# Application of the Multiple Exp-Function, Cross-Kink, Periodic-Kink, Solitary Wave Methods, and Stability Analysis for the CDG Equation

Authors: Haifa Bin Jebreen; Yurilev Chalco-Cano
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this article, the exact wave structures are discussed to the Caudrey-Dodd-Gibbon equation with the assistance of Maple based on the Hirota bilinear form. It is investigated that the equation exhibits the trigonometric, hyperbolic, and exponential function solutions. We first construct a combination of the general exponential function, periodic function, and hyperbolic function in order to derive the general periodic-kink solution for this equation. Then, the more periodic wave solutions are presented with more arbitrary autocephalous parameters, in which the periodic-kink solution localized in all directions in space. Furthermore, the modulation instability is employed to discuss the stability of the available solutions, and the special theorem is also introduced. Moreover, the constraint conditions are also reported which validate the existence of solutions. Furthermore, 2-dimensional graphs are presented for the physical movement of the earned solutions under the appropriate selection of the parameters for stability analysis. The concluded results are helpful for the understanding of the investigation of nonlinear waves and are also vital for numerical and experimental verification in engineering sciences and nonlinear physics.


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## Body

## 1. Introduction

It is known that these exact solutions of nonlinear evolution equations (NLEEs), especially the soliton solutions [1–3], can be given by using a variety of different methods [4, 5], such as Jacobi elliptic function expansion method [6], inverse scattering transformation (IST) [7, 8], Darboux transformation (DT) [9], extended generalized DT [10], Lax pair (LP) [11], Lie symmetry analysis [12], Hirota bilinear method [13], and others [14, 15]. The Hirota bilinear method is an efficient tool to construct exact solutions of NLEEs, and there exists plenty of completely integrable equations which are studied in this way. For instance, generalized bilinear equations [16], the lump-type solutions in a homogenous-dispersive medium [17], the (2+1)-dimensional KdV equation [18], the (3+1)-D potential-YTSF equation [19], the generalized BKP equation [20], the (3+1)-dimensional BKP-Boussinesq equation [21], the (3+1) dimensional generalized KP-Boussinesq equation [22], the (2+1)-dimensional integrable Boussinesq model [23], and the (2+1)-dimensional Breaking Soliton equation [24, 25]. More recently, lump waves and rogue waves have attracted a growing amount of attention, and many theoretical and experimental studies of lump waves are mentioned [26–33]. A novel method for finding the special rogue waves with predictability of NLEEs is proposed by using the Hirota bilinear method by powerful researchers in Refs. [34, 35], in which some results are very helpful for us to study some physical phenomena in engineering.The Caudrey-Dodd-Gibbon equation introduced by Aiyer et al. [36] who describes the inelastic interactions between the solitary waves under strong physical contexts in certain integrable or nonintegrable systems and has been investigated the related dynamic behavior [37], which reads
(1)Φt+Φxxxxx+30ΦΦxxx+30ΦxΦxx+180Φ2Φx=0.In 2006, the tanh solutions of the equation [38] and, in 2008, the multiple-soliton solutions utilizing the Hirota bilinear method combined with the simplified Hereman method [39] for the above equation are derived by Wazwaz. Also, the physical comprehension of Equation (1) was demonstrated by plenty of scholars who investigated its solitary type solutions and given in Refs. [40] and [41]. The homotopy perturbation method has been utilized to find solutions for the aforementioned equation [42–44]. Based on the obtained transformation of integrating Equation (1), we get to the following nonlinear PDE [45]:
(2)Λt+60Λx3+30ΛxΛxxx+Λxxxxx=0.According to [46], the Hirota bilinear from of the CDG equation reads
(3)Λ=2lnΓx,Φ=Λxand, also by applying the dependent variable transformation, turns into the Hirota bilinear form
(4)Dxt+Dx6Γ.Γ=0,where D is a bilinear operator. By deeming the D-operator defined with the aid of the functions Γ1 and Γ2, we get to the following relation:
(5)Dxπ1Dtπ2Γ1.Γ2=∂∂x−∂∂x′π1∂∂t−∂∂t′π2Γ1x,t.Γ2x′,t′x′=x,t′=t.With the help of the transformation Equation (3), the general periodic-kink solutions of Equation (1) can be given. We get to the bilinear form the of Γ as
(6)ΓΓxt−ΓxΓt+ΓΓxxxxxx−6ΓxΓxxxxx+15ΓxxΓxxxx−10Γxxx2=0.Moreover, the stability analysis and the more general periodic-wave solutions and special rogue waves with predictability are investigated in our paper, which have never been studied. Various types of studies were investigated by capable authors in which some of them can be mentioned, for example, the Caudrey–Dodd–Gibbon equation [47], the pZK equation using Lie point symmetries [48], group-invariant solutions of the (3+1)-dimensional generalized KP equation [49], optimal system and dynamics of solitons for a higher-dimensional Fokas equation [50], dynamics of solitons for (2+1)-dimensional NNV equations [51], the combined MCBS-nMCBS equation [52], Lie symmetry reductions for (2+1)-dimensional Pavlov equation [53], Schrödinger-Hirota equation with variable coefficients [54], the (2+1)-dimension paraxial wave equation [55], the fractional Drinfeld-Sokolov-Wilson equation [56], the (3+1)-dimensional extended Jimbo-Miwa equations [57], and a high-order partial differential equation with fractional derivatives [58]. In the valuable work, the capable authors studied the periodic wave solutions and stability analysis for the KP-BBM equation [59] and breather and periodic wave solution for generalized Bogoyavlensky-Konopelchenko equation [60] with the aid of Hirota operator.To make this paper more self-contained, a combination of general exponential function, periodic function, and hyperbolic function of the (3+1)-dimensional CDG equation is constructed with the help of a bilinear operator, which is crucial to obtain the periodic-wave solution of Equation (1). Based on the Hirota bilinear form Equation (6), the general periodic-wave solution is derived in Section 2 and the novel periodic solutions which can be arisen with twenty one classes. In Section 4, we shall investigate the stability analysis to obtain the modulation stability spectrum of this equation. The final section will be reserved for the conclusions and the discussions.


