# Around Chaotic Disturbance and Irregularity for Higher Order Traveling Waves

Authors: Emile F. Doungmo Goufo; Ignace Tchangou Toudjeu
Journal: Journal of Mathematics
 (2018)

---

## Abstract

Many unknown features in the theory of wave motion are still captivating the global scientific community. In this paper, we consider a model of seventh order Korteweg–de Vries (KdV) equation with one perturbation level, expressed with the recently introduced derivative with nonsingular kernel, Caputo-Fabrizio derivative (CFFD). Existence and uniqueness of the solution to the model are established and proven to be continuous. The model is solved numerically, to exhibit the shape of related solitary waves and perform some graphical simulations. As expected, the solitary wave solution to the model without higher order perturbation term is shown via its related homoclinic orbit to lie on a curved surface. Unlike models with conventional derivative (γ=1) where regular behaviors are noticed, the wave motions of models with the nonsingular kernel derivative are characterized by irregular behaviors in the pure factional cases (γ<1). Hence, the regularity of a soliton can be perturbed by this nonsingular kernel derivative, which, combined with the perturbation parameter ζ of the seventh order KdV equation, simply causes more accentuated irregularities (close to chaos) due to small irregular deviations.


---

## Body

## 1. Introduction

In the last decade, a great number of researchers have paid a particular attention to the study of solitary wave equations that undergo the influence of external perturbations. Most of physical dynamics related to the movement of liquids and waves are governed by Korteweg–de Vries (KdV) equation and its variants. Hence, for this particular equation, Cao et al. [1] as well as Grimshaw and Tian [2] have recently shown that a force combined together with dissipation can provoke a chaotic behavior usually detectable by other analysis like phase plane analysis or nonnegative Liapunov exponents. KdV equation and its variants are of infinite dimension and their use to address traveling waves or chaotic dynamics of low dimension is facilitated by numerical approximations, which have proven that correlation dimension established via Grassberger-Procaccia technique and information dimension obtained from formula of Kaplan-Yorke are both between two and three for steady traveling waves [3].However, many authors (like, e.g., [4–7]) preferred to use numerical approach to analyze the KdV equation or its variants, especially the one with many levels of perturbations. Hence, it was shown in [4] that there is no periodic waves for the autonomous Korteweg–de Vries–Burgers equation of dimension two. We follow, in this paper, the same trend of numerical approach by making use of the recently developed fractional derivative with nonsingular kernel [8–13], to express a seventh order Korteweg–de Vries (KdV) equation with one perturbation level. This is the first instance where such a model is extended to the scope of fractional differentiation and fully investigated. We prove existence and uniqueness of a continuous solution. Before that, we shall give in the following section a brief review of the recent developments done in the theory of fractional differentiation.


## 2. Around the Nonsingular Kernel Differentiation with Fractional Order

The concept of fractional order derivative is seen by many authors as a great endeavor to ameliorate nonlinear mathematical models, widen their analysis, and expand their interpretation. Today’s literature of the concept has been enriched with many innovative definitions more related to the complexity and diversity of natural phenomena surrounding us. There are fractional order derivatives of local type and also of nonlocal type [8, 14–17]. The Caputo’s definition(1)Dtγut=1Γn-γ∫0tt-τn-γ-1ddτnuτdτ,n-1<γ≤n. remains the most commonly used in the applied science, followed by Riemann-Liouville’s version given by(2)Dtγut=1Γn-γddtn∫0tt-τn-γ-1uτdτ,n-1<γ≤n.Recent observations by Caputo and Fabrizio [8] stated that the two definitions above better describe physical processes, related to fatigue, damage, and electromagnetic hysteresis, but do not genuinely depict some behavior taking place in multiscale systems and in materials with massive heterogeneities. Hence, the same authors introduced the following new version of fractional order derivative with no singular kernel:Definition 1 (Caputo-Fabrizio fractional order derivative (CFFD)).
Letu be a function in H1(a;b); b>a; γ∈[0;1]; then, the Caputo-Fabrizio fractional order derivative (CFFD) is defined as(3)Dcftγut=Mγ1-γ∫0tu˙τexp⁡-γt-τ1-γdτ,where M(γ) is a normalization function such that M(0)=M(1)=1.Remark 2.
Caputo and Fabrizio [8] substituted the kernel 1/(t-τ)γ appearing in (1) when n=1 by the function exp-γ(t-τ)/(1-γ) and 1/Γ1-γ by M(γ)/1-γ. This immediately removes the singularity at t=τ that exists in the previous Caputo’s expression.For the function that does not belong toH1(a;b), the CFFD is given by(4)Dcftγut=γMγ1-γ∫0tut-uτexp⁡-γt-τ1-γdτ.Losada and Nieto [13] upgraded this definition of CFFD by proposing the following:(5)Dcftγut=2-γMγ21-γ∫0tu˙τexp⁡-γt-τ1-γdτ.Unlike the classical version of Caputo fractional order derivative [14, 18], the CFFD with no singular kernel appears to be easier to handle. Furthermore, the CFFD verifies the following equalities:(6)limγ→1Dcftγut=u˙t,(7)limγ→0Dcftγut=ut-ua,with u any suitable function and a the starting point of the integrodifferentiation. The fractional integral related to the CFFD and proposed by Losada and Nieto reads as(8)Icftγut=21-γ2-γMγut+2γ2-γMγ∫0tuτdτ,γ∈[0,1]t≥0. This antiderivative represents sort of average between the function u and its integral of order one. The Laplace transform of the CFFD reads(9)LDtγut,s=su~x,s-u0xs+γ1-s,where u~(x,s) is the Laplace transform L(u(x,t),s) of u(x,t).Definition 3 (New Riemann-Liouville fractional order derivative (NRLFD)).
As a response to the CFFD and being aware of the conflicting situations that exist between the classical Riemann-Liouville and Caputo derivatives, the classical Riemann-Liouville definition was modified [9, 10] in order to propose another definition known as the new Riemann-Liouville fractional derivative (NRLFD) without singular kernel and expressed for γ∈[0,1] as(10)Datγut=Mγ1-γddt∫atuτexp⁡-γ1-γt-τdτ.Again, the NRLFD is without any singularity at t=τ compared to the classical Riemann-Liouville fractional order derivative and also verifies(11)limγ→1Datγut=u˙t,limγ→0Datγut=ut.Compared to (7), we note here the exact correspondence with u at γ→0. The Laplace transform of the NRLFD reads as [9, 10](12)LDt-γut,s=sMγs+γ1-sLut,s.Other versions and innovative definitions of fractional derivatives have since been introduced. This paper however uses the CFFD, so for more details about those recent definitions, please feel free to consult the articles and works mentioned above and also the references mentioned therein.


