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
(a)
(b)
(a)
(b)
4.2. Shape of Solitary Waves for the Higher Order Approximation
Now, the perturbation term is considered to have (48):subject to localized initial condition
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 obtainLet ; then, Application of inverse Laplace transform on both sides yields
This leads to the following recursive systemthen making the solution to be Numerical approximations are performed according to the following steps: (i) is considered as initial input: Choose as the number of terms to compute. Name to be the approximate solution. Set and (ii)For the other terms use . Compute . to finally get
where Note we can use , 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 () and the standard case ()).Note that, for , we recover for the conventional model (49) the following well known resultThis 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 admits the closed form found to beThe shape of numerical approximations is depicted in Figures 3–6, plotted for different values of the derivative order and showing a spot of irregular perturbed motion in the pure fractional case () compared to the conventional case (). 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.
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 term 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was partially supported by the National Research Foundation (NRF) of South Africa, Grant no. 105932.