## 2. Multiple Exp-Function Method

In this section, according to [61–63] so that it can be further employed to the nonlinear partial differential equation (NLPDE) in order to furnish its exact solutions, it can be presented as:Step 1.
The following is the NLPDE:(7)Px,y,t,Ψ,Ψx,Ψy,Ψz,Ψt,Ψxx,Ψtt,⋯=0.We commence a sequence of new variablesξi=ξix,t,1≤i≤n, by solvable PDEs, for example, the linear ones,
(8)ξi,x=αiξi,ξi,t=δiξi,1≤i≤n,where αi,1≤i≤n, is the angular wave number and δi,1≤i≤n, is the wave frequency. It should be pointed that this is frequently the initiating step for constructing the exact solutions to NLPDEs, and moreover, solving such linear equations redounds to the exponential function solutions:
(9)ξi=ϖieθi,θi=αix−δit,1≤i≤n,in which ϖi,1≤i≤n, are undetermined constants.Step 2.
Determine the solution of Equation (7) as the following form in terms of the new variables ξi,1≤i≤n:
(10)Ψx,t=Δξ1,ξ2,⋯,ξnΩξ1,ξ2,⋯,ξn,Δ=∑r,s=1n∑i,j=0MΔrs,ijξriξsj,Ω=∑r,s=1n∑i,j=0NΩrs,ijξriξsj,in which Δrs,ij and Ωrs,ij are the amounts to be settled. Appending Equation (10) into Equation (7) and ordering the numerator of the rational function to zero, we can achieve a series of nonlinear algebraic equations about the variables αi,δi,Δrs,ij and Ωrs,ij. Solving the solutions for these nonlinear algebraic equations and putting these solutions into Equation (10), the multiple soliton solutions to Equation (7) can be obtained in the below form as
(11)Ψx,t=Δϖ1eα1x−δ1t,⋯,ϖneαnx−δntΩϖ1eα1x−δ1t,⋯,ϖneαnx−δnt,in which Ω≠0, and also, we have
(12)Δt=∑i=1nΔξiξi,t,Ωt=∑i=1nΩξiξi,t,Δx=∑i=1nΔξiξi,x,Ωx=∑i=1nΩξiξi,x,Ψt=Ω∑i=1nΔξiξi,t−Δ∑i=1nΩξiξi,tΩ2,Ψx=Ω∑i=1nΔξiξi,x−Δ∑i=1nΩξiξi,xΩ2.


## 3. Multiple Soliton Solutions for the CDG Equation

### 3.1. Set I: One-Wave Solution

We start up with one-wave function based on the explanation in Step2 in the previous section, we deem that Equation (1) has the below form of one-wave solution as
(13)Ψx,t=η1η2,η1=σ1+σ2eα1x−δ1t,η2=1+ρ1+ρ2eα1x−δ1t,in which ρ1,ρ2,σ1, and σ2 are the unfound constants. Plugging (13) into Equation (1), we get to the following cases:Case 1.
(14)α1=α1,β1=β1,ρ1=ρ2σ1−σ2σ2,ρ2=ρ2,σ1=σ1,σ2=σ2,δ1=α15.Case 2.
(15)α1=α1,β1=β1,ρ1=−1,ρ2=ρ2,σ1=σ1,σ2=σ2,δ1=α15.Case 3.
(16)α1=α1,β1=β1,ρ1=ρ1,ρ2=σ21+ρ1σ1,σ1=σ1,σ2=σ2,δ1=δ1.
For example, the resulting one-wave solution for Cases1 to 3 will be read, respectively, as
(17)Ψ1x,t=σ1+σ2e−tα15+xα11+ρ2σ1−σ2/σ2+ρ2e−tα15+xα1,Ψ2x,t=σ1+σ2e−tα15+xα1ρ2e−tα15+xα1,Ψ1x,t=σ1+σ2e−tδ1+xα11+ρ1+σ21+ρ1e−tδ1+xα1/σ1.


### 3.2. Set II: Two-Wave Solutions

We start up with two-wave functions based on the explanations in Step2 in the previous section; we deem that Equation (1) has the bellow form of two-wave solutions as
(18)Ψx,t=η1η2,(19)η1=ρ0+ρ1eα1x−δ1t+ρ2eα2x−δ2t+ρ1ρ2ρ12eα1+α2x−δ1+δ2t,(20)η2=1+σ1eα1x−δ1t+σ2eα2x−δ2t+σ1σ2σ12eα1+α2x−δ1+δ2t.Plugging (18) along with (19) into Equation (1), we gain the following cases:Case 1.
(21)α1=0,α2=α2,δ1=δ1,δ1=δ1,ρ0=ρ2σ2,ρ1=0,ρ2=ρ2,ρ12=ρ12,σ1=σ1,σ2=σ2,σ12=1.Case 2.
(22)α1=0,α2=α2,δ1=δ1,δ2=δ2,ρ0=1ρ12,ρ1=ρ1,ρ2=σ2ρ12,ρ12=ρ12,σ1=0,σ2=σ2,σ12=σ12.Case 3.
(23)α1=0,α2=α2,δ1=δ1,δ2=α25,ρ0=1ρ12,ρ1=ρ1,ρ2=ρ2,ρ12=ρ12,σ1=σ1,σ2=0,σ12=σ12.Case 4.
(24)α1=0,α2=α2,δ1=δ1,δ2=δ2,ρ0=ρ2σ2,ρ1=ρ1,ρ2=ρ2,ρ12=σ2ρ2,σ1=σ1,σ2=σ2,σ12=1.Case 5.
(25)α1=α1,α2=α2,δ1=α15,δ2=α25,ρ0=α14−3α13α2+4α12α22−3α1α23+α24ρ12α12+α1α2+α22α1+α22,ρ1=ρ1,ρ2=ρ2,ρ12=ρ12,σ1=0,σ2=0,σ12=σ12.Case 6.
(26)α1=α1,α2=α2,δ1=α15,δ2=δ2,ρ0=1ρ12,ρ1=ρ1,ρ2=σ2ρ12,ρ12=ρ12,σ1=0,σ2=σ2,σ12=σ12.Case 7.
(27)α1=α1,α2=α2,δ1=δ1,δ2=δ2,ρ0=ρ2σ2,ρ1=ρ2σ1σ2,ρ2=ρ2,ρ12=σ2σ12ρ2,σ1=0,σ2=σ2,σ12=σ12.
For example, the resulting two-wave solution for Cases1–7 will be read, respectively, as
(28)Ψ1x,t=ρ2/σ2+ρ2e−tδ2+xα21+σ1e−tδ1+σ2e−tδ2+xα2+σ1σ2e−tδ1−tδ2+xα2,Ψ2x,t=ρ12−1+ρ1e−tδ1+σ2e−tδ2+xα2/ρ12+ρ1σ2e−tδ1−tδ2+xα21+σ2e−tδ2+xα2,Ψ3x,t=ρ12−1+ρ1e−tδ1+ρ2e−tα25+xα2+ρ1ρ2ρ12e−tα25−tδ1+xα21+σ1e−tδ1,Ψ4x,t=ρ2/σ2+ρ1e−tδ1+ρ2e−tδ2+xα2+ρ1σ2e−tδ1−tδ2+xα21+σ1e−tδ1+σ2e−tδ2+xα2+σ1σ2e−tδ1−tδ2+xα2,Ψ5x,t=α14−3α13α2+4α12α22−3α1α23+α24ρ12α12+α1α2+α22α1+α22+ρ1e−tα15+xα1+ρ2e−tα25+xα2+ρ1ρ2ρ12e−tα15−tα25+xα1+xα2,Ψ6x,t=ρ12−1+ρ1e−tα15+xα1+σ2e−tδ2+xα2/ρ12+ρ1σ2e−tα15−tδ2+xα1+xα21+σ2e−tδ2+xα2,Ψ7x,t=ρ2/σ2+ρ2σ1e−tδ1+xα1/σ2+ρ2e−tδ2+xα2+ρ2σ1σ12e−tδ1−tδ2+xα1+xα21+σ1e−tδ1+xα1+σ2e−tδ2+xα2+σ1σ2σ12e−tδ1−tδ2+xα1+xα2.


