Dynamical Behaviour of Conformable Time-Fractional Coupled Konno-Oono Equation in Magnetic Field

Elbrolosy, M. E.; Elmandouh, A. A.

doi:https://doi.org/10.1155/2022/3157217

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Fractional-Order Systems: Control Theory and Applications 2022

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Research Article | Open Access

Volume 2022 | Article ID 3157217 | https://doi.org/10.1155/2022/3157217

Dynamical Behaviour of Conformable Time-Fractional Coupled Konno-Oono Equation in Magnetic Field

M. E. Elbrolosy^1,2and A. A. Elmandouh^1,3

Academic Editor: Abdellatif Ben Makhlouf

Received05 May 2022

Accepted24 May 2022

Published28 Jun 2022

Abstract

Our aim is to construct distinct types of novel traveling wave solutions for the nonlinear conformable time-fractional coupled Konno–Oono equation in a magnetic field utilizing the qualitative analysis of the dynamical system that corresponds to the Hamiltonian system. The influences of fractional-order derivative on the wave solutions in addition to equation parameters on the waveform are investigated. Furthermore, He’s variational technique has been applied to seek some new solutions. 2D and 3D graphics of the obtained solutions are presented to clarify the physical and mathematical meaning of the waves.

1. Introduction

Models based on fractional derivatives have been successful in describing nonlinear physical phenomena. A continuing interest in fractional calculus resides in its applicability. In fields such as physics, mechanics, chemistry, and biology, fractional calculus is extremely useful [1–12]. The properties of derivatives in classical Riemann–Liouville fractional calculus, however, are more complicated than those in usual derivatives. Leibnitz’s rule, for instance, cannot be applied. Tarasov had proved several important theorems, in the field of theory, to support his claim that, in general, Leibnitz’s rule does not apply for Riemann–Liouville’s fractional derivatives [13]. On the other hand, there are some serious mistakes in existed theory such as Jumarie’s fractional derivative. Lately, Liu supplied essential development in the field since he provided counterexamples to show that Jumarie’s basic formulas are not correct [14, 15]. Respectively, all those results based on Jumarie’s formulas are also incorrect. Moreover, Liu [14] verified strictly that the local fractional derivative did not exist at all on any interval. Taking into account these significant results, we should neatly verify the fundamental formulas when the fractional derivative is considered. Conformal fractional derivatives are introduced to overcome the limitations of usual fractional derivatives, which satisfy routine derivative rules and therefore have attracted the attention of numerous scientists. This derivative is particularly useful in describing some local power-law behaviors. Hence, we can learn a lot by studying solutions of conformable fractional differential equations. In other words, the usage of the conformable fractional operator overcomes some restrictions of the different fractional operator’s properties such as the chain rule, the derivative of the quotient of two functions, the product of two functions, and the mean value theorem. Thus, it becomes more interesting in describing many physical problems. Now, let us give a short note about the conformable time fractional.

