# Abundant Bounded and Unbounded Solitary, Periodic, Rogue-Type Wave Solutions and Analysis of Parametric Effect on the Solutions to Nonlinear Klein–Gordon Model

Authors: Mohammad Mobarak Hossain; Alrazi Abdeljabbar; Harun-Or Roshid; Md. Mamunur Roshid; Abu Naim Sheikh
Journal: Complexity
 (2022)

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## Abstract

This paper exploits the modified simple equation and dynamical system schemes to integrate the Klein–Gordon (KG) model amid quadratic nonlinearity arising in nonlinear optics, quantum theories, and solid state physics. By implementing the modified simple equation (MSE) technique, we develop some disguise adaptation of analytical solutions in terms of hyperbolic, exponential, and trigonometric functions with some special parameters. We apply the dynamical system to bifurcate the model and draw distinct phase portraits on unlike parametric constraints. Following each orbit of all phase portraits, we originate bounded and unbounded solitary, periodic, and periodic rogue-type wave solutions of the KG model. These two schemes extract widespread classes of solitary, periodic, and periodic rogue-type wave solutions for the KG model jointly due to restrictions on parameters. We also analyze the effect of parameters on the obtained wave solutions and discuss why and when it changes its nature. We illustrate some dynamical features of the acquired solutions via the 3D, 2D, and contour graphics.


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## Body

## 1. Introduction

Complex phenomena customarily turned into nonlinear differential equations (NLDEs) by a youngest researcher. Consequently, the study of NLDEs has sustained to attract much effort in the last few years. Many scientific experimental models are employed in nonlinear differential form from the phenomena of nonlinear fiber optics, high-amplitude waves, fluids, plasma, solid state particle motions, etc. Surveying literature, we realized ideas that many scientists worked to disclose innovative, efficient techniques for explaining internal behaviors of NLDEs with constant coefficients that are significant to elucidate different intricate problems such as a discrete algebraic framework [1], IRM-CG method [2], transformed rational function scheme [3], fractional residual method [4], new multistage technique [5], new analytical technique [6], extended tanh approach [7], Hirota-bilinear approach [8–10], multi exp-expansion method [11, 12], Jacobi elliptic expansion method [13, 14], Lie approach [15], Lie symmetry analysis techniques [16], generalized Kudryashov scheme [17, 18], generalized exponential rational function scheme [19], MSE method [20–22], and many more. Such or similar schemes are also used to solve the model with variable coefficients to visualize various new nonlinear dynamics [23–25]. Recently, the nature of rogue waves and diverse dynamical interaction solutions has been studied in numerous fields. There are many researchers who have investigated rogue waves in different fields of mathematical physics and engineering branches [26–32]. Hossain explicated some natures of the solitary and rogue waves with interacting observable facts [26], Ali investigated rogue wave solution from coupled Schrödinger equations [27], and Ohta found general rogue ripple to the discrete nonlinear Schrödinger model [28]. Zhang et al. found rogue wave envelops of (3 + 1)-D Jimbo–Miwa model [29], while Lu et al. established manifold rogue wave envelops for the cKP model [30]. Rogue wave solutions are investigated in the coupled nonlinear Schrödinger models by Degasperis [31], while Ankiewicz found rogue signal elucidations of integrable nonlinear Schrödinger hierarchy [32]. Due to the sensitive effect of a rogue wave, scholars  are being highly interested in deriving rogue and such type of colliding wave solutions in the recent exploration [33–36]. Moreover, dynamical researchers have investigated new rogon waves [37], optical M-shaped solitons with interaction with shock waves [38], orbital stability of solitary waves [39], and the nonexistence of global solutions of the time-fractional model [40] newly.In this survey, we shed light on the quadratic nonlinear dynamics structure, namely, KG, a succeeding structure, which frequently arises in optics, quantum, and solid state physics:(1)utt−α2uxx+βu−γu2=0.To analyze the dynamics behavior of solitons in quantum meadow theories, solid state, and nonlinear optical physics, this model is mostly investigated [41–44]. In the last centuries, different dominant and influential schemes have been suggested to execute solutions of the KG equation, such as the Adomian decomposition method [41], auxiliary equation method [42], method of normal forms of Shatah [43], exp-function method [44], and many more. All of the above techniques took the help of an auxiliary equation. But, we need to investigate the internal characteristics of the nonlinear model without the use of other helping equations. Among the above integral techniques, the MSE is better as it never takes any help from the auxiliary equation and can easily solve any nonlinear model with fewer efforts by direct integrations. Besides this, the dynamical system approach is also a directed integral technique that constructs exact solutions according to each energy orbit of its phase portrait. To the best of our knowledge, this considered model was not investigated by such direct integral schemes still now.Due to this fact, we aim to implement modified simple equation [20–22] and dynamical system schemes [45] on the KG equation to search different types of bounded and unbounded solitary, bright and dark bell envelop, periodic wave, and periodic rouge type wave’s solutions of this model with the adequate condition on the exit’s parameters.The rest of the article is organized as follows: in Section2, we demonstrated analytic solutions of the KG model amid quadratic nonlinearity using the modified simple equation method. Reduction in the dynamical system, bifurcation analysis, and derivation of solutions according to the dynamical scheme is uttered in Section 3. Then, in Section 4, all the deliberated results of the two schemes are illustrated with numerical graphics and explained briefly. In Section 5, we incorporated comparisons and a few remarks with other solutions. The main points of this study are to obtain rogue waves, lump solution, and solitary waves according to energy orbits which are discussed in the last conclusion section.


