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

Authors: M. E. Elbrolosy; A. A. Elmandouh
Journal: Mathematical Problems in Engineering
 (2022)

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## 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.


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## Body

## 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]).
Letg:0,∞⟶ℝ be a function, then the conformable fractional derivative of order α is defined as(1)Tαgt=limσ⟶0gt+σt1−α−gtσ,for all t>0 and 0<α≤1.
In what follows, we present some significant properties of the conformable derivatives. Let functionsg1,g2 be two α−conformable differential functions. For t>0 and two constants a and b, we have the following properties:(1)
Tαag1+bg2=aTαg1+bTαg2(2)
Tαtρ=ρtρ−α for all ρ∈ℝ(3)
Tαg1g2=g1Tαg2+g2Tαg1(4)
Tαg1/g2=1/g22g2Tαg1−g1Tαg1(5)
Tαgt=t1−αdg/dtt(6)
Ifg:0,∞⟶ℝ is a map which is differentiable and α− differentiable and f is another function which is defined in the range of g, then Tαg°f=t1−αf′tg′ft
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(2)Dtαpxx,t−2px,tvx,t=0,Dtαvx,t+2px,tpxx,t=0,where Dtα 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)pxtx,t−2px,tvx,t=0,vtx,t+2px,tpxx,t=0.
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 G′/G−expansion method [24], the modified exp−Ωξ− 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., [36–38]).
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: Section2 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 transformation(4)px,t=ψη,η=ax−ωαtα,(5)vx,t=ϕη,where a and ω refer to a wave number and a wave velocity respectively. One can easily get(6)Dtαpxx,t=aDtaψ′η=−a2ωψ′′η,(7)Dtαvxx,t=−aωϕ′η,where ′ refers to the differentiation with respect to η. Substituting into (2), we obtain(8)a2ωψ′′η+2ψηϕη=0,(9)−aωϕ′η+2aψηψ′η=0.By integrating (9), we have(10)ϕη=ψ2η−Cω,where C is an integral constant. Substituting in (8), it follows(11)a2ω2ψ′′η+2ψ3η−2Cψη=0.Eq. (11) can be expressed as a Hamilton system in the form(12)ψ′η=zη,z′η=−2a2ω2ψ3η+2Ca2ω2ψη,given that aω≠0. System (12) is a one-dimensional Hamiltonian system with a Hamiltonian function(13)Hψ,z=z2η2+12a2ω2ψ4η−Ca2ω2ψ2η.Due to the Hamiltonian function (13) does not rely explicitly on η, it is a conserved quantity. Thus, we have(14)z2η2+12a2ω2ψ4η−Ca2ω2ψ2η=h,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.(15)Vψ=12a2ω2ψ4η−Ca2ω2ψ2η.Notice, the expression (14) consists of two parts. The first is 1/2z2η which is explained physically as the kinetic energy per unit mass of a particle, and the second term is Vη 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.(16)dψ−ψ4+2Cψ2+2a2ω2h=±dηaω.The integration of Eq. (16) requires appropriate domains of the constants C, a, ω, and h. The domains of these constants can be found by applying distinct methods such as a complete discriminant method [60] for the quartic polynomial −ψ4+2Cψ2+2a2ω2h 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 ψ′η=0 and z′η=0; that is, they are ψ0,0, where ψ0 is the possible real solution for ψ02ψ02−C=0. Hence, the number of equilibrium points for system (12) is determined according to the value of C.(i)
IfC≤0, there is a unique equilibrium point for system (12) which is O=0,0(ii)
IfC>0, there are three equilibrium points for system (12) which are O and P±=±C,0Secondly, 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(a). If C>0, the equilibrium point O=0,0 is a saddle point while P±=±C,0 are center points as in Figure 1(b).Figure 1
Phase portrait for the dynamical system (12) in the phase plane (ψ, z). The solid circles indicate the equilibrium points. (a) C<0. (b) C>0.
(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.Figure 2
The potential of the dynamical system (12). (a) C<0. (b) C>0.
(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 thataω≠0andC∈ℝ. IfC≤0, system (12) has a family of periodic closed orbits around the center pointOas in Figure 1(a), while, forC>0, the system has the following:(i)
Two homoclinic orbits to the saddle pointO, in red, around the two center point pointsP−andP+forh=0(ii)
Two separated families of periodic closed orbits, in green, around the center pointsP−andP+, inside the homoclinic orbits, forh∈h1,0, whereh1=−C2/2a2ω2, and(iii)
A family of super-periodic closed orbits, in blue, outside the homoclinic orbits, forh>0, 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 solutionsψηandϕηare periodic wave solutions ifC,h∈−∞,0×ℝ+∪ℝ+×h1,0∪ℝ+×0,∞, whereh1=−C2/2a2ω2. They are solitary wave solutions ifC,h∈ℝ+×0.Taking into account the bifurcation’s constraints on the two parametersC, h in the last theorem and the differential form (16), we obtain the following.Figure 3
Solution (18) for various values of α. (a) α=0.1. (b) α=0.4. (c) α=1. (d) 2−D graph.
(a)(b)(c)(d)Figure 4
Solution (19) for various values of α. (a) α=0.1. (b) α=0.4. (c) α=1. (d) 2−D graph.
(a)(b)(c)(d)Figure 5
Solution (21) for various values of α. (a) α=0.1. (b) α=0.4. (c) α=1. (d) 2−D graph.
(a)(b)(c)(d)Figure 6
Solution (22) for various values of α. (a) α=0.1. (b) α=0.4. (c) α=1. (d) 2−D graph.
(a)(b)(c)(d)


