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

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.

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 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][43][44]. Based on the obtained transformation of integrating Equation (1), we get to the following nonlinear PDE [45]: According to [46], the Hirota bilinear from of the CDG equation reads and, also by applying the dependent variable transformation, turns into the Hirota bilinear form where D is a bilinear operator. By deeming the D-operator defined with the aid of the functions Γ 1 and Γ 2 , we get to the following relation: 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 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], groupinvariant 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.

Multiple Exp-Function Method
In this section, according to [61][62][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: We commence a sequence of new variables ξ i = ξ i ðx, tÞ, 1 ≤ i ≤ n, by solvable PDEs, for example, the linear ones, where α i , 1 ≤ i ≤ n, is the angular wave number and δ i , 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: in which ϖ i , 1 ≤ i ≤ n, are undetermined constants.
Step 2. Determine the solution of Equation (7) as the following form in terms of the new variables ξ i , 1 ≤ i ≤ n: in which Δ rs,ij and Ω rs,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 2 Advances in Mathematical Physics a series of nonlinear algebraic equations about the variables α i , δ i , Δ rs,ij and Ω rs,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 in which Ω ≠ 0, and also, we have

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 Step 2 in the previous section, we deem that Equation (1) has the below form of one-wave solution as in which ρ 1 , ρ 2 , σ 1 , and σ 2 are the unfound constants. Plugging (13) into Equation (1), we get to the following cases: Case 2.
For example, the resulting one-wave solution for Cases 1 to 3 will be read, respectively, as 3.2. Set II: Two-Wave Solutions. We start up with two-wave functions based on the explanations in Step 2 in the previous section; we deem that Equation (1) has the bellow form of two-wave solutions as Plugging (18) along with (19) into Equation (1), we gain the following cases: Case 2.

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Case 5.

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 where the relation constant θ is a steady state solution of Equation (66). Appending (66) into Equation (1), one can obtain By linearization of Equation (67), we get 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 Proof. By appending the equality (69) in the linear PDE (68), we obtain By solving and simplifying, we can find the value of βðαÞ as follows: After that, we get to the needed solution. Hence, the proof of the theorem is complete.
In Figures 3-5, it can be seen that when the sign of βðαÞ is positive for all amounts of α, then any superposition of solutions of the form e iðαx+βtÞ will come to ascent, while the sign of βðαÞ is negative for all amounts of α, then any superposition of solutions of the form e iðαx+βtÞ will come to decay and the steady condition is stable. After that, in Figures 3 and 4, it can be observed that if the βðαÞ 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 βðαÞ is positive for all amounts of α, then any superposition of solutions of the form e iðαx+βtÞ will come to ascent, while the sign of βðαÞ is negative for all amounts of α, then any superposition of solutions of the form e iðαx+βtÞ will come to decay and the steady condition is stable.

Conclusion
In this work, the multiple exp-function, cross-kink, periodickink, 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.

Data Availability
The datasets supporting the conclusions of this article are included within the article and its additional file.

Conflicts of Interest
The authors declare that they have no conflicts of interest.