## 3. Existence and Uniqueness

In this section we prove the existence and uniqueness results for the seventh order Korteweg–de Vries equation (KdV) with one perturbation level, expressed with the CFFD and given by(13)Dcftγux,t=-6uux-uxxx+uxxxxx-ζuxxxxxxx,assumed to satisfy the initial condition(14)ux,0=fx,where ζ is the perturbation parameter and Dcftγ is the Caputo-Fabrizio fractional order derivative (CFFD) given in (5). Existence results for the model (13)-(14) here above are established by making use of the expression of the antiderivative (8). This yields (15)ux,t-ux,0=Icftγ-6uux-uxxx+uxxxxx-ζuxxxxxxx.This can be rewritten as(16)ux,t-ux,0=21-γ2-γMγ-6uux-uxxx+uxxxxx-ζuxxxxxxx+2γ2-γMγ∫0t-6uux-uxxx+uxxxxx-ζuxxxxxxxdτ.Set now(17)Yx,t,u,ζ=-6uux-uxxx+uxxxxx-ζuxxxxxxx.The next step is to look for a real constant K≥0 such that (18)Yx,t,u,ζ-Yx,t,v,ζ≤Ku-v.In fact (19)Yx,t,u,ζ-Yx,t,v,ζ=-6uux-uxxx+uxxxxx-ζuxxxxxxx--6vvx-vxxx+vxxxxx-ζvxxxxxxx=6vvx-uux+vxxx-uxxx+uxxxxx-vxxxxx+ζvxxxxxxx-uxxxxxxx.Well known properties for the norms give(20)Yx,t,u,ζ-Yx,t,v,ζ=6vvx-uux+vxxx-uxxx+uxxxxx-vxxxxx+ζvxxxxxxx-uxxxxxxx≤6vvx-uux+vxxx-uxxx+uxxxxx-vxxxxx+ζvxxxxxxx-uxxxxxxx≤6∂xv2-u2+∂xxxv-u+∂xxxxxu-v+ζ∂xxxxxxxv-u.Keeping in mind that u and v are bounded functions, then there exists real numbers r1>0 and r2>0 such that(21)u≤r1,v≤r2.Set r=max⁡(r1,r2); hence,(22)u≤r,v≤r.Therefore, the Lipschitz condition holds for the partial derivatives ∂xu and ∂xv and there exists a real constant K1≥0 such that(23)Yx,t,u,ζ-Yx,t,v,ζ≤6K1v2-u2+K13v-u+K15u-v+ζK17v-u≤6K1u+v·u-v+K13v-u+K15u-v+ζK17v-u≤12rK1+K13+K15+ζK17u-v,where the bounded condition (14) has been exploited, whence (24)Yx,t,u,ζ-Yx,t,v,ζ≤Ku-vwith(25)K=12rK1+K13+K15+ζK17.This proves that Y satisfies the Lipschitz condition and then, it allows us to state the following proposition.Proposition 4.
If the condition1>2(1-γ)/(2-γ)M(γ)K+2tKγ/(2-γ)M(γ) holds, then, there exists a unique and continuous solution to the seventh order Korteweg–de Vries equation with one perturbation level expressed with the CFFD given in (5):(26)Dcftγux,t=-6uux-uxxx+uxxxxx-ζuxxxxxxx,ux,0=fx.Proof.
Let us go back to the model (16) rewritten as(27)ux,t-ux,0=21-γ2-γMγYx,t,u,ζ+2γ2-γMγ∫0tYx,τ,u,ζdτwhich yields the recurrence formulation given as follows:(28)u0x,t=ux,0,unx,t=21-γ2-γMγYx,t,un-1,ζ+2γ2-γMγ∫0tYx,τ,un-1,ζdτ.Let(29)u_x,t=limn→∞unx,t,and it can be shown that u_(x,t)=u(x,t) is a continuous solution. Indeed, if we take(30)Bnx,t=unx,t-un-1x,tthen, it is straightforward to see that (31)unx,t=∑p=0nBpx,t.More explicitly, we have(32)Bnx,t=21-γ2-γMγYx,t,un-1,ζ-Yx,t,un-2,ζ+2γ2-γMγ∫0tYx,τ,un-1,ζ-Yx,τ,un-2,ζdτ.Passing this equation to the norm yields(33)Bnx,t=unx,t-un-1x,t≤21-γ2-γMγYx,t,un-1,ζ-Yx,t,un-2,ζ+2γ2-γMγ∫0tYx,τ,un-1,ζ-Yx,τ,un-2,ζdτ≤21-γ2-γMγYx,t,un-1,ζ-Yx,t,un-2,ζ+2γ2-γMγ∫0tYx,τ,un-1,ζ-Yx,τ,un-2,ζdτ.Applying the Lipschitz condition to Y gives (34)Bnx,t≤21-γ2-γMγKun-1-un-2+2Kγ2-γMγ∫0tun-1-un-2dτwhich can be rewritten as(35)Bnx,t≤21-γ2-γMγKBn-1+2Kγ2-γMγ∫0tBn-1dτ.After integrating, we make use of well known properties of the recursive technique from (35) to have (36)Bnx,t≤21-γ2-γMγKn+2tKγ2-γMγnfx,with f(x)=u(x,0), which explicitly shows the existence of the solution and that it is continuous.
The step forward is to prove that the solution of the model (26) is given by the function (37)ux,t=limn→∞unx,t.For that, let (38)Qnx,t=u_x,t-unx,twithn∈N.Making use of (29), we should have limn→∞Qn=0. In other terms, the gap that exists between u_(x,t) and un(x,t) should vanish as n→∞. Consider (39)u_-un-1=21-γ2-γMγYx,t,u,ζ-Yx,t,un,ζ+2γ2-γMγ∫0tYx,τ,u,ζ-Yx,τ,un,ζdτ,giving(40)u_x,t-un+1≤21-γ2-γMγYx,t,u,ζ-Yx,t,un,ζ+2γ2-γMγ∫0tYx,τ,u,ζ-Yx,τ,un,ζdτ≤21-γ2-γMγKu-un+2Kγ2-γMγ∫0tu-undτ≤21-γ2-γMγKQn+2Kγ2-γMγ∫0tQndτ.Hence, limn→∞Qn=0 and from the right hand side, we have (41)limn→∞un=u_.Just take u(x,t)=u_(x,t) as the solution of (26) that is continuous. Moreover, the Lipschitz condition for Y yields(42)ux,t-21-γ2-γMγYx,t,u,ζ-2γ2-γMγ∫0tYx,t,u,ζdτ=Bnx,t+21-γ2-γMγYx,τ,un-1,ζ-Yx,t,u,ζ+2γ2-γMγ∫0tYx,τ,un-1,ζ-Yx,t,u,ζdτ.This yields(43)ux,t-21-γ2-γMγYx,t,u,ζ-2γ2-γMγ∫0tYx,t,u,ζdτ=Bnx,t+21-γ2-γMγ+2γ2-γMγBn-1x,tconsidering the initial condition and taking the limit as n→0 gives (44)ux,t=fx+21-γ2-γMγYx,t,u,ζ+2γ2-γMγ∫0tYx,t,u,ζdτ.
Uniqueness. To prove that the solution is unique, we take two different functions u and v that satisfy the model (26); then,(45)u-v≤21-γ2-γMγKu-v+2tKγ2-γMγu-v,equivalently (46)u-v1-21-γ2-γMγK-2tKγ2-γMγ≤0.This yields u=v if (47)1-21-γ2-γMγK-2tKγ2-γMγ>0,where we have used the Lipschitz condition for Y and this ends the proof.