### 3.3. Set III: Three-Wave Solutions

We start up with three-wave functions according to the given explanations in Step2 in the previous section, we deem that Equation (1) has the below form of three-wave solutions as
(29)Ψx,t=dfη1/dxη1,(30)η1=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,in which Λi=αix−δit,i=1,2,3. Appending (29) along with (30) into Equation (1), we obtain the following case:
(31)αi=αi,δi=αi5,i=1,2,3,ρij=ρij,i,j=1,2,3,i≠j.Therefore, three-wave solution will be as(32)Ψ1x,t=ρ1α1e−tα15+xα1+ρ2α2e−tα25+xα2+ρ3α3e−tα35+xα3+ρ1ρ2ρ12α1+α2e−tα15−tα25+xα1+xα2+ρ1ρ3ρ13α1+α3e−tα15−tα35+xα1+xα3+ρ2ρ3ρ23α2+α3e−tα25−tα35+xα2+xα3+ρ1ρ2ρ3ρ12ρ13ρ23α1+α2+α3e−tα15−tα25−tα35+xα1+xα2+xα3/1+ρ1e−tα15+xα1+ρ2e−tα25+xα2+ρ3e−tα35+xα3+ρ1ρ2ρ12e−tα15−tα25+xα1+xα2+ρ1ρ3ρ13e−tα15−tα35+xα1+xα3+ρ2ρ3ρ23e−tα25−tα35+xα2+xα3+ρ1ρ2ρ3ρ12ρ13ρ23e−tα15−tα25−tα35+xα1+xα2+xα3.


### 3.4. Cross-Kink Solutions

Here, we will consider the cross-kink wave solution with selecting the below function which for Equation (1) has been taken as
(33)f=expτ1+θ10exp−τ1+sinhτ2+sinτ3+θ11,τ1=θ1x+θ2t+θ3,τ2=θ4x+θ5t+θ6,τ3=θ7x+θ8t+θ9,(34)Ψx,t=lnfxx,where θi,i=1,⋯,11, are undetermined amounts which should be detected. Appending (34) into Equation (1) and afterwards collecting the coefficients, we obtain the following consequences:Case 1.
(35)θ1=θ4=iθ7,θ2=θ5=−16iθ75,θ8=−16θ75,θ11=0,i2=−1.
Substituting (35) into (33) and (34), we achieve a cross-kink wave solution of Equation (1) as follows:
(36)f=e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3−sinh16iθ75t−iθ7x−θ6−sin16θ75t−θ7x−θ9,Ψ1=−θ72e−16iθ75t+iθ7x+θ3−θ10θ72e16iθ75t−iθ7x−θ3+sinh16iθ75t−iθ7x−θ6θ72+sin16θ75t−θ7x−θ9θ72e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3−sinh16iθ75t−iθ7x−θ6−sin16θ75t−θ7x−θ9−iθ7e−16iθ75t+iθ7x+θ3−iθ10θ7e16iθ75t−iθ7x−θ3+icosh16iθ75t−iθ7x−θ6θ7+cos16θ75t−θ7x−θ9θ72e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3−sinh16iθ75t−iθ7x−θ6−sin16θ75t−θ7x−θ92.Case 2.
(37)θ1=3θ7=i3θ4,θ2=−3θ8=163iθ45,θ5=−16θ45,θ10=0,θ11=0,i2=−1.
Substituting (37) into (33) and (34), we achieve a cross-kink wave solution of Equation (1) which can be written as follows:
(38)f=e163θ45it+3iθ4x+θ3−sinh16tθ45−xθ4−θ6−sin16θ45it−iθ4x−θ9,Ψ1=3i2θ42e163θ45it+3iθ4x+θ3−sinh16tθ45−xθ4−θ6θ42+sin16θ45it−iθ4x−θ9i2θ42e163θ45it+3iθ4x+θ3−sinh16tθ45−xθ4−θ6−sin16θ45it−iθ4x−θ9−3iθ4e163θ45it+3iθ4x+θ3+cosh16tθ45−xθ4−θ6θ4+cos16θ45it−iθ4x−θ9iθ42e163θ45it+3iθ4x+θ3−sinh16tθ45−xθ4−θ6−sin16θ45it−iθ4x−θ92.


