# Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example

Authors: Sheng Zhang; Yuanyuan Wei; Bo Xu
Journal: Complexity
 (2019)

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## Abstract

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 fractionaln-soliton solution. In order to gain more insights into the fractionaln-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.


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## Body

## 1. Introduction

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. [1] investigated soliton propagation of an extended nonlinear Schröinger equation with fractional dispersion term and fractional nonlinearity term. In 2014, Yang et al. [2] used a local fractional KdV equation to model fractal waves on shallow water surfaces.In the field of nonlinear mathematical physics, the spectral transform [3] with the reputation of nonlinear Fourier transform is a famous analytical method for constructing exact and explicitn-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:(1)Dtαu+6uux+uxxx=0,0<α≤1.Here we note that if α=1 then eq. (1) becomes the celebrated KdV equation ut+6uux+uxxx=0. In eq. (1), the local time-fractional derivative Dtαu at the point t=t0 is defined as [30](2)Dtαux,t0=∂αux,t∂tαt=t0=limt→t0⁡Δαux,t-ux,t0t-t0α,where Δα(u(x,t)-u(x,t0))≅Γ(1+α)(u(x,t)-u(x,t0)); some useful properties [33] of the local time-fractional derivative have been used in this paper.The rest of this paper is organized as follows. In Section2, we derive the local tfKdV eq. (1) by introducing a linear spectral problem equipped with local time-fractional derivative. In Section 3, we construct fractionaln-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