## 2. Solutions of KG Model via MSE

In this division, the acquired solutions are an abundant form of traveling wave solutions to the KG model by means of the MSE method [20–22], which might be caring to investigate the different nonlinear representations of turbulence, the wave motions, and a lot of fields such as nonlinear optic, solid state, and plasma physics.Let us undertake the KG equation with quadratic nonlinearity in the succeeding system [41]. Utilizing the wave transformation relationξ=kx−ωtand ux,t=Uξ into Equation (1)which convert the ordinary differential form utt=ω2U″ and uxx=k2U″. Then using the facts in (1), it reduces as follows:(2)ω2−α2k2U″+βU−γU2=0,whereU″=d2Udξ2.Now, we allow the MSE method [20–22] to set the solution of (2) as uξ=∑k=0k=lakϕ′ξ/ϕξk, where akk=0,1,2,…,l are free parameters and al≠0 with l∈ℜ+ are urbanized from the balancing between the highest order and nonlinear terms eminences in (2).Here, the power of the series isl=2 obtained from the equilibrium between U″ and U2. Then, the solution of (2) is studied as(3)Uξ=a0+a1ϕ′ξϕξ+a2ϕ′ξϕξ2.At this instant, includinguξ and its derivative form in (2) and considering the coefficient of ϕξ−k (k=0,1,2……) equal to zero, it leads to the structure of the algebraic system:(4)ϕ0ξ:βa0−γa02=0,ϕ−1ξ:ω2a1−α2k2a1ϕ″′+βa1−2a1a0ϕ′=0,ϕ−2ξ:−3ω2a1S+3α2k2a1ϕ′+2ω2a2−3α2k2a2ϕ″′ϕ″′+2ω2a2−γa1−2α2k2a2ϕ″′−2γa1a0+βa1ϕ′ϕ′=0,ϕ−3ξ:−2γa1a2+2ω2a1−2α2k2a1ϕ′+10α2k2a2−10ω2a2ϕ″=0,ϕ−4ξ:6ω2a2−6α2k2a2−γa22ϕiv=0.The coefficient ofϕ0ξ and ϕ−4ξ brings out two set solutions,Set-01:a0=0,a2=6ω2−6α2k2/γ.Set-02:a0=β/γ,a2=6ω2−6α2k2/γNow, we insert the value ofa0 and a2 from set-01 in the remaining equations and resolve them with the assistance of Maple which gives(5)ϕξ=c1+c2expa1γξ−6ω2+6α2k2,Aω=±−β−36βα2k2+a12γ26β,a1=a1,k=k,(6)ϕξ=c1+c2expa1γξ−6ω2+6α2k2,a1=6α2k2−ω2γ2,ω=ω,k=k.Case-01: while equation (4) is treated and making use of the values in equation (3), acquiesce(7)Uξ=6βc1c2eθγc1+c2eθ2,where θ=−a1γξ/6ω2−6α2k2, ξ=kx−−β−36βα2k2+a12γ2/6βt, and a1,k,β,α, and γ are arbitrary constant.Wheneverc1≠c2, and the condition β<0 and a12γ2<36βα2k2 or β>0 and a12γ2>36βα2k2, then solution (7) can be expressed in trigonometric function solution in the following form:(8)Uξ=6βc1c2γ2c1c2+cosθc12+c22−isinθc12−c22.Insertingc1=±c2 in (8), it develops in the succeeding form:(9)Uξ=±3β2γsec2θ2.Settingc1=±ic2 in (8), then the solution becomes(10)uξ=3βγi±sinθ.Wheneverc1≠c2 and reflects on the condition β<0 and a12γ2>36βα2k2, the solution (7) gathers in hyperbolic form as(11)Uξ=6βc1c2γ2c1c2+coshθc12+c22−sinhθc12−c22.Insertingc1=±c2 in (11), then the solution reduces to(12)Uξ=±3β2γsech2θ2.Substituting setc1=±ic2, then (11) is(13)Uξ=±3βγ1∓isinhθ.Case-02: while equation (5) is treated and making use of the values in equation (3), we acquiesce(14)Uξ=6βc1c2eθγc1+c2eθ2,where θ=γ2ξ/α2k2−ω2, ξ=kx−ωt, and γ,α,k,ω are arbitrary constant.Making use ofc1≠c2 and α2k2<ω2 in (14), then the solution is expressed in trigonometric function solution in the following form:(15)Uξ=6βc1c2γ2c1c2+cosθc12+c22−isinθc12−c22.Insertingc1=±c2 in (15), the solution is(16)Uξ=±3β2γsec2θ2.Employingc1=±ic2 in (15), then the solution develops as(17)Uξ=3βγi±sinθ.Making use ofc1≠c2 and α2k2>ω2 in (14), then the solution is expressed in hyperbolic function solution in the following form:(18)Uξ=6βc1c2γ2c1c2+coshθc12+c22+sinhθc12−c22.Insertingc1=±c2 in (18), the solution takes the form(19)Uξ=±3β2γsech2θ2.Employingc1=±ic2 in (18), then the solution develops as(20)Uξ=3βγi∓sinhθ,where θ=β/α2k2−ω2ξ,ξ=kx−ωt,α,k,ω,β, and γ are arbitrary constant.Set-02: ifa0=β/γ,a2=6ω2−6α2k2/γ.Inserting the value ofa0 and a2 from set-02 in the enduring equations and resolving them, then we gather the solution in the following form:(21)ϕξ=c1+c2expξ,k=ω2−βα,a1=±6βγ.Now, the above values are placed in solution (3).(22)Uξ=βγ−6βγc2eξc1+c2eξ+6βγc2eξc1+c2eξ2,where ξ=ω2−β/αx−ωt, and β,ω,γ,α are arbitrary constants.Wheneverc1≠c2 and the condition ω2<β, then solution (22) brings out in the trigonometric form(23)Uξ=βγ−6βc1c2γ2c1c2+cosξc12+c22+isinξc12−c22,where ξ=−ω2+β/α+iωt and β,k,γ,α are arbitrary constant.Insertingc1=±c2 in (23), then the solution develops as(24)Uξ=βγ∓3β2γsec2ξ2.Puttingc1=±ic2 in (23), then the solution is(25)Uξ=βγ∓3βγi±sinξ.Wheneverc1≠c2 and the condition β<ω2, then (22) can be transcribed in hyperbolic form(26)Uξ=βγ−6βc1c2γ2c1c2+coshξc12+c22+sinhξc12−c22.Making use ofc1=±c2 into (26) yields(27)Uξ=βγ∓3β2γsech2ξ2.Ifc1=±ic2, then solution (26) is(28)Uξ=βγ∓3βγ1±isinhξ.


## 3. Bifurcation Analysis and Solutions for Each Orbit of the Phase Portraits

In this part, we shed light on bifurcating the KG model due to the involve parameters, and various phase portraits are derived depending on dissimilar conditions of the parameters. We also establish diverse soliton, periodic, and superperiodic solutions according to each energy orbit of the phase portrait. Here, we draw this action spiting the work into two subsections as follows.