## 4. Shape of Solitary Waves via Numerical Approximations

### 4.1. Shape of Solitary Waves for the Lower-Order Approximation

In this section, we are interested in waves traveling to a specific direction, and then, we consider solutions to the seventh order KdV equation(48)Dcftγux,t=-6uux-uxxx+uxxxxx-ζuxxxxxxx.We start by considering the conventional case where γ=1 to have, using Dcft1f(t)=f′(t), the following model:(49)∂ux,t∂t=-6uux-uxxx+uxxxxx-ζuxxxxxxx.We investigate the traveling waves taking the form u(x,t)=u(x+ct)=u(η), where c is the speed of the wave. We assume that u does not depend on x independently from t but rather depends on the combined variable η=x+ct. We also assume that the wave dies at infinity, meaning(50)limη→±∞uη=limη→±∞uηη=⋯=limη→±∞uηηηηηηηη=0.Now, it is possible to transform the seventh order KdV equation (49) into an ordinary differential equation (ODE) by making use of the basic properties of differentiation. Then, ux=uη·ηx=uη and ut=uη·ηt=cuη which yield the following ODE:(51)-cuη-6uuη-uηηη+uηηηηη-ζuηηηηηηη=0.The η-integration of this equation once gives(52)-cu-3u2-uηη+uηηηη-ζuηηηηηη=0,where we have ignored the constant of η-integration that is null due to boundary conditions (50). If the higher order perturbation term ζuηηηηηη is ignored, then we have(53)-cu-3u2-uηη+uηηηη=0.Then we can solve numerically this equation by transforming it into a system of four ODEs of order one as follows:(54)uη=w,uηη=wη=y,uηηη=yη=z,uηηηη=zη=cu+3u2+y.Numerical simulations are done in the phase-space (u,uη,uηη) as shown in Figure 1. Let us now come back to the full model (48) with the nonsingular kernel derivative CFFD (given in (5)) and with no higher order perturbation parameter ζ, given as(55)Dcftγux,t=-6uux-uxxx+uxxxxx.This fractional equation is solved numerically by making use of the Adams-Bashforth-Moulton type method also known as predictor-corrector (PECE) technique and is fully detailed in the article by Diethelm et al. [19]. Figures 1 and 2 represent the numerical simulations of solutions to the fractional model (55) with different values of the derivative order γ. They clearly point out relative irregularities when the derivative order of the CFFD is γ<1.Figure 1
Numerical plot in the phase-space(u,uη,uηη) for the model KdV model (55) or model (48) with no higher order perturbation term, for γ=1 (conventional case). As expected in (a), the soliton solution is shown via its related homoclinic orbit to lie on a curved surface. In (b), the projection on the plane (u,uη).
(a) (b)Figure 2
Numerical plot in the phase-space(u,uη,uηη) for the model KdV model (55) with no higher order perturbation term, for γ=0.75. As expected in (a), even in the pure fractional case, the soliton solution is shown via its related homoclinic orbit to lie on a curved surface, with some irregular movements compared to Figure 1. In (b), the projection on the plane (u,uη).
(a) (b)