For the local tfKdV eq. (1), we have the following Theorem 1.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:(3)ϕxx=η-uϕ,η=-k2,(4)Dtαϕ=ux+ϑϕ+4k2-2uϕx,where ϕ=ϕ(x,t) and u=u(x,t) are all differentiable functions with respect to x and t, the spectral parameter η is independent of t, and ϑ is an arbitrary constant.Proof.
Taking the time-fractional derivative of eq. (3) yields(5)Dtαϕxx=-ϕDtαu-u+k2Dtαϕ,Substituting eq. (4) into eq. (5), we have(6)Dtαϕxx=-Dtαu+ux+ϑu+k2ϕ-4k2-2uu+k2ϕx,Taking the derivative of eq. (4) with respect to x twice gives(7)Dtαϕxx=uxxxϕ-3ux-ϑϕxx+4k2-2uϕxxx,With the help of eq. (3), from eq. (7) we have (8)Dtαϕxx=uxxx+5u-k2ux-ϑu+k2ϕ-4k2-2uu+k2ϕx.
On the other hand, at the aribitrary pointt=t0 we have(9)Dtαϕxxx,t0=limt→t0⁡Δαϕxxx,t-ϕxxx,t0t-t0α,Dtαϕx,t0xx=limt→t0⁡Δαϕx,t-ϕx,t0xxt-t0α.
Finally, 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 x 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).For the direct scattering problem, we translate some necessary results and definitions [9] to the local tfKdV eq. (1).Lemma 2.
If the real potentialu(x) defined in the whole real axis -∞<x<∞ and its various derivatives are differentiable functions which vanishes rapidly as x→±∞ and satisfies(10)∫-∞+∞xjuxdx<+∞,j=0,1,2,then the linear spectral problem (3) has a set of basic solutions called Jost solutions ϕ+(x,k) and ϕ-(x,k), and they are not only bounded for all values of x but also analytic for Imk>0 and continuous for Imk≥0 and have the following asymptotic properties:(11)ϕ+x,k→eikx,x→+∞,(12)ϕ-x,k→e-ikx,x→-∞.Lemma 3.
Define the Wronskian(13)Wϕ+x,k,ϕ-x,k=ϕx+x,kϕ-x,k-ϕ+x,kϕx-x,k,and let (14)ϕ-x,k=akϕ+x,-k+bkϕ+x,k.Then(15)ak=12ikWϕ-x,k,ϕ+x,k,bk=12ikWϕ+x,-k,ϕ-x,k,where a(k) is analytic for Imk>0 and continuous for Imk≥0, b(k) is defined only on the real axis Imk=0, and the analytic function a(k) has a finite number of simple zeros km=iκm(κm>0,m=1,2,…,n).Lemma 4.
For the linear spectral problem (3), there exists a constant bm, such that(16)ϕ-x,iκm=bmϕ+x,iκm,(17)∫-∞+∞cm2ϕ+2x,iκmdx=1,cm2=-ibmakiκm.Definition 5.
The constantcm satisfying eq. (17) is named the normalization constant for the eigenfunction ϕ+(x,iκm), and cmϕ+(x,iκm) is named normalization eigenfunction.Definition 6.
The set(18)kImk=0,Rk=bkak,iκm,cm,m=1,2,…,nis named the scattering data of the linear spectral problem (3).Lemma 7.
If the eigenfunctionϕ(x,k) satisfies the linear spectral problem (3), then (19)P=Dtαϕ-ux+ϑϕ-4k2-2uϕxsolves eq. (3) as well.Proof.
A direct computation on eq. (19) tells that(20)Pxx=Dtαϕxx-uxxxϕ-4k2-2uϕxxx+3ux-ϑϕxx.With the help of eqs. (3) and (20), we have(21)Pxx+k2+uP=-ϕDtαu+6uux+uxxx=0.Thus, the proof is finished.For the time dependence of the scattering data, we have the following Theorem8.Theorem 8.
If the time evolution ofu(x,t) obeys the local tfKdV eq. (1), then the scattering data (18) for the linear spectral problem (3) possess the following time dependences: (22)Dtακmt=0,Dtαcmt=4κm3tcmt,(23)Dtαak=0,Dtαbk=8ik3bk.Proof.
Substituting eqs. (12) and (14) into eq. (3) and using the asymptotic properties of eqs. (11) and (12) as x→+∞ and x→-∞, respectively, we have(24)ϑ-4ik3e-ikx=0,(25)Dtαake-ikx+Dtαbkeikx=ϑake-ikx+bkeikx+4k2-ikake-ikx+ikbkeikx.Namely,(26)ϑ=4ik3,(27)Dtαak=akϑ-4ik3=0,(28)Dtαbk=bkϑ+4ik3=8ik3bk.It is easy to see that all the zeros of a(k) are independent of t because of Dt(α)a(k)=0. Therefore, we arrive at Dt(α)κm(t)=0.
Similarly, substituting eq. (16) into eq. (3) and using the asymptotic property of eq. (11) as x→+∞, we have (29)Dtαbmt=ϑ+4κm3tbmt=8κm3tbmt.In view of eqs. (17) and (28), we obtain(30)Dtαcm2t=-Dtαibmtakiκm=8κm3tcm2t,which 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 Theorem9.Theorem 9.
The local tfKdV eq. (1) has an exact solution of the form(31)ux,t=2ddxKx,x,t,where K(x,y,t) satisfies the Gel’fand-Levitan-Marchenko (GLM) integral equation:(32)Kx,y,t+Fx+y,t+∫x∞Kx,z,tFx+z,tdz=0,with (33)Fx,t=12π∫-∞∞Rk,teikxdk+∑m=1ncm2eiκmx,and R(k,t), cm, and κm are determined by eqs. (22) and (23).Proof.