### 3.1. The Periodic Wave Solutions

(a)
WhenC,h∈−∞,0×ℝ+, system (12) admits periodic orbits, given in Figure 1(a) in blue. The periodic wave solutions after integrating (16) are obtained as(17)ψη=±aωhC2+2a2ω2hsd2C2+2a2ω2haωη,r42C2+2a2ω2h,wherer4 is the positive root of the polynomial −ψ4+2Cψ2+2a2ω2h=0 with C≤0 and h>0 (see, e.g., [62]). Therefore, the solutions of system (2) corresponding to the periodic orbits are given by(18)px,t=±aωhC2+2a2ω2hsd2C2+2a2ω2hxω−tαα,r42C2+2a2ω2h,and(19)vx,t=a2ωhC2+2a2ω2hsd22C2+2a2ω2hxw−tαα,r42C2+2a2ω2h−C.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 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).(b)
WhenC,h∈ℝ+×h1,0, system (12) admits periodic orbits around the equilibrium points P− and P+, given in Figure 1(b) in green. The periodic wave solutions after integrating (16) are obtained as(20)ψη=±r2dnr2aωη,1−r12r22,wherer1 and r2 are the positive roots of the polynomial −ψ4+2Cψ2+2a2ω2h=0, with r1<r2, C>0, and h1<h<0. Therefore, the solutions of system (2) corresponding to these periodic orbits are given by(21)px,t=±r2dnr2xw−tαα,1−r12r22,and(22)vx,t=1ωr22dn2r2xw−tαα,1−r12r22−C.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 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)
WhenC,h∈ℝ+×0,∞, system (12) admits super periodic orbits, given in Figure 1(b) in blue. The periodic wave solutions after integrating (16) are obtained as(23)ψη=±r3cn2Caωη,r32C,wherer3 is the positive root of the polynomial −ψ4+2Cψ2+2a2ω2h=0 with C>0 and h>0. Therefore, the solutions of system (2) corresponding to these periodic orbits are given by(24)px,t=±r3cn2Cxw−tαα,r32C,and(25)vx,t=1ωr32cn22Cxw−tαα,r32C−C.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 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.


### 3.2. The Solitary Wave Solutions

IfC,h∈ℝ+×0, system (12) admits a homoclinic orbit, given in Figure 1(b) in red. The periodic wave solutions after integrating (16) are obtained as(26)ψη=±2Csech2Caωη.Therefore, the solitary wave solutions of system (2) corresponding to homoclinic orbit are given by(27)px,t=±2Csech2Cxw−tαα,and(28)vx,t=Cω2sech22Cxw−tαα−1.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 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.Figure 7
Solution (27) for various values of α. (a) α=0.1. (b) α=0.4. (c) α=1. (d) 2−D graph.
(a)(b)(c)(d)Figure 8
Solution (28) for various values of α. (a) α=0.1. (b) α=0.4. (c) α=1. (d) 2−D graph.
(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 Figure1(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 r1=0,r2=2C, 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 Figure1(b) when h⟶0. If h⟶0, we have r3=2C. 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 parameterhwhich 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 makingh∈h1,0, we have a periodic solution, and for other initial conditions makingh=0, we have a solitary wave solution. Notice that both obtained solutions are completely different from the mathematical and physics point of view.