### 4.2. Shape of Solitary Waves for the Higher Order Approximation

Now, the perturbation termζuηηηηηηη is considered to have (48):(56)Dcftγux,t=-6uux-uxxx+uxxxxx-ζuxxxxxxxsubject to localized initial condition(57)ux,0=12dksech2xk.Although we proved existence and uniqueness for this nonlinear problem, it still faces the challenge of providing an explicit expression of exact solution or approximated solution. It is almost impossible to use some analytical methods, like for instance, integral transform methods, the Green function technique, or the technique separation of variables. Hence, a semianalytical method like the Laplace iterative method can be a valuable tool to provide a special solution to the model (56)-(57) as shown in the following lines.We start by applying the Laplace transform (9) on both sides of (56) to obtain(58)su~x,s-u0xs+γ1-s=L-6uux-uxxx+uxxxxx-ζuxxxxxxx,s.Let θ(γ,s)=s+γ(1-s); then, (59)u~x,s=ux,0s+θγ,ssL-6uux-uxxx+uxxxxx-ζuxxxxxxx,s.Application of inverse Laplace transform L-1 on both sides yields (60)ux,t=L-1ux,0,ts+L-1θγ,ssL-6uux-uxxx+uxxxxx-ζuxxxxxxx,s,t.This leads to the following recursive system(61)un+1x,t=unx,t+L-1θγ,ssL-6ununx-unxxx+unxxxxx-ζunxxxxxxx,s,t,u0x,t=ux,0,then making the solution to be u(x,t)=limn→∞un(x,t). Numerical approximations are performed according to the following steps: (i)
u0(x,t)=u(x,0) is considered as initial input:Choosej as the number of terms to compute.Nameuapp to be the approximate solution.Setuapp=u(x,0)=12d/ksech2x/k and uapp=uapp.(ii)
For the other terms useun+1(x,t)=un(x,t)+L-1θ(γ,s)/sL-6un(un)x-(un)xxx+(un)xxxxx-ζ(un)xxxxxxx,s,t.ComputeXn(x,t)=Xn-1(x,t)+uapp.to finally getuapp(x,t)=Xn(x,t)+uappwhere(62)Xnx,t=L-1θγ,ssL-6ununx-unxxx+unxxxxx-ζunxxxxxxx,s,t.Note we can use θ(γ,s), to obtain following results and some numerical approximations with related absolute errors are summarized in Tables 1 and 2 (which present some numerical approximations with related absolute errors as given by the sequences (63) and (65), respectively, representing the pure fractional case (γ=0.8) and the standard case (γ=1)).(63)ux,0=12dksech2xk,u1x,t=ux,0+1.900×10-3tdk3/21.233dk1/2-3.467k2-293k4-98933ζk6+-2.257k2-135.200k4+99631ζk6cosh⁡2xk+-0.102k2+173.700k4+14681ζk6cosh⁡4xk+0.572k2-13.730k4+496ζk6cosh⁡6xksech12xktanhxk+1-γΓγtγ-1dk0.225dk-3.830k1/4-352.303ζk1/6+-0.057k1/2-3.520k1/4+685ζk1/6cosh⁡2xk+-0.030k1/2+3.820k1/4+641ζk1/6cosh⁡4xk+0.072k1/2-0.770k1/4+9.650ζk1/6cosh⁡6xksech6x/ktanhx/k2γ-γ,u2x,t=u1x,0+⋯,⋮.Note that, for γ=1, we recover for the conventional model (49) (64)∂ux,t∂t=-6uux-uxxx+uxxxxx-ζuxxxxxxx,the following well known result(65)ux,0=12dksech2xk,u1x,t=ux,0+1.900×10-3tdk3/21.233dk1/2-3.467k2-293k4-98933ζk6+-2.257k2-135.200k4+99631ζk6cosh⁡2xk+-0.102k2+173.700k4+14681ζk6cosh⁡4xk+0.572k2-13.730k4+496ζk6cosh⁡6xksech6xktanh⁡xk.This result corresponds to the similar one obtained in [20, 21] or in [22] via Adomian decomposition method. The other terms were computed following the same iterative approach and then, the functions u(x,t) admits the closed form found to be(66)ux,t=12dksech21kx-dk5/2t.The shape of numerical approximations is depicted in Figures 3–6, plotted for different values of the derivative order γ and showing a spot of irregular perturbed motion in the pure fractional case (γ=0.80) compared to the conventional case (γ=1). Before going to conclusions, we can summarize the physical aspects of this work (issued from the mathematical analysis and simulations) as follows: analysis performed on the KdV model with no higher order perturbation term has shown the soliton solution via its related homoclinic orbit lying on a curved surface (Figures 1 and 2), and this remains true in the conventional case as well as the pure fractional case. However, irregular movements are more important in the latter case. Analysis performed on the seventh order KdV equation with one perturbations level has shown the shape of solitary waves characterized by motions with more irregular behaviors tending to become chaotic. The chaos is more likely to happen in pure fractional case compared to the conventional case (Figures 3–6). Hence, the main results of this paper, reflected by the six figures, insinuate that the regularity of a soliton can be perturbed by the nonsingular kernel derivative, which, combined with the perturbation parameter of the KdV model, may lead to chaos.Table 1