### 3.1. Bifurcations and Phase Portraits of the KG Model

Recall (2) that can be rewritten into a system of dynamical form(29)U′=V=FU,V,V′=−βω2−α2k2U+γω2−α2k2U2=GU,V,which has Hamiltonian energy states as(30)HU,V=V22+βω2−α2k2U22−γω2−α2k2U33=ℏ.Next, we make an effort to perceive phase orbits of (29) with various situations on the parameters α,β,γ,k,ω. Deriving critical points in an equilibrium situation, we have to consider U′=0 and V′=0, then the prototype (29) will provide two equilibrium points σ0,0 and τβ/γ,0, if β≠0. Besides this, the dynamical archetype (29) acquiesced only one critical point σ0,0 for β=0. The Jacobian of each critical points is as follows (Figure 1(a)).(31)Jσ0,0=βω2−α2k2,JτU,0=−βω2−α2k2,TraceAσ=0,TraceAτ=0.Figure 1
(a) Phase portrait onβ>0,ω2−α2k2>0,γ>0 or β<0,ω2−α2k2<0,γ<0. (b) Phase portrait on β>0,ω2−α2k2>0,γ<0 or β<0,ω2−α2k2<0,γ>0. (c) Phase portrait from Hamiltonian for α=1,ω=2,β=4,γ=1,k=1.
(a)(b)(c)According to the bifurcation theorem [45] and our inspection, we acquire the subsequent annotations.Cluster-1 For bothβ>0,ω2−α2k2>0 and β<0,ω2−α2k2<0, we present a nature with the origin σ0,0 which is a stable center point and τβ/γ,0 is a saddle point, whereas the corresponding bifurcations of phase portraits of the prototype (29) are visualized in Figure 1(a) and 1(b), respectively. Phase portraits (Figure 1(a) and 1(b)) are drawn for α=1,ω=2,β=4,γ=1,k=1 and α=1,ω=2,β=4,γ=−1,k=1, respectively. Besides this, γ>0 and γ<0 just alter the directions of flow.Cluster-2: forβ>0,ω2−α2k2<0 and β<0,ω2−α2k2>0, we present a nature with the origin σ0,0 which is a saddle point and τβ/γ,0 is a stable center point, whereas the corresponding bifurcations of phase portraits of the prototype (29) are depicted in Figure 2(b), respectively. Phase portraits (Figures 2(a) and 2(b)) are drawn for α=1,ω=2,β=−4,γ=1,k=1 and α=1,ω=2,β=−4,γ=−1,k=1, respectively. Besides this, γ>0 and γ<0 just alter the directions of flow.Cluster-3: forβ=0, the nature at origin Ν00,0 is a cusp point, whereas the corresponding bifurcations of phase portraits of the prototype (29) are depicted, respectively. In this case, both γ>0,ω2−α2k2>0 and γ<0,ω2−α2k2<0 exhibit the same directions of flow, but an opposite sign that is γ>0,ω2−α2k2<0 or β<0,ω2−α2k2>0 exhibits altered directions of flow. Phase portraits (Figures 3(a) and 3(b)) are drawn for α=1,ω=2,γ=1,k=1 and α=1,ω=2,γ=−1,k=1, respectively.Figure 2
(a) Phase portrait onβ<0,ω2−α2k2>0,γ>0 or β>0,ω2−α2k2<0,γ<0. (b) Phase portrait on β<0,ω2−α2k2>0,γ<0 or β>0,ω2−α2k2<0,γ>0..
(a)(b)Figure 3
(a) Whenω2−α2k2>0,γ>0 or ω2−α2k2<0,γ<0. (b) When ω2−α2k2>0,γ<0 or ω2−α2k2<0,γ>0.
(a)(b)


### 3.2. Precise Solutions for Model (1)

Here, we develop a variety of precise parametric representations of wave solutions of the model (1). For facility, we first draw out the energy stages via Hamiltonian of the model, which is identified by equation (29) with separated various regions due to the energy level of the critical points (Figures 2(a), 3(a), and 3(b):(32)ℏσ=H0,0=0,ℏτ=Hβγ,0=β36γ2ω2−α2k2.


#### 3.2.1. Solutions for Cluster-1

(i)
On the parametric situationβ>0,ω2−α2k2>0,γ>0 or β<0,ω2−α2k2<0,γ<0, the linked homoclinic orbit (see blue curve in Figure 1(a) at τβ/γ,0 is illustrated via HU,V=ℏτ, which suggests a valley type smooth solitary wave solution. The orbit HU,V=ℏτ=β3/6γ2ω2−α2k2 in equation (29) yields(33)V=±2γ3ω2−α2k2U−βγU+β2γ.We merge the first equation of (29) and (33) with integration; we attain a valley-type smooth solitary wave solution(34)ux,t=β2γ−1+3tanh2β4ω2−α2k2ξ,whereξ=kx−ωt.(iii)
On the parametric situationβ>0,ω2−α2k2>0,γ<0 or β<0,ω2−α2k2<0,γ>0 the linked homoclinic orbit (see blue curve in Figure 1(b) at τβ/γ,0 is illustrated via HU,V=ℏτ, which suggests a valley-type smooth solitary wave solution. The orbit HU,V=ℏτ=β3/6γ2ω2−α2k2 in equation (29) yields(35)V=±2γ3α2k2−ω2U+βγβ2γ−U.We merge the first equation of (29) and (35) with integration; we attain a valley-type smooth solitary wave solution(36)ux,t=β2γ1−3tanh2β4ω2−α2k2ξ,whereξ=kx−ωt.(iii)
On the parametric situationβ>0,ω2−α2k2>0,γ>0 or β<0,ω2−α2k2<0,γ<0, the model (1) corresponds to a family of periodic orbits that provide periodic wave solutions expressed by HU,V=ℏτ,ℏ∈0,ℏτ (see the green color orbit in Figure 1(a) or phase portrait in Figure 1(c)). For this situation, equation (29) of the closed domain can be written as follows:V=±2γ/3ω2−α2k2U−ℓ1ℓ2−Uℓ3−U, (36), where ℓ1,0,ℓ2,0, and ℓ3,0 are the cutting points by the orbits HU,V=ℏ,ℏ∈0,ℏτ on the U-axis that preserve the condition −β/2γ≤ℓ1<U<ℓ2≤β/γ<ℓ3. Merging the first equation of (28) and (36), we gain the periodic solution as follows:(37)ux,t=ℓ1+ℓ−2ℓ1sn2γℓ3−ℓ16ω2−α2k2ξ,ℓ1−ℓ2ℓ1−ℓ3.On the parametric situationβ<0,ω2−α2k2>0,γ<0 or β>0,ω2−α2k2<0,γ>0, similar investigation on Figure 1(b) provide us with three intersecting points ℓ4,0,ℓ5,0, and ℓ6,0 identified by the orbits HU,V=ℏ,ℏ∈ℏτ,0 on the U-axis that preserve the condition ℓ4<β/γ≤ℓ5<U<ℓ6;γ<0.We acquire the parametric formulation of the periodic solution as follows:(38)ux,t=ℓ4+ℓ4−ℓ5ℓ6−ℓ4ℓ6−ℓ5sn2γℓ4−ℓ6/6ω2−α2k2ξ,ℓ6−ℓ5/ℓ6−ℓ4−ℓ6−ℓ4.


