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

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.


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 elds such as physics, mechanics, chemistry, and biology, fractional calculus is extremely useful [1][2][3][4][5][6][7][8][9][10][11][12]. e 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 eld 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 eld 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] veri ed strictly that the local fractional derivative did not exist at all on any interval. Taking into account these signi cant 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.
is derivative is particularly useful in describing some local power-law behaviors. Hence, we can learn a lot by studying solutions of conformable fractional di erential equations. In other words, the usage of the conformable fractional operator overcomes some restrictions of the di erent 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. us, it becomes more interesting in describing many physical problems. Now, let us give a short note about the conformable time fractional.
In what follows, we present some significant properties of the conformable derivatives. Let functions g 1 , g 2 be two α−conformable differential functions. For t > 0 and two constants a and b, we have the following properties: and α− differentiable and f is another function which is defined in the range of g, 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 form where D α t indicates the conformable time derivative of order α. When α ⟶ 1, system (2) is reduced to the integer-order derivative which has been introduced by Konno and Oono [17]. (3) 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][18][19][20][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 (G′/G)−expansion method [24], the modified exp(−Ω(ξ))− 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 (G ′ /G, 1/G)− expansion methods [32], and a new extended direct algebraic method [33]. Recently, some techniques have been imposed such as a generalized (G ′ /G)− 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][37][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][40][41][42][43][44][45][46][47][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. is 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][54][55][56][57][58][59] provide other uses of He's variational technique. e 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 timefractional 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.

Bifurcation Analysis
We consider the wave transformation where a and ω refer to a wave number and a wave velocity respectively. One can easily get where ′ refers to the differentiation with respect to η. Substituting into (2), we obtain By integrating (9), we have where C is an integral constant. Substituting in (8), it follows Eq. (11) can be expressed as a Hamilton system in the form given that aω ≠ 0. System (12) is a one-dimensional Hamiltonian system with a Hamiltonian function Due to the Hamiltonian function (13) does not rely explicitly on η, it is a conserved quantity. us, we have where h 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. e first is 1/2z 2 (η) which is explained physically as the kinetic energy per unit mass of a particle, and the second term is V(η) which is the potential per unit mass of the particle. Hence, the expression (14) characterizes the total energy per unit mass, and consequently, h 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.
e integration of Eq. (16) requires appropriate domains of the constants C, a, ω, and h. e domains of these constants can be found by applying distinct methods such as a complete discriminant method [60] for the quartic polynomial −ψ 4 + 2Cψ 2 + 2a 2 ω 2 h and bifurcation analysis [39][40][41][42][43][44][45][46][47][48]. e 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). is 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 ψ ′ (η) � 0 and z ′ (η) � 0; that is, they are (ψ 0 , 0), where ψ 0 is the possible real solution for Hence, the number of equilibrium points for system (12) is determined according to the value of C.
Secondly, we will determine the type of the equilibrium points and explain the phase plane for the Hamiltonian system (see, e.g., [61]). If C ≤ 0, the equilibrium point O � (0, 0) is center as in Figure 1 , 0) are center points as in Figure 1(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.
It is noted that all trajectories of the phase plot are bounded. e following theorem describes the phase portrait of system (12). Theorem 1. Assume that aω ≠ 0 and C ∈ R. If C ≤ 0, system (12) has a family of periodic closed orbits around the center point O as in Figure 1(a), while, for C > 0, the system has the following: (i) Two homoclinic orbits to the saddle point O, in red, around the two center point points P − and P + for h � 0 (ii) Two separated families of periodic closed orbits, in green, around the center points P − and P + , inside the homoclinic orbits, for

and (iii) A family of super-periodic closed orbits, in blue,
outside the homoclinic orbits, for h > 0, as in Figure 1(b)

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: Mathematical Problems in Engineering Theorem 2.
e solutions ψ(η) and ϕ(η) are periodic wave Taking into account the bifurcation's constraints on the two parameters C, h in the last theorem and the differential form (16), we obtain the following.

e Periodic Wave Solutions
where r 4 is the positive root of the polynomial −ψ 4 + 2Cψ 2 + 2a 2 ω 2 h � 0 with C ≤ 0 and h > 0 (see, e.g., [62]). erefore, the solutions of system (2) corresponding to the periodic orbits are given by and e 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 α 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 α takes values different from one, the periodic solutions lose their periodicity as shown in Figures 3(d) and 4(d). (12) admits periodic orbits around the equilibrium points P − and P + , given in Figure 1(b) in green. e periodic wave solutions after integrating (16) are obtained as where r 1 and r 2 are the positive roots of the polynomial −ψ 4 + 2Cψ 2 + 2a 2 ω 2 h � 0, with r 1 < r 2 , C > 0, and h 1 < h < 0. erefore, the solutions of system (2) corresponding to these periodic orbits are given by and e 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 α ⟶ 1, solutions (21) and (22) are also new solutions for the time-integer derivative of coupled Konno-Oono equation (2). It is noticeable that when α takes values different from one, the periodic solutions lose their periodicity as shown in Figures 5(d) and 6(d).
(c) When (C, h)∈ R + ×]0, ∞[, system (12) admits super periodic orbits, given in Figure 1(b) in blue. e periodic wave solutions after integrating (16) are obtained as where r 3 is the positive root of the polynomial −ψ 4 + 2Cψ 2 + 2a 2 ω 2 h � 0 with C > 0 and h > 0. erefore, the solutions of system (2) corresponding to these periodic orbits are given by and e expressions in (24) and (25) are new elliptic function solutions of conformable time-fractional coupled Konno-Oono equation (2). It is remarkable that, if α 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 α ≠ 1, these solutions also lose its periodicity.

e Solitary Wave Solutions. If
admits a homoclinic orbit, given in Figure 1 e periodic wave solutions after integrating (16) are obtained as erefore, the solitary wave solutions of system (2) corresponding to homoclinic orbit are given by and Mathematical Problems in Engineering e 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 α 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.

Remark 1. One of the most advantages of 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 h tends to zero. Consequently, the corresponding solutions to these orbits also must have this property. When h ⟶ 0, we have r 1 � 0, r 2 � �� � 2C √ , and hence, the periodic solution (21) and (22) (27) and (28). It is worth mentioning that the degeneracy property also proves the consistency of the obtained solutions and its validity.
(ii) e superperiodic orbits in blue will reduce to the homoclinic orbit in red as clarified by Figure 1 . Consequently, the periodic solution (24) and (25) degenerates into the solitary solution (27) and (28).

Remark 2.
e type of wave solutions for Equation (2) depends on the initial conditions. e degeneracy is always studied from the parameter h which is always determined from the initial conditions from the conserved quantity (14). is shows the dependence of the solutions on the initial conditions. For more clarifications, for certain initial conditions making h ∈ (h 1 , 0), we have a periodic solution, and for other initial conditions making h � 0, we have a solitary wave solution. Notice that both obtained solutions are completely different from the mathematical and physics point of view.

Variational Principal
e variational principle of Eq.(11) can be found by using the semiinverse method [49,[53][54][55], which has been used widely to construct the needed variational formulations by     Mathematical Problems in Engineering presenting an undetermined function as a trial functional [56][57][58][59].

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 form where p 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 q, we get Hence, the bright soliton solution in Eq.(30) takes the form Hence, system (2) admits the solutions

Periodic Solution.
We assume the solution is in the form ψ(η) � q cos(η).

Conclusion
is work is interested in studying the conformable timefractional 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. is 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. is study has enabled us to prove eorem 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. ree 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 α. Or, equivalently, we have outlined the effect of the fractional-order derivative α 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. e 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