Some values for numerical and exact solutions to the seventh order KdV equation expressed with the CFFD given in (5) at k=2.5, d=4.0, and γ=0.80.
Forγ=0.80 Time (t) Spatial (x) Exact value Numerical value Error made 0.5 −15.0 0.000098126 0.000224130 1.26 × 1 0 - 4 −7.0 0.056620000 0.056514000 1.06 × 1 0 - 4 0.0 0.100372300 0.100470000 1.00 × 1 0 - 4 +7.0 0.070555000 0.069355000 1.20 × 1 0 - 3 +15.0 0.000118180 0.000120180 2.00 × 1 0 - 6  1.0 −15.0 0.000091592 0.000284080 1.20 × 1 0 - 4 −7.0 0.054779000 0.054779000 00 0.0 0.100390000 0.100380000 1.00 × 1 0 - 5 +7.0 0.075546000 0.071755000 1.20 × 1 0 - 4 +15.0 0.000126620 0.000073380 2.00 × 1 0 - 4  2.0 −15.0 0.000081235 0.000181240 1.00 × 1 0 - 4 −7.0 0.048620000 0.069620000 2.10 × 1 0 - 2 0.0 0.100410000 0.100510000 1.01 × 1 0 - 4 +7.0 0.085083000 0.084954000 1.29 × 1 0 - 4 +15.0 0.000142760 0.000242760 1.00 × 1 0 - 4  4.0 −15.0 0.000065920 0.000132590 1.26 × 1 0 - 3 −7.0 0.039495000 0.039495000 00 0.0 0.100430000 0.090430000 1.00 × 1 0 - 2 +7.0 0.104610000 0.102610000 2.00 × 1 0 - 3 +15.0 0.000175920 0.000175920 00  8.0 −15.0 0.000045819 0.000224130 1.26 × 1 0 - 3 −7.0 0.027489000 0.058768000 1.6 × 1 0 - 3 0.0 0.090400000 0.100400000 1 × 1 0 - 2 +7.0 0.149720000 0.071755000 1.2 × 1 0 - 3 +15.0 0.000253100 0.000120180 1.01 × 1 0 - 5  10.0 −15.0 0.000038771 0.000224130 1.26 × 1 0 - 4 −7.0 0.023272000 0.058768000 1.06 × 1 0 - 3 0.0 0.090402000 0.091402000 1 × 1 0 - 3 +7.0 0.176380000 0.176380000 00 +15.0 0.000299110 0.00012018 2.1 × 1 0 - 5Table 2
Some values for numerical and exact solutions to the seventh order KdV equation expressed with the CFFD given in (5) at k=2.5, d=4.0, and γ=1.
Forγ=1 Time (t) Spatial (x) Exact value Numerical value Error made 0.5 −15.0 0.000099315 0.0001 094 20 1.01 × 1 0 - 5 −7.0 0.593670000 0.593670000 00 0.0 0.151107000 0.151107000 00 +7.0 0.06971 8000 0.06971 8000 00 +15.0 0.000 167 700 0.00009 6770 2.00 × 1 0 - 5  1.0 −15.0 0.00009 159 2 0.00008 857 2 3.02 × 1 0 - 6 −7.0 0.05477 9000 0.05477 9000 00 0.0 0.151907000 0.151907000 00 +7.0 0.07 554 6000 0.07 554 6000 00 +15.0 0.0001 266 20 0.00002 6620 1.00 × 1 0 - 4  2.0 −15.0 0.00007790 1 0.000237900 1.60 × 1 0 - 4 −7.0 0.04 6635000 0.04663 5000 00 0.0 0.152007000 0.152140000 1.31 × 1 0 - 4 +7.0 0.08 868 8000 0.08 868 8000 00 +15.0 0.0001 488 70 0.0001591 70 1.03 × 1 0 - 5  4.0 −15.0 0.00005 635 2 0.001316 400 1.26 × 1 0 - 3 −7.0 0.03 378 5000 0.03 378 5000 00 0.0 0.152060000 0.162060000 1.00 × 1 0 - 2 +7.0 0.122 130000 0.120 120000 2.01 × 1 0 - 3 +15.0 0.0002057 90 0.0002057 90 00  8.0 −15.0 0.00002 948 8 0.001 289 500 1.26 × 1 0 - 3 −7.0 0.01 771 1000 0.01 931 1000 1.6 × 1 0 - 3 0.0 0.152060000 0.162060000 1 × 1 0 - 2 +7.0 0.230 430000 0.231 630000 1.2 × 1 0 - 3 +15.0 0.0003 932 70 0.0004 033 70 1.01 × 1 0 - 5  10.0 −15.0 0.00002 133 1 0.00002 007 1 1.26 × 1 0 - 6 −7.0 0.01 281 9000 0.01283 0000 1.06 × 1 0 - 5 0.0 0.161760000 0.162760000 1.00 × 1 0 - 3 +7.0 0.315 320000 0.315 320000 00 +15.0 0.0005 436 40 0.0005 436 40 00Figure 3
The shape of approximated solutionuapp=u3 in the conventional case (γ=1). It is plotted with respect to different fixed values of time and space at k=2.5, d=4.0, ζ=0.5. The motion is quite standard and regular.Figure 4
The shape of approximated solutionuapp=u3 in the pure fractional case (γ=0.80), plotted with respect to different fixed values of time and space at k=2.5, d=4.0, ζ=0.5. The motion is shown to behave irregularly compared to the conventional case and the dynamic tending to a chaotic one.Figure 5
The shape of approximated solutionuapp=u3 in the conventional case (γ=1). It is plotted with respect to different fixed values of time and space at k=2.5, d=4.0, ζ=0.01. The motion is quite standard and regular.Figure 6
The shape of approximated solutionuapp=u3 in the pure fractional case (γ=0.80), plotted with respect to different fixed values of time and space at k=2.5, d=4.0, ζ=0.01. The motion is shown to behave irregularly compared to the conventional case and the dynamic tending to a chaotic one.