## 3.1. The Periodic Wave Solutions

(a)
WhenC,h∈−∞,0×ℝ+, system (12) admits periodic orbits, given in Figure 1(a) in blue. The periodic wave solutions after integrating (16) are obtained as(17)ψη=±aωhC2+2a2ω2hsd2C2+2a2ω2haωη,r42C2+2a2ω2h,wherer4 is the positive root of the polynomial −ψ4+2Cψ2+2a2ω2h=0 with C≤0 and h>0 (see, e.g., [62]). Therefore, the solutions of system (2) corresponding to the periodic orbits are given by(18)px,t=±aωhC2+2a2ω2hsd2C2+2a2ω2hxω−tαα,r42C2+2a2ω2h,and(19)vx,t=a2ωhC2+2a2ω2hsd22C2+2a2ω2hxw−tαα,r42C2+2a2ω2h−C.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 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).(b)
WhenC,h∈ℝ+×h1,0, system (12) admits periodic orbits around the equilibrium points P− and P+, given in Figure 1(b) in green. The periodic wave solutions after integrating (16) are obtained as(20)ψη=±r2dnr2aωη,1−r12r22,wherer1 and r2 are the positive roots of the polynomial −ψ4+2Cψ2+2a2ω2h=0, with r1<r2, C>0, and h1<h<0. Therefore, the solutions of system (2) corresponding to these periodic orbits are given by(21)px,t=±r2dnr2xw−tαα,1−r12r22,and(22)vx,t=1ωr22dn2r2xw−tαα,1−r12r22−C.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 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)
WhenC,h∈ℝ+×0,∞, system (12) admits super periodic orbits, given in Figure 1(b) in blue. The periodic wave solutions after integrating (16) are obtained as(23)ψη=±r3cn2Caωη,r32C,wherer3 is the positive root of the polynomial −ψ4+2Cψ2+2a2ω2h=0 with C>0 and h>0. Therefore, the solutions of system (2) corresponding to these periodic orbits are given by(24)px,t=±r3cn2Cxw−tαα,r32C,and(25)vx,t=1ωr32cn22Cxw−tαα,r32C−C.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 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.


## 3.2. The Solitary Wave Solutions

IfC,h∈ℝ+×0, system (12) admits a homoclinic orbit, given in Figure 1(b) in red. The periodic wave solutions after integrating (16) are obtained as(26)ψη=±2Csech2Caωη.Therefore, the solitary wave solutions of system (2) corresponding to homoclinic orbit are given by(27)px,t=±2Csech2Cxw−tαα,and(28)vx,t=Cω2sech22Cxw−tαα−1.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 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.Figure 7
Solution (27) for various values of α. (a) α=0.1. (b) α=0.4. (c) α=1. (d) 2−D graph.
(a)(b)(c)(d)Figure 8
Solution (28) for various values of α. (a) α=0.1. (b) α=0.4. (c) α=1. (d) 2−D graph.
(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 Figure1(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 r1=0,r2=2C, 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 Figure1(b) when h⟶0. If h⟶0, we have r3=2C. 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 parameterhwhich 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 makingh∈h1,0, we have a periodic solution, and for other initial conditions makingh=0, 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].(29)ℒ=∫−12ψ′2η+12a2ω2ψ4η−ca2ω2ψη2dη.


### 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 form(30)ψη=qsechη,where p is a constant. Entering the expression (30) into Eq.(29), we get(31)ℒq=∫0∞−q212+Ca2ω2sech2η+q221+q2a2ω2sech4ηdη=−q26a2ω2a2ω2+6C−2q2.According to the basis of the Ritz-like method, the stationary condition for Eq.(31) gives(32)∂ℒq∂q=q3a2ω24q2−a2ω2+6C=0.Solving the last equation forq, we get(33)q=±12a2ω2+6C.Hence, the bright soliton solution in Eq.(30) takes the form(34)ψη=±12a2ω2+6Csechη.Hence, system (2) admits the solutions(35)px,t=±12a2ω2+6Csechax−ωαtα,vx,t=1ω14a2ω2+6Csech2ax−ωαtα−C.