### 3.5. Periodic-Kink Wave Solutions

Here, we will consider the periodic-kink wave solution with selecting the below function which for Equation (1) has been taken as
(39)f=expτ1+θ10exp−τ1+coshτ2+cosτ3+θ11,τ1=θ1x+θ2t+θ3,τ2=θ4x+θ5t+θ6,τ3=θ7x+θ8t+θ9,(40)Ψx,t=lnfxx,where θi,i=1,⋯,11, are undetermined amounts which should be detected. Appending (40) into Equation (1) and afterwards collecting the coefficients, we obtain the following consequences:Case 1.
(41)θ1=θ4=iθ7,θ2=θ5=−16iθ75,θ8=−16θ75,θ11=0,i2=−1.
Substituting (41) into (39) and (40), we achieve a periodic-kink wave solution of Equation (1) which can be written as follows:
(42)f=e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3+cosh−16iθ75t+iθ7x+θ6+cos−16θ75t+θ7x+θ9,Ψ2=i2θ72e−16iθ75t+iθ7x+θ3+θ10i2θ72e16iθ75t−iθ7x−θ3+cosh16iθ75t−iθ7x−θ6i2θ72−cos16tθ75−xθ7−θ9θ72e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3+cosh16iθ75t−iθ7x−θ6+cos16tθ75−xθ7−θ9−iθ7e−16iθ75t+iθ7x+θ3−θ10iθ7e16iθ75t−iθ7x−θ3−sinh16iθ75t−iθ7x−θ6iθ7+sin16tθ75−xθ7−θ9θ72e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3+cosh16iθ75t−iθ7x−θ6+cos16tθ75−xθ7−θ92.Case 2.
(43)θ1=3θ7=i3θ4,θ2=−3θ8=163iθ45,θ5=−16θ45,θ10=0,θ11=0,i2=−1.
Substituting (43) into (39) and (40), we achieve a periodic-kink wave solution of Equation (1) as follows:
(44)f=e163θ45it+3iθ4x+θ3+sinh−16tθ45+xθ4+θ6+sin−16θ45it+iθ4x+θ9,Ψ2=3i2θ42e163θ45it+3iθ4x+θ3+cosh16tθ45−xθ4−θ6θ42−cos16θ45it−iθ4x−θ9i2θ42e163θ45it+3iθ4x+θ3+cosh16tθ45−xθ4−θ6+cos16θ45it−iθ4x−θ9−3iθ4e163θ45it+3iθ4x+θ3−sinh16tθ45−xθ4−θ6θ4+sin16θ45it−iθ4x−θ9iθ42e163θ45it+3iθ4x+θ3+cosh16tθ45−xθ4−θ6+cos16θ45it−iθ4x−θ92.


### 3.6. Periodic Type Wave Solutions-I

Here, we will consider the periodic type wave solution with selecting the below function which for Equation (1) has been taken as
(45)f=θ10expτ1+θ11exp−τ2+θ12cosτ3,τ1=θ1x+θ2t+θ3,τ2=θ4x+θ5t+θ6,τ3=θ7x+θ8t+θ9,(46)Ψx,t=lnfxx,where θi,i=1,⋯,11, are undetermined amounts which should be detected. Appending (46) into Equation (1) and afterwards collecting the coefficients, we obtain the following consequences:Case 1.
(47)θ1=3θ7,θ2=163θ75,θ8=−16θ75,θ11=0.
Substituting (47) into (45) and (46), a periodic type wave solution of Equation (1) can be obtained as follows:
(48)f=θ10e16θ753t+3θ7x+θ3+θ12cos16tθ75−xθ7−θ9,Ψ1=3θ10θ72e16θ753t+3θ7x+θ3−θ12cos16tθ75−xθ7−θ9θ72θ10e16θ753t+3θ7x+θ3+θ12cos16tθ75−xθ7−θ9−θ103θ7e16θ753t+3θ7x+θ3+θ12sin16tθ75−xθ7−θ9θ72θ10e16θ753t+3θ7x+θ3+θ12cos16tθ75−xθ7−θ92.Case 2.
(49)θ1=−θ4=−3θ7,θ2=−θ5=−163θ75,θ8=−16θ75.
Substituting (49) into (45) and (46), a periodic type wave solution of Equation (1) can be obtained as follows:
(50)f=θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6+θ12cos16tθ75−xθ7−θ9,Ψ2=3θ10θ72e−16θ753t−3θ7x+θ3+3θ11θ72e−16θ753t−3θ7x−θ6−θ12cos16tθ75−xθ7−θ9θ72θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6+θ12cos16tθ75−xθ7−θ9−−θ103θ7e−16θ753t−3θ7x+θ3−θ113θ7e−16θ753t−3θ7x−θ6+θ12sin16tθ75−xθ7−θ9θ72θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6+θ12cos16tθ75−xθ7−θ92.


### 3.7. Periodic Type Wave Solutions-II

Here, we will consider the periodic wave solution with selecting the below function which for Equation (1) has been taken as
(51)f=θ10expτ1+θ11exp−τ2+θ12sinτ3,τ1=θ1x+θ2t+θ3,τ2=θ4x+θ5t+θ6,τ3=θ7x+θ8t+θ9,(52)Ψx,t=lnfxx,where θi,i=1,⋯,11, are undetermined amounts which should be detected. Appending (52) into Equation (1) and afterwards collecting the coefficients, we obtain the following consequences:Case 1.
(53)θ1=3θ7,θ2=163θ75,θ8=−16θ75,θ11=0.
Substituting (53) into (51) and (52), a periodic type wave solution of Equation (1) can be obtained as follows:
(54)f=θ10e16θ753t+3θ7x+θ3+θ12sin−16tθ75+xθ7+θ9,Ψ1=3θ11θ72e16θ753t+3θ7x+θ3+θ12sin16tθ75−xθ7−θ9θ72θ11e16θ753t+3θ7x+θ3−θ12sin16tθ75−xθ7−θ9−θ113θ7e16θ753t+3θ7x+θ3+θ12cos16tθ75−xθ7−θ9θ72θ11e16θ753t+3θ7x+θ3−θ12sin16tθ75−xθ7−θ92.Case 2.
(55)θ1=−θ4=−3θ7,θ2=−θ5=−163θ75,θ8=−16θ75.
Substituting (55) into (51) and (52), a periodic type wave solution of Equation (1) can be obtained as follows:
(56)f=θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6−θ12sin16tθ75−xθ7−θ9,Ψ2=3θ10θ72e−16θ753t−3θ7x+θ3+3θ11θ72e−16θ753t−3θ7x−θ6+θ12sin16tθ75−xθ7−θ9θ72θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6−θ12sin16tθ75−xθ7−θ9−−θ103θ7e−16θ753t−3θ7x+θ3−θ113θ7e−16θ753t−3θ7x−θ6+θ12cos16tθ75−xθ7−θ9θ72θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6−θ12sin16tθ75−xθ7−θ92.