Definition 1 (see [16]). Let be a function, then the conformable fractional derivative of order is defined asfor all and .
In what follows, we present some significant properties of the conformable derivatives. Let functions be two conformable differential functions. For and two constants and , we have the following properties:(1)(2) for all (3)(4)(5)(6)If is a map which is differentiable and differentiable and is another function which is defined in the range of , then Indeed, fractional calculus was recently utilized to model the majority of physical and engineering processes that, in some cases, provide an adequate description of such models in a comparison of integer-order derivatives. Consequently, this motivates us to study the conformable time-fractional coupled Konno–Oono equation. It takes the formwhere indicates the conformable time derivative of order . When , system (2) is reduced to the integer-order derivative which has been introduced by Konno and Oono [17].Physically, system (3) describes a current-fed string interacting with an external magnetic fed. Geometrically, it describes the parallel transport of points of the arc along the time direction such that the connection is magnetic-valued [17–21]. It has attracted great interest from mathematicians and physicists, and many mathematical approaches have been developed to investigate novel solutions to it. In [22], the modified simple equation method has been utilized for getting some properties of system (3) where some solutions have been obtained like kink solutions and bell-shaped solutions. In [23], the tanh-function method and extended tanh-function method have been applied to construct soliton solutions for system (3). Additional methods have been forced such as a new generalized expansion method [24], the modified expansion function method [25], the sine-Gordon expansion method [26], the external trial equation method [27], the generalized exp-function method [28], a modified extended exp-function method [29], the extended simplest equation method [30], the extended Jacobian elliptic function expansion method [31], the functional variable and the two variables expansion methods [32], and a new extended direct algebraic method [33]. Recently, some techniques have been imposed such as a generalized expansion method for stochastic Konno–Oono equation that is forced by multiplicative noise term [34] and a unified solver with the aid of probability function distributions [35], for more details about wave solutions for stochastic PDE (see, e.g., [36–38]).
In this work, we study the existence and the type of some novel traveling wave solutions for (2) by the qualitative theory of planar dynamical system which has been applied successfully in various works such as [39–48] before constructing them through the kind of the orbits. For instance, the existence of periodic, homoclinic, and heteroclinic orbits imply to the existence of periodic, solitary, and kink (or antikink) wave solutions. We find the wave solutions by integrating the conserved quantity along with certain parts of the phase orbits corresponding to real wave propagation to avoid the presence of complex solutions which are not desirable in the majority of the physical world. We also study the degeneracy of the solutions throughout the transmission of the phase orbits. This method is summarized in Appendix A. He who is a Chinese mathematician, in his review paper [49] and monograph [50], developed a primitive but intriguing variational approach to the search for solitary solutions to nonlinear wave equations. He’s variational method was determined to be highly simple and successful by Tao [51, 52]. References [49, 53–59] provide other uses of He’s variational technique.
The paper in the following is organized as follows: Section 2 contains the study of the bifurcation analysis for the Hamiltonian system corresponding to the conformable time-fractional coupled Konno–Oono equation. In Section 3, we construct some new bounded wave solutions for the conformable time-fractional coupled Konno–Oono equation and study the degeneracy of these solutions through the transmission between the phase plane orbits. In Section 4, we apply the variational principle to construct abundant solutions including bright soliton, bright-like soliton, kinky-bright soliton, and periodic solutions. Section 5 is the conclusion.

2. Bifurcation Analysis

We consider the wave transformationwhere and refer to a wave number and a wave velocity respectively. One can easily getwhere refers to the differentiation with respect to . Substituting into (2), we obtain

By integrating (9), we havewhere is an integral constant. Substituting in (8), it follows

Eq. (11) can be expressed as a Hamilton system in the formgiven that . System (12) is a one-dimensional Hamiltonian system with a Hamiltonian function

Due to the Hamiltonian function (13) does not rely explicitly on , it is a conserved quantity. Thus, we havewhere is a constant. Equivalently, the problem of finding a wave solution for conformable time-fractional coupled Konno–Oono equation is reduced to the problem of finding a solution of a Hamiltonian system (12) which describes the one-dimensional motion of a unit mass particle under the influence of the potential forces derived from the potential function.

Notice, the expression (14) consists of two parts. The first is which is explained physically as the kinetic energy per unit mass of a particle, and the second term is which is the potential per unit mass of the particle. Hence, the expression (14) characterizes the total energy per unit mass, and consequently, is the value of the total energy per unit mass. To obtain the traveling wave solutions of (11), we combine Eq. (14) and the first equation in (12), and we obtain the first-order separable differential equation.

The integration of Eq. (16) requires appropriate domains of the constants , , , and . The domains of these constants can be found by applying distinct methods such as a complete discriminant method [60] for the quartic polynomial and bifurcation analysis [39–48]. The bifurcation analysis is more significant because it gives the range of those parameters and specifies the kind of the solution before constructing them. For illustration, the existence of periodic, homoclinic, and heteroclinic trajectories refers to the presence of the periodic, solitary, and kink (antikink) solutions. Furthermore, it enables us to examine the dependence of the obtained wave solutions on the given initial conditions for the Hamiltonian system (12). This motivates us to study the bifurcation analysis and examine the phase portrait for the Hamiltonian system (12).

We first find the equilibrium points for the Hamiltonian system (12) by setting and ; that is, they are , where is the possible real solution for . Hence, the number of equilibrium points for system (12) is determined according to the value of .(i)If , there is a unique equilibrium point for system (12) which is (ii)If , there are three equilibrium points for system (12) which are and

Secondly, we will determine the type of the equilibrium points and explain the phase plane for the Hamiltonian system (see, e.g., [61]). If , the equilibrium point is center as in Figure 1(a). If , the equilibrium point is a saddle point while are center points as in Figure 1(b).

(a)

(b)

Notice, as a result of system (12) being a Hamiltonian, the equilibrium points appears as a critical point for the potential function (15), and their nature is specified; accordingly, these equilibrium points are either local maximum (saddle) or local minimum (center), as in Figure 2.

(a)

(b)

It is noted that all trajectories of the phase plot are bounded. The following theorem describes the phase portrait of system (12).