#### 4.1.2. Bright-Like Soliton

We assume that the solution of Eq.(11) is in the form(36)ψη=q1+coshη.Inserting Eq.(40) into Eq.(29), we obtain(37)Jq=∫0∞−q2coshη−121+coshη3+q42a2ω21+coshη4−Cq2a2ω21+coshη2dη=6q2−7a2ω2−70Cq2210a2ω2.The stationary condition for (37) is(38)∂Jq∂q=12q2−7a2ω2−70Cq105a2ω2=0,which implies to(39)q=±21a2ω2+210C6.Thus, the bright dark solution in Eq.(40) takes the form(40)ψη=±21a2ω2+210C61+coshη.Hence, system (2) admits the solution(41)px,t=±21a2ω2+210C61+coshax−ω/αtα,vx,t=1ω7a2ω2+70C121+cosh2ax−ω/αtα−C.


#### 4.1.3. Kinky-Bright Soliton

We assume a kinky-bright solitons solution of Eq.(11) in the form(42)ψη=qsech2η.Inserting the expression Eq.(51) into Eq.(29), we obtain(43)Jq=∫0∞q42a2ω2sech8η+2q2sech6η−2+Ca2ω2q2sech4ηdη=212q2−14a2ω2−35Cq2105a2ω2.Computing the stationary condition∂Jq/∂q=0, we have(44)q=±84a2ω2+210C12.Therefore, the kinky-bright soliton in Eq.(51) is(45)ψη=±84a2ω2+210C12sech2η.Hence, system (2) admits the solution(46)px,t=±84a2ω2+210C12sech2ax−ωαtα,vx,t=1ω14a2ω2+35C24sech4ax−ωαtα−C.


#### 4.1.4. Periodic Solution

We assume the solution is in the form(47)ψη=qcosη.Inserting the expression Eq.(47) into Eq.(29), we obtain(48)Jq=∫0T/4q42a2ω2cos4η+q212−Ca2ω2cos2η−q22dη,where T indicates the period of the nonlinear term of Eq.(11); that is, T=2π. Thus,(49)Jq=π3q2−4a2ω2−8Cq232a2ω2.The stationary condition∂Jq/∂q=0 implies(50)q=±6a2ω2+12C3.Therefore, the solution in Eq.(47) is(51)ψη=±6a2ω2+12C3cosη.Hence, system (2) admits the solution(52)px,t=±6a2ω2+12C3cosax−ωαtα,vx,t=1ω2a2ω2+4C3cos2ax−ωαtα−C.


## 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 form(30)ψη=qsechη,where p is a constant. Entering the expression (30) into Eq.(29), we get(31)ℒq=∫0∞−q212+Ca2ω2sech2η+q221+q2a2ω2sech4ηdη=−q26a2ω2a2ω2+6C−2q2.According to the basis of the Ritz-like method, the stationary condition for Eq.(31) gives(32)∂ℒq∂q=q3a2ω24q2−a2ω2+6C=0.Solving the last equation forq, we get(33)q=±12a2ω2+6C.Hence, the bright soliton solution in Eq.(30) takes the form(34)ψη=±12a2ω2+6Csechη.Hence, system (2) admits the solutions(35)px,t=±12a2ω2+6Csechax−ωαtα,vx,t=1ω14a2ω2+6Csech2ax−ωαtα−C.


### 4.1.2. Bright-Like Soliton

We assume that the solution of Eq.(11) is in the form(36)ψη=q1+coshη.Inserting Eq.(40) into Eq.(29), we obtain(37)Jq=∫0∞−q2coshη−121+coshη3+q42a2ω21+coshη4−Cq2a2ω21+coshη2dη=6q2−7a2ω2−70Cq2210a2ω2.The stationary condition for (37) is(38)∂Jq∂q=12q2−7a2ω2−70Cq105a2ω2=0,which implies to(39)q=±21a2ω2+210C6.Thus, the bright dark solution in Eq.(40) takes the form(40)ψη=±21a2ω2+210C61+coshη.Hence, system (2) admits the solution(41)px,t=±21a2ω2+210C61+coshax−ω/αtα,vx,t=1ω7a2ω2+70C121+cosh2ax−ω/αtα−C.


### 4.1.3. Kinky-Bright Soliton

We assume a kinky-bright solitons solution of Eq.(11) in the form(42)ψη=qsech2η.Inserting the expression Eq.(51) into Eq.(29), we obtain(43)Jq=∫0∞q42a2ω2sech8η+2q2sech6η−2+Ca2ω2q2sech4ηdη=212q2−14a2ω2−35Cq2105a2ω2.Computing the stationary condition∂Jq/∂q=0, we have(44)q=±84a2ω2+210C12.Therefore, the kinky-bright soliton in Eq.(51) is(45)ψη=±84a2ω2+210C12sech2η.Hence, system (2) admits the solution(46)px,t=±84a2ω2+210C12sech2ax−ωαtα,vx,t=1ω14a2ω2+35C24sech4ax−ωαtα−C.