## 4.1. Shape of Solitary Waves for the Lower-Order Approximation

In this section, we are interested in waves traveling to a specific direction, and then, we consider solutions to the seventh order KdV equation(48)Dcftγux,t=-6uux-uxxx+uxxxxx-ζuxxxxxxx.We start by considering the conventional case where γ=1 to have, using Dcft1f(t)=f′(t), the following model:(49)∂ux,t∂t=-6uux-uxxx+uxxxxx-ζuxxxxxxx.We investigate the traveling waves taking the form u(x,t)=u(x+ct)=u(η), where c is the speed of the wave. We assume that u does not depend on x independently from t but rather depends on the combined variable η=x+ct. We also assume that the wave dies at infinity, meaning(50)limη→±∞uη=limη→±∞uηη=⋯=limη→±∞uηηηηηηηη=0.Now, it is possible to transform the seventh order KdV equation (49) into an ordinary differential equation (ODE) by making use of the basic properties of differentiation. Then, ux=uη·ηx=uη and ut=uη·ηt=cuη which yield the following ODE:(51)-cuη-6uuη-uηηη+uηηηηη-ζuηηηηηηη=0.The η-integration of this equation once gives(52)-cu-3u2-uηη+uηηηη-ζuηηηηηη=0,where we have ignored the constant of η-integration that is null due to boundary conditions (50). If the higher order perturbation term ζuηηηηηη is ignored, then we have(53)-cu-3u2-uηη+uηηηη=0.Then we can solve numerically this equation by transforming it into a system of four ODEs of order one as follows:(54)uη=w,uηη=wη=y,uηηη=yη=z,uηηηη=zη=cu+3u2+y.Numerical simulations are done in the phase-space (u,uη,uηη) as shown in Figure 1. Let us now come back to the full model (48) with the nonsingular kernel derivative CFFD (given in (5)) and with no higher order perturbation parameter ζ, given as(55)Dcftγux,t=-6uux-uxxx+uxxxxx.This fractional equation is solved numerically by making use of the Adams-Bashforth-Moulton type method also known as predictor-corrector (PECE) technique and is fully detailed in the article by Diethelm et al. [19]. Figures 1 and 2 represent the numerical simulations of solutions to the fractional model (55) with different values of the derivative order γ. They clearly point out relative irregularities when the derivative order of the CFFD is γ<1.Figure 1
Numerical plot in the phase-space(u,uη,uηη) for the model KdV model (55) or model (48) with no higher order perturbation term, for γ=1 (conventional case). As expected in (a), the soliton solution is shown via its related homoclinic orbit to lie on a curved surface. In (b), the projection on the plane (u,uη).
(a) (b)Figure 2
Numerical plot in the phase-space(u,uη,uηη) for the model KdV model (55) with no higher order perturbation term, for γ=0.75. As expected in (a), even in the pure fractional case, the soliton solution is shown via its related homoclinic orbit to lie on a curved surface, with some irregular movements compared to Figure 1. In (b), the projection on the plane (u,uη).
(a) (b)