#### 3.2.2. Solutions for Cluster-2

(i)
On the parametric situationβ<0,ω2−α2k2>0,γ>0 or β>0,ω2−α2k2<0,γ<0, the model (1) corresponds to a homoclinic orbit (see the blue curve in Figure 2(a)) at the point σ0,0 identified by HU,V=ℏσ=0, where the prototype (28) suggests itself as a smooth solitary wave solution of valley type. The assistance of the relation HU,V=ℏσ=0 in equation (29) yields(39)V=±2γ3ω2−α2k2UU−3β2γ. Then, merging the first equation of (29) and (39) yields the valley-type smooth solitary wave solution:(40)ux,t=3β2γtanh2−β4ω2−α2k2ξ−1,whereξ=kx−ωt.(ii)
On the parametric situationβ>0,ω2−α2k2<0,γ>0 or β<0,ω2−α2k2>0,γ<0, the model (1) corresponds to a homoclinic orbit (see the blue curve in Figure 2(b)) at the point σ0,0 identified by HU,V=ℏσ=0, where the prototype (28) suggests itself as a smooth solitary wave solution of valley -type. The assistance of the relation HU,V=ℏσ=0 in equation (29) yields(41)V=±2γ3α2k2−ω2U3β2γ−U. Then, merging the first equation of (29) and (41) yields the valley-type smooth solitary wave solution:(42)ux,t=3β2γ1−tanh2−β4ω2−α2k2ξ,whereξ=kx−ωt.(iii)
On the parametric situationβ<0,ω2−α2k2>0,γ>0 or β>0,ω2−α2k2<0,γ<0, the prototype (28) has a set of smooth periodic wave solutions identified by HU,V=ℏ, which can be observed in Figure 2(a)ℏ∈ℏτ,0, and on situation β>0,ω2−α2k2<0,γ>0 or β<0,ω2−α2k2>0,γ<0, Figure 2(b) for ℏ∈0,ℏτ, respectively. For this aspect, the formulation of the periodic solution is identical to the solutions in equations (37) and (38), respectively.


#### 3.2.3. Solutions for Cluster-3

(i)
On the stateβ=0,ω2−α2k2>0,γ>0 or β=0,ω2−α2k2<0,γ<0, there is an unrestricted orbit with the alike Hamiltonian at the origin σ0,0 (Figure 3(a)). This open cusp orbit (blue color) can be précised by(43)V=±2γ3ω2−α2k2U3/2.Then, by facilitation of the first equation of the prototypes (29) and (43), we get hold of the periodic cusp wave solution as follows:(44)ux,t=6ω2−α2k2γ1ξ2,whereξ=kx−ωt.(ii)
On the stateβ=0,ω2−α2k2>0,γ<0 or β=0,ω2−α2k2<0,γ>0, there is also an unrestricted orbit with the alike Hamiltonian as the origin σ0,0 (Figure 3(b)). Due to this condition, we also catch out the similar periodic cusp wave solution in the form(45)ux,t=−6ω2−α2k2γ1ξ2,whereξ=kx−ωt.


## 3.1. Bifurcations and Phase Portraits of the KG Model

Recall (2) that can be rewritten into a system of dynamical form(29)U′=V=FU,V,V′=−βω2−α2k2U+γω2−α2k2U2=GU,V,which has Hamiltonian energy states as(30)HU,V=V22+βω2−α2k2U22−γω2−α2k2U33=ℏ.Next, we make an effort to perceive phase orbits of (29) with various situations on the parameters α,β,γ,k,ω. Deriving critical points in an equilibrium situation, we have to consider U′=0 and V′=0, then the prototype (29) will provide two equilibrium points σ0,0 and τβ/γ,0, if β≠0. Besides this, the dynamical archetype (29) acquiesced only one critical point σ0,0 for β=0. The Jacobian of each critical points is as follows (Figure 1(a)).(31)Jσ0,0=βω2−α2k2,JτU,0=−βω2−α2k2,TraceAσ=0,TraceAτ=0.Figure 1
(a) Phase portrait onβ>0,ω2−α2k2>0,γ>0 or β<0,ω2−α2k2<0,γ<0. (b) Phase portrait on β>0,ω2−α2k2>0,γ<0 or β<0,ω2−α2k2<0,γ>0. (c) Phase portrait from Hamiltonian for α=1,ω=2,β=4,γ=1,k=1.
(a)(b)(c)According to the bifurcation theorem [45] and our inspection, we acquire the subsequent annotations.Cluster-1 For bothβ>0,ω2−α2k2>0 and β<0,ω2−α2k2<0, we present a nature with the origin σ0,0 which is a stable center point and τβ/γ,0 is a saddle point, whereas the corresponding bifurcations of phase portraits of the prototype (29) are visualized in Figure 1(a) and 1(b), respectively. Phase portraits (Figure 1(a) and 1(b)) are drawn for α=1,ω=2,β=4,γ=1,k=1 and α=1,ω=2,β=4,γ=−1,k=1, respectively. Besides this, γ>0 and γ<0 just alter the directions of flow.Cluster-2: forβ>0,ω2−α2k2<0 and β<0,ω2−α2k2>0, we present a nature with the origin σ0,0 which is a saddle point and τβ/γ,0 is a stable center point, whereas the corresponding bifurcations of phase portraits of the prototype (29) are depicted in Figure 2(b), respectively. Phase portraits (Figures 2(a) and 2(b)) are drawn for α=1,ω=2,β=−4,γ=1,k=1 and α=1,ω=2,β=−4,γ=−1,k=1, respectively. Besides this, γ>0 and γ<0 just alter the directions of flow.Cluster-3: forβ=0, the nature at origin Ν00,0 is a cusp point, whereas the corresponding bifurcations of phase portraits of the prototype (29) are depicted, respectively. In this case, both γ>0,ω2−α2k2>0 and γ<0,ω2−α2k2<0 exhibit the same directions of flow, but an opposite sign that is γ>0,ω2−α2k2<0 or β<0,ω2−α2k2>0 exhibits altered directions of flow. Phase portraits (Figures 3(a) and 3(b)) are drawn for α=1,ω=2,γ=1,k=1 and α=1,ω=2,γ=−1,k=1, respectively.Figure 2
(a) Phase portrait onβ<0,ω2−α2k2>0,γ>0 or β>0,ω2−α2k2<0,γ<0. (b) Phase portrait on β<0,ω2−α2k2>0,γ<0 or β>0,ω2−α2k2<0,γ>0..
(a)(b)Figure 3
(a) Whenω2−α2k2>0,γ>0 or ω2−α2k2<0,γ<0. (b) When ω2−α2k2>0,γ<0 or ω2−α2k2<0,γ>0.
(a)(b)


## 3.2. Precise Solutions for Model (1)

Here, we develop a variety of precise parametric representations of wave solutions of the model (1). For facility, we first draw out the energy stages via Hamiltonian of the model, which is identified by equation (29) with separated various regions due to the energy level of the critical points (Figures 2(a), 3(a), and 3(b):(32)ℏσ=H0,0=0,ℏτ=Hβγ,0=β36γ2ω2−α2k2.