### 4.1.4. Periodic Solution

We assume the solution is in the form(47)ψη=qcosη.Inserting the expression Eq.(47) into Eq.(29), we obtain(48)Jq=∫0T/4q42a2ω2cos4η+q212−Ca2ω2cos2η−q22dη,where T indicates the period of the nonlinear term of Eq.(11); that is, T=2π. Thus,(49)Jq=π3q2−4a2ω2−8Cq232a2ω2.The stationary condition∂Jq/∂q=0 implies(50)q=±6a2ω2+12C3.Therefore, the solution in Eq.(47) is(51)ψη=±6a2ω2+12C3cosη.Hence, system (2) admits the solution(52)px,t=±6a2ω2+12C3cosax−ωαtα,vx,t=1ω2a2ω2+4C3cos2ax−ωαtα−C.


## 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 form(30)ψη=qsechη,where p is a constant. Entering the expression (30) into Eq.(29), we get(31)ℒq=∫0∞−q212+Ca2ω2sech2η+q221+q2a2ω2sech4ηdη=−q26a2ω2a2ω2+6C−2q2.According to the basis of the Ritz-like method, the stationary condition for Eq.(31) gives(32)∂ℒq∂q=q3a2ω24q2−a2ω2+6C=0.Solving the last equation forq, we get(33)q=±12a2ω2+6C.Hence, the bright soliton solution in Eq.(30) takes the form(34)ψη=±12a2ω2+6Csechη.Hence, system (2) admits the solutions(35)px,t=±12a2ω2+6Csechax−ωαtα,vx,t=1ω14a2ω2+6Csech2ax−ωαtα−C.


## 4.1.2. Bright-Like Soliton

We assume that the solution of Eq.(11) is in the form(36)ψη=q1+coshη.Inserting Eq.(40) into Eq.(29), we obtain(37)Jq=∫0∞−q2coshη−121+coshη3+q42a2ω21+coshη4−Cq2a2ω21+coshη2dη=6q2−7a2ω2−70Cq2210a2ω2.The stationary condition for (37) is(38)∂Jq∂q=12q2−7a2ω2−70Cq105a2ω2=0,which implies to(39)q=±21a2ω2+210C6.Thus, the bright dark solution in Eq.(40) takes the form(40)ψη=±21a2ω2+210C61+coshη.Hence, system (2) admits the solution(41)px,t=±21a2ω2+210C61+coshax−ω/αtα,vx,t=1ω7a2ω2+70C121+cosh2ax−ω/αtα−C.


## 4.1.3. Kinky-Bright Soliton

We assume a kinky-bright solitons solution of Eq.(11) in the form(42)ψη=qsech2η.Inserting the expression Eq.(51) into Eq.(29), we obtain(43)Jq=∫0∞q42a2ω2sech8η+2q2sech6η−2+Ca2ω2q2sech4ηdη=212q2−14a2ω2−35Cq2105a2ω2.Computing the stationary condition∂Jq/∂q=0, we have(44)q=±84a2ω2+210C12.Therefore, the kinky-bright soliton in Eq.(51) is(45)ψη=±84a2ω2+210C12sech2η.Hence, system (2) admits the solution(46)px,t=±84a2ω2+210C12sech2ax−ωαtα,vx,t=1ω14a2ω2+35C24sech4ax−ωαtα−C.


## 4.1.4. Periodic Solution

We assume the solution is in the form(47)ψη=qcosη.Inserting the expression Eq.(47) into Eq.(29), we obtain(48)Jq=∫0T/4q42a2ω2cos4η+q212−Ca2ω2cos2η−q22dη,where T indicates the period of the nonlinear term of Eq.(11); that is, T=2π. Thus,(49)Jq=π3q2−4a2ω2−8Cq232a2ω2.The stationary condition∂Jq/∂q=0 implies(50)q=±6a2ω2+12C3.Therefore, the solution in Eq.(47) is(51)ψη=±6a2ω2+12C3cosη.Hence, system (2) admits the solution(52)px,t=±6a2ω2+12C3cosax−ωαtα,vx,t=1ω2a2ω2+4C3cos2ax−ωαtα−C.


## 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 Theorem2 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 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. 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.



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