## 4.2. Shape of Solitary Waves for the Higher Order Approximation

Now, the perturbation termζuηηηηηηη is considered to have (48):(56)Dcftγux,t=-6uux-uxxx+uxxxxx-ζuxxxxxxxsubject to localized initial condition(57)ux,0=12dksech2xk.Although we proved existence and uniqueness for this nonlinear problem, it still faces the challenge of providing an explicit expression of exact solution or approximated solution. It is almost impossible to use some analytical methods, like for instance, integral transform methods, the Green function technique, or the technique separation of variables. Hence, a semianalytical method like the Laplace iterative method can be a valuable tool to provide a special solution to the model (56)-(57) as shown in the following lines.We start by applying the Laplace transform (9) on both sides of (56) to obtain(58)su~x,s-u0xs+γ1-s=L-6uux-uxxx+uxxxxx-ζuxxxxxxx,s.Let θ(γ,s)=s+γ(1-s); then, (59)u~x,s=ux,0s+θγ,ssL-6uux-uxxx+uxxxxx-ζuxxxxxxx,s.Application of inverse Laplace transform L-1 on both sides yields (60)ux,t=L-1ux,0,ts+L-1θγ,ssL-6uux-uxxx+uxxxxx-ζuxxxxxxx,s,t.This leads to the following recursive system(61)un+1x,t=unx,t+L-1θγ,ssL-6ununx-unxxx+unxxxxx-ζunxxxxxxx,s,t,u0x,t=ux,0,then making the solution to be u(x,t)=limn→∞un(x,t). Numerical approximations are performed according to the following steps: (i)
u0(x,t)=u(x,0) is considered as initial input:Choosej as the number of terms to compute.Nameuapp to be the approximate solution.Setuapp=u(x,0)=12d/ksech2x/k and uapp=uapp.(ii)
For the other terms useun+1(x,t)=un(x,t)+L-1θ(γ,s)/sL-6un(un)x-(un)xxx+(un)xxxxx-ζ(un)xxxxxxx,s,t.ComputeXn(x,t)=Xn-1(x,t)+uapp.to finally getuapp(x,t)=Xn(x,t)+uappwhere(62)Xnx,t=L-1θγ,ssL-6ununx-unxxx+unxxxxx-ζunxxxxxxx,s,t.Note we can use θ(γ,s), to obtain following results and some numerical approximations with related absolute errors are summarized in Tables 1 and 2 (which present some numerical approximations with related absolute errors as given by the sequences (63) and (65), respectively, representing the pure fractional case (γ=0.8) and the standard case (γ=1)).(63)ux,0=12dksech2xk,u1x,t=ux,0+1.900×10-3tdk3/21.233dk1/2-3.467k2-293k4-98933ζk6+-2.257k2-135.200k4+99631ζk6cosh⁡2xk+-0.102k2+173.700k4+14681ζk6cosh⁡4xk+0.572k2-13.730k4+496ζk6cosh⁡6xksech12xktanhxk+1-γΓγtγ-1dk0.225dk-3.830k1/4-352.303ζk1/6+-0.057k1/2-3.520k1/4+685ζk1/6cosh⁡2xk+-0.030k1/2+3.820k1/4+641ζk1/6cosh⁡4xk+0.072k1/2-0.770k1/4+9.650ζk1/6cosh⁡6xksech6x/ktanhx/k2γ-γ,u2x,t=u1x,0+⋯,⋮.Note that, for γ=1, we recover for the conventional model (49) (64)∂ux,t∂t=-6uux-uxxx+uxxxxx-ζuxxxxxxx,the following well known result(65)ux,0=12dksech2xk,u1x,t=ux,0+1.900×10-3tdk3/21.233dk1/2-3.467k2-293k4-98933ζk6+-2.257k2-135.200k4+99631ζk6cosh⁡2xk+-0.102k2+173.700k4+14681ζk6cosh⁡4xk+0.572k2-13.730k4+496ζk6cosh⁡6xksech6xktanh⁡xk.This result corresponds to the similar one obtained in [20, 21] or in [22] via Adomian decomposition method. The other terms were computed following the same iterative approach and then, the functions u(x,t) admits the closed form found to be(66)ux,t=12dksech21kx-dk5/2t.The shape of numerical approximations is depicted in Figures 3–6, plotted for different values of the derivative order γ and showing a spot of irregular perturbed motion in the pure fractional case (γ=0.80) compared to the conventional case (γ=1). Before going to conclusions, we can summarize the physical aspects of this work (issued from the mathematical analysis and simulations) as follows: analysis performed on the KdV model with no higher order perturbation term has shown the soliton solution via its related homoclinic orbit lying on a curved surface (Figures 1 and 2), and this remains true in the conventional case as well as the pure fractional case. However, irregular movements are more important in the latter case. Analysis performed on the seventh order KdV equation with one perturbations level has shown the shape of solitary waves characterized by motions with more irregular behaviors tending to become chaotic. The chaos is more likely to happen in pure fractional case compared to the conventional case (Figures 3–6). Hence, the main results of this paper, reflected by the six figures, insinuate that the regularity of a soliton can be perturbed by the nonsingular kernel derivative, which, combined with the perturbation parameter of the KdV model, may lead to chaos.Table 1
Some values for numerical and exact solutions to the seventh order KdV equation expressed with the CFFD given in (5) at k=2.5, d=4.0, and γ=0.80.