### 3.8. Solitary Wave Solutions

Here, we will consider the solitary wave solution with selecting the below function which for Equation (1) has been taken as
(57)f=θ7+θ8expτ1+θ9exp−τ2,τ1=θ1x+θ2t+θ3,τ2=θ4x+θ5t+θ6,(58)Ψx,t=lnfxx,where θi,i=1,⋯,9, are undetermined amounts which should be detected. Appending (58) into Equation (1) and afterwards collecting the coefficients, we obtain the below consequences:Case 1.
(59)θ2=−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5,θ7=0.
Substituting (59) into (57) and (58), a solitary wave solution of Equation (1) can be written as follows:
(60)f=θ8e−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5t+θ1x+θ3+θ9e−tθ5−xθ4−θ6,(61)Ψ1=θ8θ12e−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5t+θ1x+θ3+θ9θ42e−tθ5−xθ4−θ6θ8e−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5t+θ1x+θ3+θ9e−tθ5−xθ4−θ6−θ8θ1e−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5t+θ1x+θ3−θ9θ4e−tθ5−xθ4−θ62θ8e−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5t+θ1x+θ3+θ9e−tθ5−xθ4−θ62.By using suitable values of parameters, the analytical treatment of periodic wave solution is presented in Figure1 including 3D plot and 2D plot with three points of time including t=0, t=0.02, and t=0.04.Case 2.
(62)θ1=−θ4,θ2=θ45,θ5=−θ45.
Substituting (62) into (57) and (58), a solitary wave solution of Equation (1) can be obtained as follows:
(63)f=θ7+θ8etθ45−xθ4+θ3+θ9etθ45−xθ4−θ6,Ψ2=θ42θ8etθ45−xθ4+θ3+θ9etθ45−xθ4−θ6θ7θ7+θ8etθ45−xθ4+θ3+θ9etθ45−xθ4−θ62.Figure 1
The solitary wave (61) at θ1=1,θ4=2,θ3=2,θ5=.5,θ6=3,θ8=1.3,θ9=2.4.By using suitable values of parameters, the analytical treatment of periodic wave solution is presented in Figure2 including 3D plot and 2D plot with three points of time including t=0, t=0.02, and t=0.04.Case 3.
(64)θ1=−12−1±3iθ4,θ2=121±3iθ45,θ5=−θ45.
Substituting (64) into (57) and (58), we achieve a solitary wave solution of Equation (1) as follows:
(65)f=θ7+θ8e1/2tθ45i3+1−1/2xθ4i3−1+θ3+θ9etθ45−xθ4−θ6,Ψ3=1/4θ8θ42i3−12e1/2tθ45i3+1−1/2xθ4i3−1+θ3+θ9θ42etθ45−xθ4−θ6θ7+θ8e1/2tθ45i3+1−1/2xθ4i3−1+θ3+θ9etθ45−xθ4−θ6−−1/2θ8θ4i3−1e1/2tθ45i3+1−1/2xθ4i3−1+θ3−θ9θ4etθ45−xθ4−θ62θ7+θ8e1/2tθ45i3+1−1/2xθ4i3−1+θ3+θ9etθ45−xθ4−θ62.Figure 2
The solitary wave (63) at θ4=2,θ3=2,θ6=3,θ7=1.3,θ8=2,θ9=2.4.


## 3.1. Set I: One-Wave Solution

We start up with one-wave function based on the explanation in Step2 in the previous section, we deem that Equation (1) has the below form of one-wave solution as
(13)Ψx,t=η1η2,η1=σ1+σ2eα1x−δ1t,η2=1+ρ1+ρ2eα1x−δ1t,in which ρ1,ρ2,σ1, and σ2 are the unfound constants. Plugging (13) into Equation (1), we get to the following cases:Case 1.
(14)α1=α1,β1=β1,ρ1=ρ2σ1−σ2σ2,ρ2=ρ2,σ1=σ1,σ2=σ2,δ1=α15.Case 2.
(15)α1=α1,β1=β1,ρ1=−1,ρ2=ρ2,σ1=σ1,σ2=σ2,δ1=α15.Case 3.
(16)α1=α1,β1=β1,ρ1=ρ1,ρ2=σ21+ρ1σ1,σ1=σ1,σ2=σ2,δ1=δ1.
For example, the resulting one-wave solution for Cases1 to 3 will be read, respectively, as
(17)Ψ1x,t=σ1+σ2e−tα15+xα11+ρ2σ1−σ2/σ2+ρ2e−tα15+xα1,Ψ2x,t=σ1+σ2e−tα15+xα1ρ2e−tα15+xα1,Ψ1x,t=σ1+σ2e−tδ1+xα11+ρ1+σ21+ρ1e−tδ1+xα1/σ1.


## 3.2. Set II: Two-Wave Solutions

We start up with two-wave functions based on the explanations in Step2 in the previous section; we deem that Equation (1) has the bellow form of two-wave solutions as
(18)Ψx,t=η1η2,(19)η1=ρ0+ρ1eα1x−δ1t+ρ2eα2x−δ2t+ρ1ρ2ρ12eα1+α2x−δ1+δ2t,(20)η2=1+σ1eα1x−δ1t+σ2eα2x−δ2t+σ1σ2σ12eα1+α2x−δ1+δ2t.Plugging (18) along with (19) into Equation (1), we gain the following cases:Case 1.
(21)α1=0,α2=α2,δ1=δ1,δ1=δ1,ρ0=ρ2σ2,ρ1=0,ρ2=ρ2,ρ12=ρ12,σ1=σ1,σ2=σ2,σ12=1.Case 2.
(22)α1=0,α2=α2,δ1=δ1,δ2=δ2,ρ0=1ρ12,ρ1=ρ1,ρ2=σ2ρ12,ρ12=ρ12,σ1=0,σ2=σ2,σ12=σ12.Case 3.
(23)α1=0,α2=α2,δ1=δ1,δ2=α25,ρ0=1ρ12,ρ1=ρ1,ρ2=ρ2,ρ12=ρ12,σ1=σ1,σ2=0,σ12=σ12.Case 4.
(24)α1=0,α2=α2,δ1=δ1,δ2=δ2,ρ0=ρ2σ2,ρ1=ρ1,ρ2=ρ2,ρ12=σ2ρ2,σ1=σ1,σ2=σ2,σ12=1.Case 5.
(25)α1=α1,α2=α2,δ1=α15,δ2=α25,ρ0=α14−3α13α2+4α12α22−3α1α23+α24ρ12α12+α1α2+α22α1+α22,ρ1=ρ1,ρ2=ρ2,ρ12=ρ12,σ1=0,σ2=0,σ12=σ12.Case 6.
(26)α1=α1,α2=α2,δ1=α15,δ2=δ2,ρ0=1ρ12,ρ1=ρ1,ρ2=σ2ρ12,ρ12=ρ12,σ1=0,σ2=σ2,σ12=σ12.Case 7.
(27)α1=α1,α2=α2,δ1=δ1,δ2=δ2,ρ0=ρ2σ2,ρ1=ρ2σ1σ2,ρ2=ρ2,ρ12=σ2σ12ρ2,σ1=0,σ2=σ2,σ12=σ12.
For example, the resulting two-wave solution for Cases1–7 will be read, respectively, as
(28)Ψ1x,t=ρ2/σ2+ρ2e−tδ2+xα21+σ1e−tδ1+σ2e−tδ2+xα2+σ1σ2e−tδ1−tδ2+xα2,Ψ2x,t=ρ12−1+ρ1e−tδ1+σ2e−tδ2+xα2/ρ12+ρ1σ2e−tδ1−tδ2+xα21+σ2e−tδ2+xα2,Ψ3x,t=ρ12−1+ρ1e−tδ1+ρ2e−tα25+xα2+ρ1ρ2ρ12e−tα25−tδ1+xα21+σ1e−tδ1,Ψ4x,t=ρ2/σ2+ρ1e−tδ1+ρ2e−tδ2+xα2+ρ1σ2e−tδ1−tδ2+xα21+σ1e−tδ1+σ2e−tδ2+xα2+σ1σ2e−tδ1−tδ2+xα2,Ψ5x,t=α14−3α13α2+4α12α22−3α1α23+α24ρ12α12+α1α2+α22α1+α22+ρ1e−tα15+xα1+ρ2e−tα25+xα2+ρ1ρ2ρ12e−tα15−tα25+xα1+xα2,Ψ6x,t=ρ12−1+ρ1e−tα15+xα1+σ2e−tδ2+xα2/ρ12+ρ1σ2e−tα15−tδ2+xα1+xα21+σ2e−tδ2+xα2,Ψ7x,t=ρ2/σ2+ρ2σ1e−tδ1+xα1/σ2+ρ2e−tδ2+xα2+ρ2σ1σ12e−tδ1−tδ2+xα1+xα21+σ1e−tδ1+xα1+σ2e−tδ2+xα2+σ1σ2σ12e−tδ1−tδ2+xα1+xα2.


