Journal of Mathematics

Volume 2018, Article ID 2391697, 11 pages

https://doi.org/10.1155/2018/2391697

## Around Chaotic Disturbance and Irregularity for Higher Order Traveling Waves

^{1}Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa^{2}Department of Electrical Engineering, Tshwane University of Technology, Pretoria 0183, South Africa

Correspondence should be addressed to Emile F. Doungmo Goufo; ac.oohay@6002elimekcnarf

Received 27 February 2018; Accepted 29 March 2018; Published 3 June 2018

Academic Editor: S. K. Q. Al-Omari

Copyright © 2018 Emile F. Doungmo Goufo and Ignace Tchangou Toudjeu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 () 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 (). Hence, the regularity of a soliton can be perturbed by this nonsingular kernel derivative, which, combined with the perturbation parameter of the seventh order KdV equation, simply causes more accentuated irregularities (close to chaos) due to small irregular deviations.

#### 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 remains the most commonly used in the applied science, followed by Riemann-Liouville’s version given by.

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)). *Let be a function in ; ; ; then, the Caputo-Fabrizio fractional order derivative (CFFD) is defined aswhere is a normalization function such that .

*Remark 2. *Caputo and Fabrizio [8] substituted the kernel appearing in (1) when by the function and by This immediately removes the singularity at that exists in the previous Caputo’s expression.

For the function that does not belong to , the CFFD is given byLosada and Nieto [13] upgraded this definition of CFFD by proposing the following: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:with any suitable function and the starting point of the integrodifferentiation. The fractional integral related to the CFFD and proposed by Losada and Nieto reads as This antiderivative represents sort of average between the function and its integral of order one. The Laplace transform of the CFFD readswhere is the Laplace transform of

*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 asAgain, the NRLFD is without any singularity at compared to the classical Riemann-Liouville fractional order derivative and also verifiesCompared to (7), we note here the exact correspondence with at The Laplace transform of the NRLFD reads as [9, 10]

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 byassumed to satisfy the initial conditionwhere is the perturbation parameter and 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 This can be rewritten asSet nowThe next step is to look for a real constant such that In fact

Well known properties for the norms give Keeping in mind that and are bounded functions, then there exists real numbers and such thatSet ; hence,Therefore, the Lipschitz condition holds for the partial derivatives and and there exists a real constant such thatwhere the bounded condition (14) has been exploited, whence withThis proves that satisfies the Lipschitz condition and then, it allows us to state the following proposition.

Proposition 4. *If the condition 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):*

*Proof. *Let us go back to the model (16) rewritten aswhich yields the recurrence formulation given as follows:Letand it can be shown that is a continuous solution. Indeed, if we takethen, it is straightforward to see that More explicitly, we havePassing this equation to the norm yieldsApplying the Lipschitz condition to gives which can be rewritten asAfter integrating, we make use of well known properties of the recursive technique from (35) to have with , 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 For that, let Making use of (29), we should have In other terms, the gap that exists between and should vanish as Consider givingHence, and from the right hand side, we have Just take as the solution of (26) that is continuous. Moreover, the Lipschitz condition for yieldsThis yieldsconsidering the initial condition and taking the limit as gives *Uniqueness*. To prove that the solution is unique, we take two different functions and that satisfy the model (26); then,equivalently This yields if where we have used the Lipschitz condition for 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 equationWe start by considering the conventional case where to have, using , the following model:We investigate the traveling waves taking the form , where is the speed of the wave. We assume that does not depend on independently from but rather depends on the combined variable We also assume that the wave dies at infinity, meaningNow, 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, and which yield the following ODE:The -integration of this equation once giveswhere we have ignored the constant of -integration that is null due to boundary conditions (50). If the higher order perturbation term is ignored, then we haveThen we can solve numerically this equation by transforming it into a system of four ODEs of order one as follows:Numerical simulations are done in the phase-space () 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 , given asThis 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 They clearly point out relative irregularities when the derivative order of the CFFD is