Forγ=0.80 Time (t) Spatial (x) Exact value Numerical value Error made 0.5 −15.0 0.000098126 0.000224130 1.26 × 1 0 - 4 −7.0 0.056620000 0.056514000 1.06 × 1 0 - 4 0.0 0.100372300 0.100470000 1.00 × 1 0 - 4 +7.0 0.070555000 0.069355000 1.20 × 1 0 - 3 +15.0 0.000118180 0.000120180 2.00 × 1 0 - 6  1.0 −15.0 0.000091592 0.000284080 1.20 × 1 0 - 4 −7.0 0.054779000 0.054779000 00 0.0 0.100390000 0.100380000 1.00 × 1 0 - 5 +7.0 0.075546000 0.071755000 1.20 × 1 0 - 4 +15.0 0.000126620 0.000073380 2.00 × 1 0 - 4  2.0 −15.0 0.000081235 0.000181240 1.00 × 1 0 - 4 −7.0 0.048620000 0.069620000 2.10 × 1 0 - 2 0.0 0.100410000 0.100510000 1.01 × 1 0 - 4 +7.0 0.085083000 0.084954000 1.29 × 1 0 - 4 +15.0 0.000142760 0.000242760 1.00 × 1 0 - 4  4.0 −15.0 0.000065920 0.000132590 1.26 × 1 0 - 3 −7.0 0.039495000 0.039495000 00 0.0 0.100430000 0.090430000 1.00 × 1 0 - 2 +7.0 0.104610000 0.102610000 2.00 × 1 0 - 3 +15.0 0.000175920 0.000175920 00  8.0 −15.0 0.000045819 0.000224130 1.26 × 1 0 - 3 −7.0 0.027489000 0.058768000 1.6 × 1 0 - 3 0.0 0.090400000 0.100400000 1 × 1 0 - 2 +7.0 0.149720000 0.071755000 1.2 × 1 0 - 3 +15.0 0.000253100 0.000120180 1.01 × 1 0 - 5  10.0 −15.0 0.000038771 0.000224130 1.26 × 1 0 - 4 −7.0 0.023272000 0.058768000 1.06 × 1 0 - 3 0.0 0.090402000 0.091402000 1 × 1 0 - 3 +7.0 0.176380000 0.176380000 00 +15.0 0.000299110 0.00012018 2.1 × 1 0 - 5Table 2
Some values for numerical and exact solutions to the seventh order KdV equation expressed with the CFFD given in (5) at k=2.5, d=4.0, and γ=1.
Forγ=1 Time (t) Spatial (x) Exact value Numerical value Error made 0.5 −15.0 0.000099315 0.0001 094 20 1.01 × 1 0 - 5 −7.0 0.593670000 0.593670000 00 0.0 0.151107000 0.151107000 00 +7.0 0.06971 8000 0.06971 8000 00 +15.0 0.000 167 700 0.00009 6770 2.00 × 1 0 - 5  1.0 −15.0 0.00009 159 2 0.00008 857 2 3.02 × 1 0 - 6 −7.0 0.05477 9000 0.05477 9000 00 0.0 0.151907000 0.151907000 00 +7.0 0.07 554 6000 0.07 554 6000 00 +15.0 0.0001 266 20 0.00002 6620 1.00 × 1 0 - 4  2.0 −15.0 0.00007790 1 0.000237900 1.60 × 1 0 - 4 −7.0 0.04 6635000 0.04663 5000 00 0.0 0.152007000 0.152140000 1.31 × 1 0 - 4 +7.0 0.08 868 8000 0.08 868 8000 00 +15.0 0.0001 488 70 0.0001591 70 1.03 × 1 0 - 5  4.0 −15.0 0.00005 635 2 0.001316 400 1.26 × 1 0 - 3 −7.0 0.03 378 5000 0.03 378 5000 00 0.0 0.152060000 0.162060000 1.00 × 1 0 - 2 +7.0 0.122 130000 0.120 120000 2.01 × 1 0 - 3 +15.0 0.0002057 90 0.0002057 90 00  8.0 −15.0 0.00002 948 8 0.001 289 500 1.26 × 1 0 - 3 −7.0 0.01 771 1000 0.01 931 1000 1.6 × 1 0 - 3 0.0 0.152060000 0.162060000 1 × 1 0 - 2 +7.0 0.230 430000 0.231 630000 1.2 × 1 0 - 3 +15.0 0.0003 932 70 0.0004 033 70 1.01 × 1 0 - 5  10.0 −15.0 0.00002 133 1 0.00002 007 1 1.26 × 1 0 - 6 −7.0 0.01 281 9000 0.01283 0000 1.06 × 1 0 - 5 0.0 0.161760000 0.162760000 1.00 × 1 0 - 3 +7.0 0.315 320000 0.315 320000 00 +15.0 0.0005 436 40 0.0005 436 40 00Figure 3
The shape of approximated solutionuapp=u3 in the conventional case (γ=1). It is plotted with respect to different fixed values of time and space at k=2.5, d=4.0, ζ=0.5. The motion is quite standard and regular.Figure 4
The shape of approximated solutionuapp=u3 in the pure fractional case (γ=0.80), plotted with respect to different fixed values of time and space at k=2.5, d=4.0, ζ=0.5. The motion is shown to behave irregularly compared to the conventional case and the dynamic tending to a chaotic one.Figure 5
The shape of approximated solutionuapp=u3 in the conventional case (γ=1). It is plotted with respect to different fixed values of time and space at k=2.5, d=4.0, ζ=0.01. The motion is quite standard and regular.Figure 6
The shape of approximated solutionuapp=u3 in the pure fractional case (γ=0.80), plotted with respect to different fixed values of time and space at k=2.5, d=4.0, ζ=0.01. The motion is shown to behave irregularly compared to the conventional case and the dynamic tending to a chaotic one.


## 5. Concluding Remarks

We have made use of the recent version of derivative with nonsingular kernel to prove existence and uniqueness results for a model of seventh order Korteweg–de Vries (KdV) equation with one perturbation level. The unique solution is continuous. We then use numerical approximations to evaluate the behavior of the solution under the influence of external factors. It happened that when applied to equations of wave motion like the seventh order KdV equation, the new derivative acts as one of those external factors, which, combined with the perturbation termζ of the model, causes the solution to be more irregular and unpredictable. This is the first instance where such a model is fully investigated and such result is exposed. This work differs from the previous ones within introduction of the nonsingular kernel derivative into a powerful model like Korteweg–de Vries’s, which reveals another interesting feature that exists in the domain of wave motion as well as chaos theory.



---

*Source: 2391697-2018-06-03.xml*