## 3.3. Set III: Three-Wave Solutions

We start up with three-wave functions according to the given explanations in Step2 in the previous section, we deem that Equation (1) has the below form of three-wave solutions as
(29)Ψx,t=dfη1/dxη1,(30)η1=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,in which Λi=αix−δit,i=1,2,3. Appending (29) along with (30) into Equation (1), we obtain the following case:
(31)αi=αi,δi=αi5,i=1,2,3,ρij=ρij,i,j=1,2,3,i≠j.Therefore, three-wave solution will be as(32)Ψ1x,t=ρ1α1e−tα15+xα1+ρ2α2e−tα25+xα2+ρ3α3e−tα35+xα3+ρ1ρ2ρ12α1+α2e−tα15−tα25+xα1+xα2+ρ1ρ3ρ13α1+α3e−tα15−tα35+xα1+xα3+ρ2ρ3ρ23α2+α3e−tα25−tα35+xα2+xα3+ρ1ρ2ρ3ρ12ρ13ρ23α1+α2+α3e−tα15−tα25−tα35+xα1+xα2+xα3/1+ρ1e−tα15+xα1+ρ2e−tα25+xα2+ρ3e−tα35+xα3+ρ1ρ2ρ12e−tα15−tα25+xα1+xα2+ρ1ρ3ρ13e−tα15−tα35+xα1+xα3+ρ2ρ3ρ23e−tα25−tα35+xα2+xα3+ρ1ρ2ρ3ρ12ρ13ρ23e−tα15−tα25−tα35+xα1+xα2+xα3.


## 3.4. Cross-Kink Solutions

Here, we will consider the cross-kink wave solution with selecting the below function which for Equation (1) has been taken as
(33)f=expτ1+θ10exp−τ1+sinhτ2+sinτ3+θ11,τ1=θ1x+θ2t+θ3,τ2=θ4x+θ5t+θ6,τ3=θ7x+θ8t+θ9,(34)Ψx,t=lnfxx,where θi,i=1,⋯,11, are undetermined amounts which should be detected. Appending (34) into Equation (1) and afterwards collecting the coefficients, we obtain the following consequences:Case 1.
(35)θ1=θ4=iθ7,θ2=θ5=−16iθ75,θ8=−16θ75,θ11=0,i2=−1.
Substituting (35) into (33) and (34), we achieve a cross-kink wave solution of Equation (1) as follows:
(36)f=e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3−sinh16iθ75t−iθ7x−θ6−sin16θ75t−θ7x−θ9,Ψ1=−θ72e−16iθ75t+iθ7x+θ3−θ10θ72e16iθ75t−iθ7x−θ3+sinh16iθ75t−iθ7x−θ6θ72+sin16θ75t−θ7x−θ9θ72e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3−sinh16iθ75t−iθ7x−θ6−sin16θ75t−θ7x−θ9−iθ7e−16iθ75t+iθ7x+θ3−iθ10θ7e16iθ75t−iθ7x−θ3+icosh16iθ75t−iθ7x−θ6θ7+cos16θ75t−θ7x−θ9θ72e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3−sinh16iθ75t−iθ7x−θ6−sin16θ75t−θ7x−θ92.Case 2.
(37)θ1=3θ7=i3θ4,θ2=−3θ8=163iθ45,θ5=−16θ45,θ10=0,θ11=0,i2=−1.
Substituting (37) into (33) and (34), we achieve a cross-kink wave solution of Equation (1) which can be written as follows:
(38)f=e163θ45it+3iθ4x+θ3−sinh16tθ45−xθ4−θ6−sin16θ45it−iθ4x−θ9,Ψ1=3i2θ42e163θ45it+3iθ4x+θ3−sinh16tθ45−xθ4−θ6θ42+sin16θ45it−iθ4x−θ9i2θ42e163θ45it+3iθ4x+θ3−sinh16tθ45−xθ4−θ6−sin16θ45it−iθ4x−θ9−3iθ4e163θ45it+3iθ4x+θ3+cosh16tθ45−xθ4−θ6θ4+cos16θ45it−iθ4x−θ9iθ42e163θ45it+3iθ4x+θ3−sinh16tθ45−xθ4−θ6−sin16θ45it−iθ4x−θ92.