Theorem 1. Assume thatand. If, system (12) has a family of periodic closed orbits around the center pointas in Figure 1(a), while, for, the system has the following:(i)Two homoclinic orbits to the saddle point, in red, around the two center point pointsandfor(ii)Two separated families of periodic closed orbits, in green, around the center pointsand, inside the homoclinic orbits, for, where, and(iii)A family of super-periodic closed orbits, in blue, outside the homoclinic orbits, for, as in Figure 1(b)

3. Classification of All Traveling Wave Solutions

Based on the bifurcation analysis in the previous section, the types of the wave solutions can be classified according to the following theorem:

Theorem 2. The solutionsandare periodic wave solutions if, where. They are solitary wave solutions if.

Taking into account the bifurcation’s constraints on the two parameters , in the last theorem and the differential form (16), we obtain the following.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

3.1. The Periodic Wave Solutions

(a)When , system (12) admits periodic orbits, given in Figure 1(a) in blue. The periodic wave solutions after integrating (16) are obtained as where is the positive root of the polynomial with and (see, e.g., [62]). Therefore, the solutions of system (2) corresponding to the periodic orbits are given by and The expressions in (18) and (19) are new elliptic function solutions of conformable time-fractional coupled Konno–Oono equation (2) and are given geometrically in Figures 3 and 4, respectively. Notice, when the fractional-order tends to one, the expressions (18) and (19) reduce to new elliptic solutions for the integer time order derivative version of the Konno–Oono equation (2). It is noticeable that when takes values different from one, the periodic solutions lose their periodicity as shown in Figures 3(d) and 4(d).(b)When , system (12) admits periodic orbits around the equilibrium points and , given in Figure 1(b) in green. The periodic wave solutions after integrating (16) are obtained as where and are the positive roots of the polynomial , with , , and . Therefore, the solutions of system (2) corresponding to these periodic orbits are given by and The expressions in (21) and (22) are new elliptic function solutions of conformable time-fractional coupled Konno–Oono equation (2) and are given geometrically in Figures 5 and 6, respectively. It is worth mentioning that, if , solutions (21) and (22) are also new solutions for the time-integer derivative of coupled Konno–Oono equation (2). It is noticeable that when takes values different from one, the periodic solutions lose their periodicity as shown in Figures 5(d) and 6(d).(c)When , system (12) admits super periodic orbits, given in Figure 1(b) in blue. The periodic wave solutions after integrating (16) are obtained as where is the positive root of the polynomial with and . Therefore, the solutions of system (2) corresponding to these periodic orbits are given byand

The expressions in (24) and (25) are new elliptic function solutions of conformable time-fractional coupled Konno–Oono equation (2). It is remarkable that, if tends to one, the solutions (24) and (25) are also new solutions for the time-integer derivative of coupled Konno–Oono equation (2). Also, as in the two previous cases, when , these solutions also lose its periodicity.

3.2. The Solitary Wave Solutions

If , system (12) admits a homoclinic orbit, given in Figure 1(b) in red. The periodic wave solutions after integrating (16) are obtained as

Therefore, the solitary wave solutions of system (2) corresponding to homoclinic orbit are given byand

The expressions in (27) and (28) are new hyperbolic function solutions of conformable time-fractional coupled Konno–Oono equation (2) and are given geometrically in Figures 7 and 8, respectively. When approaches to one, the solutions (27) and (28) will reduce to a well-known solution for time-integer derivatives coupled Konno- Oono equation (2) [27, 31, 32]. Notice, as outlined in Figures 7(d) and 8(d), the fractional-order derivative influences the Shap of the solitary wave solutions.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

Remark 1. One of the most advantagesof the bifurcation analysis is the study of the degeneracy of the obtained solutions through the transmission between the phase orbits. Let us clarify this point.(i)It is remarkable that the two periodic families of orbits in green as outlined in Figure 1(b) will degenerate to the homoclinic orbit in red when tends to zero. Consequently, the corresponding solutions to these orbits also must have this property. When , we have , and hence, the periodic solution (21) and (22) degenerate into the solitary solution (27) and (28). It is worth mentioning that the degeneracy property also proves the consistency of the obtained solutions and its validity.(ii)The superperiodic orbits in blue will reduce to the homoclinic orbit in red as clarified by Figure 1(b) when . If , we have . Consequently, the periodic solution (24) and (25) degenerates into the solitary solution (27) and (28).