### 3.2.1. Solutions for Cluster-1

(i)
On the parametric situationβ>0,ω2−α2k2>0,γ>0 or β<0,ω2−α2k2<0,γ<0, the linked homoclinic orbit (see blue curve in Figure 1(a) at τβ/γ,0 is illustrated via HU,V=ℏτ, which suggests a valley type smooth solitary wave solution. The orbit HU,V=ℏτ=β3/6γ2ω2−α2k2 in equation (29) yields(33)V=±2γ3ω2−α2k2U−βγU+β2γ.We merge the first equation of (29) and (33) with integration; we attain a valley-type smooth solitary wave solution(34)ux,t=β2γ−1+3tanh2β4ω2−α2k2ξ,whereξ=kx−ωt.(iii)
On the parametric situationβ>0,ω2−α2k2>0,γ<0 or β<0,ω2−α2k2<0,γ>0 the linked homoclinic orbit (see blue curve in Figure 1(b) at τβ/γ,0 is illustrated via HU,V=ℏτ, which suggests a valley-type smooth solitary wave solution. The orbit HU,V=ℏτ=β3/6γ2ω2−α2k2 in equation (29) yields(35)V=±2γ3α2k2−ω2U+βγβ2γ−U.We merge the first equation of (29) and (35) with integration; we attain a valley-type smooth solitary wave solution(36)ux,t=β2γ1−3tanh2β4ω2−α2k2ξ,whereξ=kx−ωt.(iii)
On the parametric situationβ>0,ω2−α2k2>0,γ>0 or β<0,ω2−α2k2<0,γ<0, the model (1) corresponds to a family of periodic orbits that provide periodic wave solutions expressed by HU,V=ℏτ,ℏ∈0,ℏτ (see the green color orbit in Figure 1(a) or phase portrait in Figure 1(c)). For this situation, equation (29) of the closed domain can be written as follows:V=±2γ/3ω2−α2k2U−ℓ1ℓ2−Uℓ3−U, (36), where ℓ1,0,ℓ2,0, and ℓ3,0 are the cutting points by the orbits HU,V=ℏ,ℏ∈0,ℏτ on the U-axis that preserve the condition −β/2γ≤ℓ1<U<ℓ2≤β/γ<ℓ3. Merging the first equation of (28) and (36), we gain the periodic solution as follows:(37)ux,t=ℓ1+ℓ−2ℓ1sn2γℓ3−ℓ16ω2−α2k2ξ,ℓ1−ℓ2ℓ1−ℓ3.On the parametric situationβ<0,ω2−α2k2>0,γ<0 or β>0,ω2−α2k2<0,γ>0, similar investigation on Figure 1(b) provide us with three intersecting points ℓ4,0,ℓ5,0, and ℓ6,0 identified by the orbits HU,V=ℏ,ℏ∈ℏτ,0 on the U-axis that preserve the condition ℓ4<β/γ≤ℓ5<U<ℓ6;γ<0.We acquire the parametric formulation of the periodic solution as follows:(38)ux,t=ℓ4+ℓ4−ℓ5ℓ6−ℓ4ℓ6−ℓ5sn2γℓ4−ℓ6/6ω2−α2k2ξ,ℓ6−ℓ5/ℓ6−ℓ4−ℓ6−ℓ4.


### 3.2.2. Solutions for Cluster-2

(i)
On the parametric situationβ<0,ω2−α2k2>0,γ>0 or β>0,ω2−α2k2<0,γ<0, the model (1) corresponds to a homoclinic orbit (see the blue curve in Figure 2(a)) at the point σ0,0 identified by HU,V=ℏσ=0, where the prototype (28) suggests itself as a smooth solitary wave solution of valley type. The assistance of the relation HU,V=ℏσ=0 in equation (29) yields(39)V=±2γ3ω2−α2k2UU−3β2γ. Then, merging the first equation of (29) and (39) yields the valley-type smooth solitary wave solution:(40)ux,t=3β2γtanh2−β4ω2−α2k2ξ−1,whereξ=kx−ωt.(ii)
On the parametric situationβ>0,ω2−α2k2<0,γ>0 or β<0,ω2−α2k2>0,γ<0, the model (1) corresponds to a homoclinic orbit (see the blue curve in Figure 2(b)) at the point σ0,0 identified by HU,V=ℏσ=0, where the prototype (28) suggests itself as a smooth solitary wave solution of valley -type. The assistance of the relation HU,V=ℏσ=0 in equation (29) yields(41)V=±2γ3α2k2−ω2U3β2γ−U. Then, merging the first equation of (29) and (41) yields the valley-type smooth solitary wave solution:(42)ux,t=3β2γ1−tanh2−β4ω2−α2k2ξ,whereξ=kx−ωt.(iii)
On the parametric situationβ<0,ω2−α2k2>0,γ>0 or β>0,ω2−α2k2<0,γ<0, the prototype (28) has a set of smooth periodic wave solutions identified by HU,V=ℏ, which can be observed in Figure 2(a)ℏ∈ℏτ,0, and on situation β>0,ω2−α2k2<0,γ>0 or β<0,ω2−α2k2>0,γ<0, Figure 2(b) for ℏ∈0,ℏτ, respectively. For this aspect, the formulation of the periodic solution is identical to the solutions in equations (37) and (38), respectively.


### 3.2.3. Solutions for Cluster-3

(i)
On the stateβ=0,ω2−α2k2>0,γ>0 or β=0,ω2−α2k2<0,γ<0, there is an unrestricted orbit with the alike Hamiltonian at the origin σ0,0 (Figure 3(a)). This open cusp orbit (blue color) can be précised by(43)V=±2γ3ω2−α2k2U3/2.Then, by facilitation of the first equation of the prototypes (29) and (43), we get hold of the periodic cusp wave solution as follows:(44)ux,t=6ω2−α2k2γ1ξ2,whereξ=kx−ωt.(ii)
On the stateβ=0,ω2−α2k2>0,γ<0 or β=0,ω2−α2k2<0,γ>0, there is also an unrestricted orbit with the alike Hamiltonian as the origin σ0,0 (Figure 3(b)). Due to this condition, we also catch out the similar periodic cusp wave solution in the form(45)ux,t=−6ω2−α2k2γ1ξ2,whereξ=kx−ωt.