## 3.5. Periodic-Kink Wave Solutions

Here, we will consider the periodic-kink wave solution with selecting the below function which for Equation (1) has been taken as
(39)f=expτ1+θ10exp−τ1+coshτ2+cosτ3+θ11,τ1=θ1x+θ2t+θ3,τ2=θ4x+θ5t+θ6,τ3=θ7x+θ8t+θ9,(40)Ψx,t=lnfxx,where θi,i=1,⋯,11, are undetermined amounts which should be detected. Appending (40) into Equation (1) and afterwards collecting the coefficients, we obtain the following consequences:Case 1.
(41)θ1=θ4=iθ7,θ2=θ5=−16iθ75,θ8=−16θ75,θ11=0,i2=−1.
Substituting (41) into (39) and (40), we achieve a periodic-kink wave solution of Equation (1) which can be written as follows:
(42)f=e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3+cosh−16iθ75t+iθ7x+θ6+cos−16θ75t+θ7x+θ9,Ψ2=i2θ72e−16iθ75t+iθ7x+θ3+θ10i2θ72e16iθ75t−iθ7x−θ3+cosh16iθ75t−iθ7x−θ6i2θ72−cos16tθ75−xθ7−θ9θ72e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3+cosh16iθ75t−iθ7x−θ6+cos16tθ75−xθ7−θ9−iθ7e−16iθ75t+iθ7x+θ3−θ10iθ7e16iθ75t−iθ7x−θ3−sinh16iθ75t−iθ7x−θ6iθ7+sin16tθ75−xθ7−θ9θ72e−16iθ75t+iθ7x+θ3+θ10e16iθ75t−iθ7x−θ3+cosh16iθ75t−iθ7x−θ6+cos16tθ75−xθ7−θ92.Case 2.
(43)θ1=3θ7=i3θ4,θ2=−3θ8=163iθ45,θ5=−16θ45,θ10=0,θ11=0,i2=−1.
Substituting (43) into (39) and (40), we achieve a periodic-kink wave solution of Equation (1) as follows:
(44)f=e163θ45it+3iθ4x+θ3+sinh−16tθ45+xθ4+θ6+sin−16θ45it+iθ4x+θ9,Ψ2=3i2θ42e163θ45it+3iθ4x+θ3+cosh16tθ45−xθ4−θ6θ42−cos16θ45it−iθ4x−θ9i2θ42e163θ45it+3iθ4x+θ3+cosh16tθ45−xθ4−θ6+cos16θ45it−iθ4x−θ9−3iθ4e163θ45it+3iθ4x+θ3−sinh16tθ45−xθ4−θ6θ4+sin16θ45it−iθ4x−θ9iθ42e163θ45it+3iθ4x+θ3+cosh16tθ45−xθ4−θ6+cos16θ45it−iθ4x−θ92.


## 3.6. Periodic Type Wave Solutions-I

Here, we will consider the periodic type wave solution with selecting the below function which for Equation (1) has been taken as
(45)f=θ10expτ1+θ11exp−τ2+θ12cosτ3,τ1=θ1x+θ2t+θ3,τ2=θ4x+θ5t+θ6,τ3=θ7x+θ8t+θ9,(46)Ψx,t=lnfxx,where θi,i=1,⋯,11, are undetermined amounts which should be detected. Appending (46) into Equation (1) and afterwards collecting the coefficients, we obtain the following consequences:Case 1.
(47)θ1=3θ7,θ2=163θ75,θ8=−16θ75,θ11=0.
Substituting (47) into (45) and (46), a periodic type wave solution of Equation (1) can be obtained as follows:
(48)f=θ10e16θ753t+3θ7x+θ3+θ12cos16tθ75−xθ7−θ9,Ψ1=3θ10θ72e16θ753t+3θ7x+θ3−θ12cos16tθ75−xθ7−θ9θ72θ10e16θ753t+3θ7x+θ3+θ12cos16tθ75−xθ7−θ9−θ103θ7e16θ753t+3θ7x+θ3+θ12sin16tθ75−xθ7−θ9θ72θ10e16θ753t+3θ7x+θ3+θ12cos16tθ75−xθ7−θ92.Case 2.
(49)θ1=−θ4=−3θ7,θ2=−θ5=−163θ75,θ8=−16θ75.
Substituting (49) into (45) and (46), a periodic type wave solution of Equation (1) can be obtained as follows:
(50)f=θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6+θ12cos16tθ75−xθ7−θ9,Ψ2=3θ10θ72e−16θ753t−3θ7x+θ3+3θ11θ72e−16θ753t−3θ7x−θ6−θ12cos16tθ75−xθ7−θ9θ72θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6+θ12cos16tθ75−xθ7−θ9−−θ103θ7e−16θ753t−3θ7x+θ3−θ113θ7e−16θ753t−3θ7x−θ6+θ12sin16tθ75−xθ7−θ9θ72θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6+θ12cos16tθ75−xθ7−θ92.


## 3.7. Periodic Type Wave Solutions-II

Here, we will consider the periodic wave solution with selecting the below function which for Equation (1) has been taken as
(51)f=θ10expτ1+θ11exp−τ2+θ12sinτ3,τ1=θ1x+θ2t+θ3,τ2=θ4x+θ5t+θ6,τ3=θ7x+θ8t+θ9,(52)Ψx,t=lnfxx,where θi,i=1,⋯,11, are undetermined amounts which should be detected. Appending (52) into Equation (1) and afterwards collecting the coefficients, we obtain the following consequences:Case 1.
(53)θ1=3θ7,θ2=163θ75,θ8=−16θ75,θ11=0.
Substituting (53) into (51) and (52), a periodic type wave solution of Equation (1) can be obtained as follows:
(54)f=θ10e16θ753t+3θ7x+θ3+θ12sin−16tθ75+xθ7+θ9,Ψ1=3θ11θ72e16θ753t+3θ7x+θ3+θ12sin16tθ75−xθ7−θ9θ72θ11e16θ753t+3θ7x+θ3−θ12sin16tθ75−xθ7−θ9−θ113θ7e16θ753t+3θ7x+θ3+θ12cos16tθ75−xθ7−θ9θ72θ11e16θ753t+3θ7x+θ3−θ12sin16tθ75−xθ7−θ92.Case 2.
(55)θ1=−θ4=−3θ7,θ2=−θ5=−163θ75,θ8=−16θ75.
Substituting (55) into (51) and (52), a periodic type wave solution of Equation (1) can be obtained as follows:
(56)f=θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6−θ12sin16tθ75−xθ7−θ9,Ψ2=3θ10θ72e−16θ753t−3θ7x+θ3+3θ11θ72e−16θ753t−3θ7x−θ6+θ12sin16tθ75−xθ7−θ9θ72θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6−θ12sin16tθ75−xθ7−θ9−−θ103θ7e−16θ753t−3θ7x+θ3−θ113θ7e−16θ753t−3θ7x−θ6+θ12cos16tθ75−xθ7−θ9θ72θ10e−16θ753t−3θ7x+θ3+θ11e−16θ753t−3θ7x−θ6−θ12sin16tθ75−xθ7−θ92.