Remark 2. The type of wave solutions for Equation(2) depends on the initial conditions. The degeneracy is always studied from the parameterwhich is always determined from the initial conditions from the conserved quantity (14). This shows the dependence of the solutions on the initial conditions. For more clarifications, for certain initial conditions making, we have a periodic solution, and for other initial conditions making, we have a solitary wave solution. Notice that both obtained solutions are completely different from the mathematical and physics point of view.

4. Variational Principal

The variational principle of Eq.(11) can be found by using the semiinverse method [49, 53–55], which has been used widely to construct the needed variational formulations by presenting an undetermined function as a trial functional [56–59].

4.1. Abundant Solutions

4.1.1. Bright Soliton

He’s variational method, which is presented in [58], is a powerful method for finding approximated wave solutions. We assume Eq.(11) as a bright soliton solution in the formwhere is a constant. Entering the expression (30) into Eq.(29), we get

According to the basis of the Ritz-like method, the stationary condition for Eq.(31) gives

Solving the last equation for , we get

Hence, the bright soliton solution in Eq.(30) takes the form

Hence, system (2) admits the solutions

4.1.2. Bright-Like Soliton

We assume that the solution of Eq.(11) is in the form

Inserting Eq.(40) into Eq.(29), we obtain

The stationary condition for (37) iswhich implies to

Thus, the bright dark solution in Eq.(40) takes the form

Hence, system (2) admits the solution

4.1.3. Kinky-Bright Soliton

We assume a kinky-bright solitons solution of Eq.(11) in the form

Inserting the expression Eq.(51) into Eq.(29), we obtain

Computing the stationary condition , we have

Therefore, the kinky-bright soliton in Eq.(51) is

Hence, system (2) admits the solution

4.1.4. Periodic Solution

We assume the solution is in the form

Inserting the expression Eq.(47) into Eq.(29), we obtainwhere indicates the period of the nonlinear term of Eq.(11); that is, . Thus,

The stationary condition implies

Therefore, the solution in Eq.(47) is

Hence, system (2) admits the solution

5. Conclusion

This work is interested in studying the conformable time-fractional coupled Konno–Oono equation in a magnetic field. A certain transformation has been applied to convert this equation into an ordinary differential equation as well as it has been rewritten as a two-dimensional dynamical system. This system is a conservative Hamiltonian system. A qualitative analysis or the qualitative theory of the planar dynamical system has been applied to study the bifurcation and the phase space. This study has enabled us to prove Theorem 2 including the conditions on the parameters that guarantee the existence of periodic and solitary wave solutions before constructing them via the type of the phase plane orbits. Taking into account the bifurcation constraints of the system’s parameters, the conserved quantity has been utilized to construct some new solutions for the equation under consideration. Three families of periodic wave solutions and a solitary wave solution have been established. A 2D and 3D for these solutions have been clarified graphically for diverse values of the fractional-order . Or, equivalently, we have outlined the effect of the fractional-order derivative on the obtained solutions. Moreover, the obtained new solutions have been reduced to new wave solutions for the time-integer derivative coupled Konno–Oono equation. The degeneracy of Jacobi-elliptic wave solutions has been examined via the transmission between the orbits of the phase plane orbits. From another side, He’s variational method has been applied to find some new solutions which have been assorted into bright soliton, bright-like soliton, kinky-bright soliton, and periodic solutions.

Appendix

A. Wave Solutions via Conserved Quantity

Consider a conformable time-fractional system of partial differential equation taking the formwhere is the time conformal fractional derivative with order , refers to the time, and are two real-valued functions depending on the two variables . To find wave solutions for system (A.1), we follow the next steps:

Step 1. Applying the transformationwhere are arbitrary constants to the system (A.2), we obtain, after some calculations, a system of second-order ordinary differential equations of and which, by substitution, will be rewritten as one degree of freedom Hamiltonian system; that is, it takes the formwhere is a new variable introduced to convert the second-order differential equation into a dynamical system, is a real-valued function, and indicates derivative with respect to .

Step 2. Calculate the Hamiltonian function asfrom which we obtain the separable differential equation

Step 3. Find the equilibrium points for the Hamiltonian system (A.3) and apply the qualitative theory for the planar dynamical system to specify their nature

Step 4. Find the intervals of real wave propagation which is equivalent to the intervals of real motion of a particle described by the Hamiltonian system (A.3). Integrating both sides of (A.5) along certain intervals of real propagation gives different types of wave solutions and hence follows.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors acknowledge the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research at King Faisal University, Saudi Arabia, for financial support under the annual funding track (GRANT847).

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Copyright © 2022 M. E. Elbrolosy and A. A. Elmandouh. 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.

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