The process of the proof of Theorem9 is similar to that of the classical KdV equation [9] with integer order, and the only differences are the scattering data. To avoid unnecessary repetition, we omit it here.For the fractionaln-soliton solution, we have the following Theorem 10.Theorem 10.
In the case of reflectionless potential, the local tfKdV eq. (1) has fractionaln-soliton solution of the form(34)ux,t=2d2dx2ln⁡detDx,t,where(35)Dx,t=djmx,tn×n,djmx,t=δjm+cjtcmtκj0+κm0e-κj0+κm0x,(36)cjt=cj0Eα4κj30tα,cmt=cm0Eα4κm30tα.In eq. (36), Eα(·) is the Mittag-Leffler function [33].Proof.
Firstly, we further determine the scattering data. Solving eqs. (22) and (23) yields (37)κmt=κm0,cmt=cm0Eα4κm30tα,(38)ak,t=ak,0,bk,t=bk,0Eα8iκ30tα.
Secondly, we letR(k,t)=0. In this case of reflectionless, eq. (32) reduces to(39)Kx,y,t+∑m=1ncm2te-κm0x+y+∑m=1ncm2te-κm0y∫x∞Kx,z,te-κm0zdz=0.
Suppose that eq. (39) has a separation solution(40)Kx,y,t=∑j=1ncjthjxe-κj0y,where hj(x) is an undetermined function which can be determined by the substitution of eq. (40) into eq. (39). With the determined function hj(x), we have(41)Kx,y,t=ddxln⁡detDx,t.Finally, 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 fractionaln-soliton solution (34), we consider the cases of n=1,2,3.Whenn=1, we have(42)detDx,t=1+c120Eα8κ130tα2κ10e-2κ10x,and we, hence, obtain, from eq. (34), the fractional one-soliton solution:(43)u=2κ120sech2⁡κ10x-12ln⁡c120Eα8κ130tα2κ10.Similarly, whenn=2 we obtain the fractional two-soliton solution:(44)u=2ln⁡1+c120Eα8κ130tα2κ10e-2κ10x+c220Eα8κ230tα2κ20e-2κ20x+c120c220κ10-κ202Eα8κ10tαEα8κ230tα4κ10κ20κ10+κ202e-2κ10+κ20xxx.Whenn=3, we obtain the fractional three-soliton solution:(45)u=2ln⁡1+c120Eα8κ130tα2κ10e-2κ10x+c220Eα8κ230tα2κ20e-2κ20x+c320Eα8κ330tα2κ30e-2κ30x+c120c220κ10-κ202Eα8κ10tαEα8κ230tα4κ10κ20κ10+κ202e-2κ10+κ20x+c120c320κ10-κ302Eα8κ10tαEα8κ330tα4κ10κ30κ10+κ302e-2κ10+κ30x+c220c320κ20-κ302Eα8κ20tαEα8κ330tα4κ20κ30κ20+κ302e-2κ20+κ30x+c120c220c320κ10-κ202κ10-κ302κ20-κ3028κ10κ20κ30κ10+κ202κ10+κ302κ20+κ302Eα8κ10tαEα8κ20tαEα8κ330tαe-2κ10+κ20+κ30xxx.In Figure1, we simulate the fractional one-soliton solution (43) with different values of α, where the parameters are selected as κ1(0)=0.8 and c1(0)=0.2. With the help of velocity images in Figure 2 and the formula of velocity(46)v=32ακ150tα-1,we can see that the bell-shaped solitons have different velocities depending on the values of α. At the initial stage, the smaller the value of α is selected, the faster the soliton propagates. But soon it was the opposite; for more details see Figures 3 and 4.Figure 1
Fractional one-solitons determined by solution (43) at time t=1.Figure 2
Velocity images of fractional one-solitons determined by solution (43).Figure 3
Fractional one-solitons determined by solution (43) at time t=80.Figure 4
Fractional one-solitons determined by solution (43) at time t=195.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 κ1(0)=0.5, c1(0)=1.5, κ2(0)=1, and c2(0)=-0.2. While the parameters in Figure 7 are selected as κ1(0)=1, c1(0)=1, κ2(0)=1.1, c2(0)=0.5, κ3(0)=1.3, c3(0)=2.Figure 5
Dynamical evolutions of fractional two-solitons determined by solution (44) at different times.
(a)
t = 0 (b)
t = 0.1 (c)
t = 0.5 (d)
t = 0.7Figure 6
Fractional two-solitons determined by solution (44) at t=3.Figure 7
Fractional three-solitons determined by solution (45) at t=1.


## 5. Conclusion

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 [33] 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 fractionaln-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 constructingn-soliton solution of the local tfKdV eq. (1) can be extended to some other integrable local time-fractional PDEs.



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