## 3.2.1. Solutions for Cluster-1

(i)
On the parametric situationβ>0,ω2−α2k2>0,γ>0 or β<0,ω2−α2k2<0,γ<0, the linked homoclinic orbit (see blue curve in Figure 1(a) at τβ/γ,0 is illustrated via HU,V=ℏτ, which suggests a valley type smooth solitary wave solution. The orbit HU,V=ℏτ=β3/6γ2ω2−α2k2 in equation (29) yields(33)V=±2γ3ω2−α2k2U−βγU+β2γ.We merge the first equation of (29) and (33) with integration; we attain a valley-type smooth solitary wave solution(34)ux,t=β2γ−1+3tanh2β4ω2−α2k2ξ,whereξ=kx−ωt.(iii)
On the parametric situationβ>0,ω2−α2k2>0,γ<0 or β<0,ω2−α2k2<0,γ>0 the linked homoclinic orbit (see blue curve in Figure 1(b) at τβ/γ,0 is illustrated via HU,V=ℏτ, which suggests a valley-type smooth solitary wave solution. The orbit HU,V=ℏτ=β3/6γ2ω2−α2k2 in equation (29) yields(35)V=±2γ3α2k2−ω2U+βγβ2γ−U.We merge the first equation of (29) and (35) with integration; we attain a valley-type smooth solitary wave solution(36)ux,t=β2γ1−3tanh2β4ω2−α2k2ξ,whereξ=kx−ωt.(iii)
On the parametric situationβ>0,ω2−α2k2>0,γ>0 or β<0,ω2−α2k2<0,γ<0, the model (1) corresponds to a family of periodic orbits that provide periodic wave solutions expressed by HU,V=ℏτ,ℏ∈0,ℏτ (see the green color orbit in Figure 1(a) or phase portrait in Figure 1(c)). For this situation, equation (29) of the closed domain can be written as follows:V=±2γ/3ω2−α2k2U−ℓ1ℓ2−Uℓ3−U, (36), where ℓ1,0,ℓ2,0, and ℓ3,0 are the cutting points by the orbits HU,V=ℏ,ℏ∈0,ℏτ on the U-axis that preserve the condition −β/2γ≤ℓ1<U<ℓ2≤β/γ<ℓ3. Merging the first equation of (28) and (36), we gain the periodic solution as follows:(37)ux,t=ℓ1+ℓ−2ℓ1sn2γℓ3−ℓ16ω2−α2k2ξ,ℓ1−ℓ2ℓ1−ℓ3.On the parametric situationβ<0,ω2−α2k2>0,γ<0 or β>0,ω2−α2k2<0,γ>0, similar investigation on Figure 1(b) provide us with three intersecting points ℓ4,0,ℓ5,0, and ℓ6,0 identified by the orbits HU,V=ℏ,ℏ∈ℏτ,0 on the U-axis that preserve the condition ℓ4<β/γ≤ℓ5<U<ℓ6;γ<0.We acquire the parametric formulation of the periodic solution as follows:(38)ux,t=ℓ4+ℓ4−ℓ5ℓ6−ℓ4ℓ6−ℓ5sn2γℓ4−ℓ6/6ω2−α2k2ξ,ℓ6−ℓ5/ℓ6−ℓ4−ℓ6−ℓ4.


## 3.2.2. Solutions for Cluster-2

(i)
On the parametric situationβ<0,ω2−α2k2>0,γ>0 or β>0,ω2−α2k2<0,γ<0, the model (1) corresponds to a homoclinic orbit (see the blue curve in Figure 2(a)) at the point σ0,0 identified by HU,V=ℏσ=0, where the prototype (28) suggests itself as a smooth solitary wave solution of valley type. The assistance of the relation HU,V=ℏσ=0 in equation (29) yields(39)V=±2γ3ω2−α2k2UU−3β2γ. Then, merging the first equation of (29) and (39) yields the valley-type smooth solitary wave solution:(40)ux,t=3β2γtanh2−β4ω2−α2k2ξ−1,whereξ=kx−ωt.(ii)
On the parametric situationβ>0,ω2−α2k2<0,γ>0 or β<0,ω2−α2k2>0,γ<0, the model (1) corresponds to a homoclinic orbit (see the blue curve in Figure 2(b)) at the point σ0,0 identified by HU,V=ℏσ=0, where the prototype (28) suggests itself as a smooth solitary wave solution of valley -type. The assistance of the relation HU,V=ℏσ=0 in equation (29) yields(41)V=±2γ3α2k2−ω2U3β2γ−U. Then, merging the first equation of (29) and (41) yields the valley-type smooth solitary wave solution:(42)ux,t=3β2γ1−tanh2−β4ω2−α2k2ξ,whereξ=kx−ωt.(iii)
On the parametric situationβ<0,ω2−α2k2>0,γ>0 or β>0,ω2−α2k2<0,γ<0, the prototype (28) has a set of smooth periodic wave solutions identified by HU,V=ℏ, which can be observed in Figure 2(a)ℏ∈ℏτ,0, and on situation β>0,ω2−α2k2<0,γ>0 or β<0,ω2−α2k2>0,γ<0, Figure 2(b) for ℏ∈0,ℏτ, respectively. For this aspect, the formulation of the periodic solution is identical to the solutions in equations (37) and (38), respectively.


## 3.2.3. Solutions for Cluster-3

(i)
On the stateβ=0,ω2−α2k2>0,γ>0 or β=0,ω2−α2k2<0,γ<0, there is an unrestricted orbit with the alike Hamiltonian at the origin σ0,0 (Figure 3(a)). This open cusp orbit (blue color) can be précised by(43)V=±2γ3ω2−α2k2U3/2.Then, by facilitation of the first equation of the prototypes (29) and (43), we get hold of the periodic cusp wave solution as follows:(44)ux,t=6ω2−α2k2γ1ξ2,whereξ=kx−ωt.(ii)
On the stateβ=0,ω2−α2k2>0,γ<0 or β=0,ω2−α2k2<0,γ>0, there is also an unrestricted orbit with the alike Hamiltonian as the origin σ0,0 (Figure 3(b)). Due to this condition, we also catch out the similar periodic cusp wave solution in the form(45)ux,t=−6ω2−α2k2γ1ξ2,whereξ=kx−ωt.


