly as and satisfiesthen the linear spectral problem (3) has a set of basic solutions called Jost solutions and , and they are not only bounded for all values of but also analytic for and continuous for and have the following asymptotic properties:
Lemma 3. Define the Wronskianand let Thenwhere is analytic for and continuous for , is defined only on the real axis , and the analytic function has a finite number of simple zeros .
Lemma 4. For the linear spectral problem (3), there exists a constant , such that
Definition 5. The constant satisfying eq. (17) is named the normalization constant for the eigenfunction , and is named normalization eigenfunction.
Definition 6. The setis named the scattering data of the linear spectral problem (3).
For the time dependence of the scattering data, we have the following Theorem 8.
Proof. Substituting eqs. (12) and (14) into eq. (3) and using the asymptotic properties of eqs. (11) and (12) as and , respectively, we haveNamely,It is easy to see that all the zeros of are independent of because of . Therefore, we arrive at .
Similarly, substituting eq. (16) into eq. (3) and using the asymptotic property of eq. (11) as , we have In view of eqs. (17) and (28), we obtainwhich can be finally reduced to the second term of eq. (22). Then we finish the proof.
For the inverse scattering problem, we have the following Theorem 9.
Proof. The process of the proof of Theorem 9 is similar to that of the classical KdV equation  with integer order, and the only differences are the scattering data. To avoid unnecessary repetition, we omit it here.
For the fractional n-soliton solution, we have the following Theorem 10.
Proof. Firstly, we further determine the scattering data. Solving eqs. (22) and (23) yields Secondly, we let . In this case of reflectionless, eq. (32) reduces toSuppose that eq. (39) has a separation solutionwhere is an undetermined function which can be determined by the substitution of eq. (40) into eq. (39). With the determined function , we haveFinally, from eqs. (31), (37), (38), and (41) we obtain eq. (34). Therefore, the proof is over.
4. Fractional Soliton Dynamics
In order to gain more insights into the soliton dynamics of the obtained fractional n-soliton solution (34), we consider the cases of .
When , we haveand we, hence, obtain, from eq. (34), the fractional one-soliton solution:
Similarly, when we obtain the fractional two-soliton solution:
When , we obtain the fractional three-soliton solution:
In Figure 1, we simulate the fractional one-soliton solution (43) with different values of , where the parameters are selected as and . With the help of velocity images in Figure 2 and the formula of velocitywe can see that the bell-shaped solitons have different velocities depending on the values of . At the initial stage, the smaller the value of is selected, the faster the soliton propagates. But soon it was the opposite; for more details see Figures 3 and 4.
For the fractional two-solitons and three-solitons determined, respectively, by solutions (44) and (45), similar features shown in Figures 5–7 are observed. In Figures 5 and 6, we select the parameters as , , , and . While the parameters in Figure 7 are selected as , , , , , .
In summary, we have derived and solved the local tfKdV eq. (1) in the fractional framework of the spectral transform method. This is due to the linear spectral problem (3) equipped with the local time-fractional evolution (4). As for the fractional derivatives, there are many definitions  except the local fractional derivative, such as Grünwald–Letnikov fractional derivative, Riemann-Liouville fractional derivative and Caputo’s fractional derivative. Generally speaking, whether or not the spectral transform can be extended to some other nonlinear evolution equations with another type of fractional derivative depends on whether fractional derivative has the good properties required by the spectral transform method. To the best of our knowledge, combined with the Mittag-Leffler functions the obtained exact solution (31), the fractional n-soliton solution (34), and its special cases, the fractional one-, two-, and three-soliton solutions (43)-(45), are all new, and they have not been reported in literature. It is graphically shown that the fractional order of the local tfKdV eq. (1) influences the velocity of the fractional one-soliton solution (43) with Mittag-Leffler function in the process of propagations. More importantly, the fractional scheme of the spectral transform presented in this paper for constructing n-soliton solution of the local tfKdV eq. (1) can be extended to some other integrable local time-fractional PDEs.
The data in the paper are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
This work was supported by the Natural Science Foundation of China (11547005), the Natural Science Foundation of Liaoning Province of China (20170540007), the Natural Science Foundation of Education Department of Liaoning Province of China (LZ2017002), and Innovative Talents Support Program in Colleges and Universities of Liaoning Province (LR2016021).
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