## 3.8. Solitary Wave Solutions

Here, we will consider the solitary wave solution with selecting the below function which for Equation (1) has been taken as
(57)f=θ7+θ8expτ1+θ9exp−τ2,τ1=θ1x+θ2t+θ3,τ2=θ4x+θ5t+θ6,(58)Ψx,t=lnfxx,where θi,i=1,⋯,9, are undetermined amounts which should be detected. Appending (58) into Equation (1) and afterwards collecting the coefficients, we obtain the below consequences:Case 1.
(59)θ2=−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5,θ7=0.
Substituting (59) into (57) and (58), a solitary wave solution of Equation (1) can be written as follows:
(60)f=θ8e−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5t+θ1x+θ3+θ9e−tθ5−xθ4−θ6,(61)Ψ1=θ8θ12e−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5t+θ1x+θ3+θ9θ42e−tθ5−xθ4−θ6θ8e−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5t+θ1x+θ3+θ9e−tθ5−xθ4−θ6−θ8θ1e−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5t+θ1x+θ3−θ9θ4e−tθ5−xθ4−θ62θ8e−θ15−5θ14θ4−10θ13θ42−10θ12θ43−5θ1θ44−θ45−θ5t+θ1x+θ3+θ9e−tθ5−xθ4−θ62.By using suitable values of parameters, the analytical treatment of periodic wave solution is presented in Figure1 including 3D plot and 2D plot with three points of time including t=0, t=0.02, and t=0.04.Case 2.
(62)θ1=−θ4,θ2=θ45,θ5=−θ45.
Substituting (62) into (57) and (58), a solitary wave solution of Equation (1) can be obtained as follows:
(63)f=θ7+θ8etθ45−xθ4+θ3+θ9etθ45−xθ4−θ6,Ψ2=θ42θ8etθ45−xθ4+θ3+θ9etθ45−xθ4−θ6θ7θ7+θ8etθ45−xθ4+θ3+θ9etθ45−xθ4−θ62.Figure 1
The solitary wave (61) at θ1=1,θ4=2,θ3=2,θ5=.5,θ6=3,θ8=1.3,θ9=2.4.By using suitable values of parameters, the analytical treatment of periodic wave solution is presented in Figure2 including 3D plot and 2D plot with three points of time including t=0, t=0.02, and t=0.04.Case 3.
(64)θ1=−12−1±3iθ4,θ2=121±3iθ45,θ5=−θ45.
Substituting (64) into (57) and (58), we achieve a solitary wave solution of Equation (1) as follows:
(65)f=θ7+θ8e1/2tθ45i3+1−1/2xθ4i3−1+θ3+θ9etθ45−xθ4−θ6,Ψ3=1/4θ8θ42i3−12e1/2tθ45i3+1−1/2xθ4i3−1+θ3+θ9θ42etθ45−xθ4−θ6θ7+θ8e1/2tθ45i3+1−1/2xθ4i3−1+θ3+θ9etθ45−xθ4−θ6−−1/2θ8θ4i3−1e1/2tθ45i3+1−1/2xθ4i3−1+θ3−θ9θ4etθ45−xθ4−θ62θ7+θ8e1/2tθ45i3+1−1/2xθ4i3−1+θ3+θ9etθ45−xθ4−θ62.Figure 2
The solitary wave (63) at θ4=2,θ3=2,θ6=3,θ7=1.3,θ8=2,θ9=2.4.


## 4. Stability Analysis of CDG Equation

According to [59], in order to analyze the propagation characteristics of the rogue wave in detail, we choose the linear stability analysis for the CDG equation via the following function along with appropriate parameters:
(66)Φx,t=θ+δΩx,t,where the relation constant θ is a steady state solution of Equation (66). Appending (66) into Equation (1), one can obtain
(67)δ∂∂tΩx,t+δ∂5∂x5Ωx,t+30δ∂3∂x3Ωx,tθ+30δ2∂3∂x3Ωx,tΩx,t+30δ2∂∂xΩx,t∂2∂x2Ωx,t+180δ∂∂xΩx,tθ2+360δ2∂∂xΩx,tθΩx,t+180δ3∂∂xΩx,tΩx,t2=0.By linearization of Equation (67), we get
(68)δ∂∂tΩx,t+δ∂5∂x5Ωx,t+30δ∂3∂x3Ωx,tθ+180δ∂∂xΩx,tθ2=0.Theorem 1.
Presume that the solution of Equation (68) has the following form:
(69)Ωx,t=ρ1eiαx+βt,where α,β are the normalized wave numbers, by putting (69) into Equation (68), then by solving for β, we can achieve the following form
(70)βα=−α5+30α3θ−180αθ2.Proof.
By appending the equality (69) in the linear PDE (68), we obtain
(71)δ∂∂tΩx,t+δ∂5∂x5Ωx,t+30δ∂3∂x3Ωx,tθ+180δ∂∂xΩx,tθ2=ieiαx+βtδρ1α5−30α3θ+180αθ2+β=0.
By solving and simplifying, we can find the value ofβα as follows:
(72)βα=−α5+30α3θ−180αθ2.
After that, we get to the needed solution. Hence, the proof of the theorem is complete.In Figures3–5, it can be seen that when the sign of βα is positive for all amounts of α, then any superposition of solutions of the form eiαx+βt will come to ascent, while the sign of βα is negative for all amounts of α, then any superposition of solutions of the form eiαx+βt will come to decay and the steady condition is stable. After that, in Figures 3 and 4, it can be observed that if the βα is positive or negative for some amounts of α, then with increasing time some components of a superposition will become descent, and the steady condition is stable. Finally, in Figure 5, it can be perceived that when the sign of βα is positive for all amounts of α, then any superposition of solutions of the form eiαx+βt will come to ascent, while the sign of βα is negative for all amounts of α, then any superposition of solutions of the form eiαx+βt will come to decay and the steady condition is stable.Figure 3
The behavior of stability analysisβα and wave number α with the disparate amounts θ=1,θ=2,θ=3.Figure 4
The behavior of stability analysisβα and wave number α with the disparate amounts θ=−5,θ=−3,θ=−1.Figure 5
The behavior of stability analysisβα and wave number α with the disparate amounts θ=−1,θ=0,θ=1.


## 5. Conclusion

In this work, the multiple exp-function, cross-kink, periodic-kink, and solitary wave methods with predictability of the (1+1)-dimensional CDG equation are investigated with more arbitrary autocephalous parameters. It is not hard to see that the general periodic-kink solution is an algebraically wave solution, and we noticed that some obtained solutions are singular periodic solitary wave solution which is periodic wave or periodic-kink, or solitary wave solutions in x−t direction. Also, the other presented solution is a breather type of two-solitary wave solution which contains a periodic wave and two solitary waves, whose amplitude periodically oscillates with the evolution of time. Moreover, the kink and periodic solutions were analyzed and investigated. In addition, the periodic-kink waves appeared when the periodic solution cut by a stripe soliton before or after a special time. Meanwhile, the modulation instability was applied to discuss the stability of earned solutions. Finally, we show some graphs to explain these solutions.



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