## 4. Results and Discussion

### 4.1. Solutions via Modified Simple Equation Method

For the assorted values of the appreciate constants, we achieved rogue wave multilump wave, dark and bright bell shape solution, singular and antisingular soliton, and m-type and anti-m-type periodic solutions. Those solutions are obliging to investigate a different nonlinear model of turmoil, the wave motions, and a lot of fields such as nonlinear optic, solid, and plasma state physics. We also set the parametric conditions for which the same solutions behave dissimilarly on the change of their parametric values.Here are the solutions (7), (8), (9), (13), (14), (15), (22), (23), and (24) which arrived in the form of a trigonometric function, and the solutions (10), (11), (12), (17), (18), (19), (25), (26), and (27) arrived in the form of hyperbolic function under some conditions.Solution (7) is a complex-valued function for β<0 and a12γ2<36βα2k2 or β>0 and a12γ2>36βα2k2 and it can be expressed as the trigonometric function in equations (7)–(9). The solutions (8) and (9) give periodic rogue waves solution as shown in Figure 4 for c1=1,c2=2,k=α=0.5,a1=3,γ=1 and c1=1,c2=2,k=α=0.5,a1=3,γ=1, respectively.Figure 4
This is shape periodic rogue-type lump wave of the real part of the solution (7) through the value of parameters c1=1,c2=2,k=α=0.5,a1=3,γ=1,β=1. The periodicity exists along the t-axis: (i) 3D contour plot; (ii) density plot; (iii) 2D plot.
(a)(b)(c)(d)Forβ<0 and a12γ2>36βα2k2 or β<0 and a12γ2<36βα2k2, the solution (7) is a real-valued function solution, and it can be expressed as the hyperbolic function in equations (10)–(12). These solutions always provide different types of solitary waves.Solution (14) is complex if α2k2<ω2 and it can be expressed as the trigonometric function in (15) and (16). The profile in Figures 5(a) and 5(d) is a 2D plot and 3D control plot of the real slice equation (14) for c1=γ=1,c2=2,k=α=ω=0.5,β=3 which is called a periodic solution. The 2D plot and 3D control plot of the real portion of equation (14) are a stable periodic solution for c1=γ=1,c2=2,k=α=ω=0.5, and β=3 in Figures 5(b) and 5(e). The profile in Figures 5(c) and 5(d) is the periodicity of the real portion of equation (14) for β=3.c1=γ=1,c2=2,k=α=ω=0.5, with 2D plot and 3D control plot.Figure 5
Periodic solution of equation (14) and the periodicity exist along the x-axis: (i) dark wave: 3D (upper) and contour (lower) plots, (iv) 2D plot of (i); (ii) bright wave: 3D (upper) and contour (lower) plots, (v) 2D plot of (ii); (iii) dark-Bright wave: 3D (upper) and contour (lower) plots, (vi) 2D plot of (iii).
(a)(b)(c)(d)(e)(f)For the conditionα2k2>ω2, solution (14) can be expressed in the hyperbolic form in (18), (19), and (20). The solutions in (18) and (19) and the real part of (19) present bright and dark shape solution in Figures 6(a) and 6(c) and Figures 6(b) and 6(d) for k=α=1,γ=0.05,β=1,ω=0.5, taking the positive and negative signs of solutions, respectively.Figure 6
Profile of solution of equation (18): (i) 3D (upper) and density (lower) plot as bright bell wave, (iii) 2D plot of (i); (ii) 3D (upper) and density (lower) plots as dark bell wave, (iii) 2D plots of (ii).
(a)(b)(c)(d)The profile in Figure7 represents density, three dimensional control plot, and 2D plot of the imaginary chunk of (20) for c1=Ic2=1,k=a1=α=1,γ=−1,β=−0.05. Figure 4Figure 7
The profile is the single soliton as a dark-bright kink wave of the imaginary portion of equation (19) for c1=1,k=α=1,ω=0.5,γ=0.05,β=1: (i) 3D plot (upper) and contour plot (lower), (ii) 2D plot.
(a)(b)Forω2<β, the solution (22) is a complex valued function and it can be expressed as the trigonometric function in equations (22), (23), and (24). Solution (24) represents periodic rogue waves solution for ω=0.5,α=2,γ=1,β=1 and has a nature like Figure 4. The profile in Figure 8 is 3D contour plot and density plot of multitype lump wave solution of (25) for ω=0.5,α=2,γ=1,β=1. Figures 5–8.Figure 8
Multilump wave solution of solution (24) for γ=1,β=1.k=α=0.5. The periodicity exists along the x-axis: (i) 3D plot (upper) and density plot (lower), (ii) 2D plot.
(a)(b)


### 4.2. Solutions via Dynamical System Scheme

In this subsection, we are going to visualize the nature of the solution that is achieved for each orbit with assorted values of the appreciate parameters. We see from the MSE scheme that a single solution changes into three different types of solutions due to a change in parametric conditions. But, this method initially bifurcated the phase orbits, and different types of orbits provided different types of solutions. The solutions (33), (35), (40), and (41) are consistent with homoclinic orbits and exhibit bell wave solutions, but (34) and (40) yield dark bell, while (36) and (41) yield bright bell wave envelops. The profile of dark bell solitary wave solution via (34) is shown in Figure 9. The profile of bright bell solitary wave solution via (36) is shown in Figure 10. Solutions (37) and (38) exhibit a different type of periodic wave solutions which are depicted by Figures 11 and 12 respectively. Besides this, cusp orbit provided us cusp type peaked wave solutions (44) and (45) which are exploited in Figures 13 and 14, respectively. They exhibit upward and downward peaks as directions of orbits are opposite.Figure 9
Shapes of the solution of equation (33): (a) 3D shape (upper), density (lower) plots of dark bell wave, and (b) 2D shape for α=2,β=−2,γ=−1,ω=k=1.
(a)(b)Figure 10
Shapes of the solution of equation (35): (a) 3D shape (upper), density (lower) plots of bright bell wave, and (b) 2D shape for α=2,β=−2,γ=ω=k=1.
(a)(b)Figure 11
Shapes of the solution of equation (38): (a) 3D shape (upper), density (lower) plots of periodic wave, and (b) 2D shape for α=β=γ=k=1,ω=2,l1=0,l2=1,l3=2.
(a)(b)Figure 12
Shapes of the solution of equation (37): (a) 3D shape (upper), density (lower) plots of periodic wave, and (b) 2D shape for α=2,β=γ=k=ω=1,l4=0.8,l5=1.1,l6=3.
(a)(b)Figure 13
Shapes of the solution of equation (44): (a) 3D shape (upper), density (lower) plots of periodic wave, and (b) 2D shape for α=3,β=2,γ=−4,k=ω=1.
(a)(b)Figure 14
Shapes of the solution of equation (45): (a) 3D shape (upper), density (lower) plots of periodic wave, and (b) 2D shape for α=3,β=2,γ=4,k=ω=1.
(a)(b)


## 4.1. Solutions via Modified Simple Equation Method

For the assorted values of the appreciate constants, we achieved rogue wave multilump wave, dark and bright bell shape solution, singular and antisingular soliton, and m-type and anti-m-type periodic solutions. Those solutions are obliging to investigate a different nonlinear model of turmoil, the wave motions, and a lot of fields such as nonlinear optic, solid, and plasma state physics. We also set the parametric conditions for which the same solutions behave dissimilarly on the change of their parametric values.Here are the solutions (7), (8), (9), (13), (14), (15), (22), (23), and (24) which arrived in the form of a trigonometric function, and the solutions (10), (11), (12), (17), (18), (19), (25), (26), and (27) arrived in the form of hyperbolic function under some conditions.Solution (7) is a complex-valued function for β<0 and a12γ2<36βα2k2 or β>0 and a12γ2>36βα2k2 and it can be expressed as the trigonometric function in equations (7)–(9). The solutions (8) and (9) give periodic rogue waves solution as shown in Figure 4 for c1=1,c2=2,k=α=0.5,a1=3,γ=1 and c1=1,c2=2,k=α=0.5,a1=3,γ=1, respectively.Figure 4
This is shape periodic rogue-type lump wave of the real part of the solution (7) through the value of parameters c1=1,c2=2,k=α=0.5,a1=3,γ=1,β=1. The periodicity exists along the t-axis: (i) 3D contour plot; (ii) density plot; (iii) 2D plot.
(a)(b)(c)(d)Forβ<0 and a12γ2>36βα2k2 or β<0 and a12γ2<36βα2k2, the solution (7) is a real-valued function solution, and it can be expressed as the hyperbolic function in equations (10)–(12). These solutions always provide different types of solitary waves.Solution (14) is complex if α2k2<ω2 and it can be expressed as the trigonometric function in (15) and (16). The profile in Figures 5(a) and 5(d) is a 2D plot and 3D control plot of the real slice equation (14) for c1=γ=1,c2=2,k=α=ω=0.5,β=3 which is called a periodic solution. The 2D plot and 3D control plot of the real portion of equation (14) are a stable periodic solution for c1=γ=1,c2=2,k=α=ω=0.5, and β=3 in Figures 5(b) and 5(e). The profile in Figures 5(c) and 5(d) is the periodicity of the real portion of equation (14) for β=3.c1=γ=1,c2=2,k=α=ω=0.5, with 2D plot and 3D control plot.Figure 5
Periodic solution of equation (14) and the periodicity exist along the x-axis: (i) dark wave: 3D (upper) and contour (lower) plots, (iv) 2D plot of (i); (ii) bright wave: 3D (upper) and contour (lower) plots, (v) 2D plot of (ii); (iii) dark-Bright wave: 3D (upper) and contour (lower) plots, (vi) 2D plot of (iii).
(a)(b)(c)(d)(e)(f)For the conditionα2k2>ω2, solution (14) can be expressed in the hyperbolic form in (18), (19), and (20). The solutions in (18) and (19) and the real part of (19) present bright and dark shape solution in Figures 6(a) and 6(c) and Figures 6(b) and 6(d) for k=α=1,γ=0.05,β=1,ω=0.5, taking the positive and negative signs of solutions, respectively.Figure 6
Profile of solution of equation (18): (i) 3D (upper) and density (lower) plot as bright bell wave, (iii) 2D plot of (i); (ii) 3D (upper) and density (lower) plots as dark bell wave, (iii) 2D plots of (ii).
(a)(b)(c)(d)The profile in Figure7 represents density, three dimensional control plot, and 2D plot of the imaginary chunk of (20) for c1=Ic2=1,k=a1=α=1,γ=−1,β=−0.05. Figure 4Figure 7
The profile is the single soliton as a dark-bright kink wave of the imaginary portion of equation (19) for c1=1,k=α=1,ω=0.5,γ=0.05,β=1: (i) 3D plot (upper) and contour plot (lower), (ii) 2D plot.
(a)(b)Forω2<β, the solution (22) is a complex valued function and it can be expressed as the trigonometric function in equations (22), (23), and (24). Solution (24) represents periodic rogue waves solution for ω=0.5,α=2,γ=1,β=1 and has a nature like Figure 4. The profile in Figure 8 is 3D contour plot and density plot of multitype lump wave solution of (25) for ω=0.5,α=2,γ=1,β=1. Figures 5–8.Figure 8
Multilump wave solution of solution (24) for γ=1,β=1.k=α=0.5. The periodicity exists along the x-axis: (i) 3D plot (upper) and density plot (lower), (ii) 2D plot.
(a)(b)


## 4.2. Solutions via Dynamical System Scheme

In this subsection, we are going to visualize the nature of the solution that is achieved for each orbit with assorted values of the appreciate parameters. We see from the MSE scheme that a single solution changes into three different types of solutions due to a change in parametric conditions. But, this method initially bifurcated the phase orbits, and different types of orbits provided different types of solutions. The solutions (33), (35), (40), and (41) are consistent with homoclinic orbits and exhibit bell wave solutions, but (34) and (40) yield dark bell, while (36) and (41) yield bright bell wave envelops. The profile of dark bell solitary wave solution via (34) is shown in Figure 9. The profile of bright bell solitary wave solution via (36) is shown in Figure 10. Solutions (37) and (38) exhibit a different type of periodic wave solutions which are depicted by Figures 11 and 12 respectively. Besides this, cusp orbit provided us cusp type peaked wave solutions (44) and (45) which are exploited in Figures 13 and 14, respectively. They exhibit upward and downward peaks as directions of orbits are opposite.Figure 9
Shapes of the solution of equation (33): (a) 3D shape (upper), density (lower) plots of dark bell wave, and (b) 2D shape for α=2,β=−2,γ=−1,ω=k=1.
(a)(b)Figure 10
Shapes of the solution of equation (35): (a) 3D shape (upper), density (lower) plots of bright bell wave, and (b) 2D shape for α=2,β=−2,γ=ω=k=1.
(a)(b)Figure 11
Shapes of the solution of equation (38): (a) 3D shape (upper), density (lower) plots of periodic wave, and (b) 2D shape for α=β=γ=k=1,ω=2,l1=0,l2=1,l3=2.
(a)(b)Figure 12
Shapes of the solution of equation (37): (a) 3D shape (upper), density (lower) plots of periodic wave, and (b) 2D shape for α=2,β=γ=k=ω=1,l4=0.8,l5=1.1,l6=3.
(a)(b)Figure 13
Shapes of the solution of equation (44): (a) 3D shape (upper), density (lower) plots of periodic wave, and (b) 2D shape for α=3,β=2,γ=−4,k=ω=1.
(a)(b)Figure 14
Shapes of the solution of equation (45): (a) 3D shape (upper), density (lower) plots of periodic wave, and (b) 2D shape for α=3,β=2,γ=4,k=ω=1.
(a)(b)


## 5. Remarks and Comparison

It is worth mentioning that both techniques we used here are very simple and direct and take less computational effort than the other methods [42, 44], and these techniques can be handled without help of the auxiliary equation to exhibit the internal mechanism of the model. Sirendaoreji [42] used the auxiliary method to solve the quadratic nonlinear KG model and extracted solitary, periodic, and singular solutions in real form only by getting the help of the auxiliary equation. Besides this, Zhang [44] solved the same model by using the exp-function method and extracted a few solutions, including solitary as well as periodic waves only in real form. But, our solutions obtained by the two methods found solitary, periodic, and singular solutions in both real and complex forms, which cover all of their solutions. In addition, we derived our results by direct integration from its Hamiltonian following each energy orbit of its phase portraits. We extracted different bounded and unbounded periodic waves, dark and bright bell shape, dark-bright and bright-dark kink wave, polynomial solution, disguise version of soliton, and periodic rouge wave resolutions of the KG model which was not reported in [42, 44].


## 6. Conclusion

In this research, we successfully investigated the quadratic nonlinear KG model with bifurcation analysis. The MSE and dynamical system schemes are employed in the model and extracted different bounded and unbounded periodic waves, dark and bright bell shape, dark-bright and bright-dark kink wave, disguise version of soliton, and periodic rouge wave resolutions of the KG model. We apply the dynamical system to bifurcate the model and draw distinct phase portraits on unlike parametric constraints. Following each orbit of all phase portraits, we originate bounded and unbounded solitary, periodic and periodic rogue-type wave solutions of the KG model. We also analyze the effect of parameters on the obtained wave solutions and discuss why and when it changes its self-nature. Numerical illustrations of the derived solutions with 3D contour, density, and 2D plots are presented by the arbitrary picking of parameters allied with conditions. Lastly, we think that the realized schemes are burly and more skilled than other schemes and the attained solutions are reliable.



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