A new study of soliton solutions for the variable coefficients Caudrey–Dodd–Gibbon equation

We present a paper on the different methods for finding solutions to the Caudrey–Dodd–Gibbon equation with variable coefficients, which has wide applications in quantum mechanics and nonlinear optics. We have studied the equation analytically by using the unified method and the modified Kudryashov method. With the aid of symbolic computation and several types of auxiliary equations, we have obtained soliton solutions and other solutions. Then, we assign different values to the parameters, generating two-dimensional and three-dimensional graphics of the solutions, and discuss the interaction of several groups of solion solutions to form periodic wave solutions and kink solitary wave solutions.


Introduction
Nonlinear partial differential equations play a crucial role in the study of physical phenomena. Through symbolic calculation, we cannot only discover the properties of nonlinear partial differential equations [1][2][3], but also get the exact solutions about them. Therefore, solving partial differential equations is a significant research subject. With the rapid development of the scientific society, there have been some methods proposed to deal with the nonlinear partial differential equations. For example, a history approach [4], the fixed point technique [5], the new method [6], the modified (G /G)-expansion method [7,8], the novel generalized (G /G) -expansion technique [9], the modified Kudryashov method [10][11][12][13], the extended simplest equation [14][15][16], the homotopy analysis transform method [17] and the united method [18] and so on.
In this paper, we will deal with the Caudrey-Dodd-Gibbon equation with variable coefficients (vcCDG) using the unified method and the modified Kudryashov method of the above methods. Up to now, Mickle methods are applied to Equation (1) as yet, such as Hirota's bilinear method [19] is used to obtain some breather wave and lumps solutions to the CDG equation, the expfunction method [20] and the (G /G)-expansion method [21] find generalized solitary solutions and exact solutions respectively, the invariance properties, optimal system and group invariant solutions are investigated in [22], the approximate solutions are obtained by the variational iteration method [23], and the improved generalized Riccati equation mapping method [24] figures out exact solutions. Nevertheless, the unified method and the modified Kudryashov method have not been applied to the equation.
In fact, it has been shown that some known methods (the (G /G)-expansion method, the F-expansion method, the exp-function method and the rational expansion method and others) are special cases of the united method, which was established in 2012. The emergence of the unified method has solved a great many of equation problems, such as, the thermophoretic motion equation [25], the coupled Burgers equations [26], the Zakharov-Kuznetsov equation [27], the Benjamin-Bona-Mahony-Peregrine equation [28] and the generalized (2 + 1)-dimensional Boussinesq equation [29]. Moreover, the modified Kudryashov method is also an important method for solving partial differential equation problems, the generalized Schrödinger-Boussinesq equations [10], the Klein-Gordon equations [11], the Benjamin-Bona-Mahony-Peregrine equation [28] and the KdV-KZK equation [30] are all solved by the modified Kudryashov method.
Here, the vcCDG equation [31,32] reads, where u = u(x, t), uu xxx and u xxxxx are dispersive terms, σ i (t) (i = 1, 2, 3, 4) are functions with t. As one of the fifth-order KdV, it can also describe the nonlinear phenomena in fluids or plasmas and is completely integrable [32]. Compared with existing articles, breather wave and lumps solutions, solitary solutions, exact solutions, group invariant solutions, and approximate solutions have been obtained [19][20][21][22][23][24]. Our primary mission is committed to using the unified method and the modified Kudryashov method to look for polynomial solutions, rational function solutions and travelling wave solutions so as to get more relevant properties of the equation. Some significant examples are given below. When is completely integrable and has soliton solutions. When are used to model nonlinear dispersive waves such as laser optics and plasma physics.
The remaining sections of this paper are textured in what follows. In Section 2, we use the unified method to receive the solutions of Equation (1) and draw graphs for analysis. In Section 3, we are devoted to the utilization of the modified Kudryashov method to the vcCDG equation. Finally, conclusions will be given in Section 4.

The unified method
In view of the operation steps of the method, the following hypothesis is framed: (4) where α 1 , α 2 , β 1 and β 2 are arbitrary constants. With the above travelling wave transformation, Equation (1) becomes the following differential equation: where U 1 = dU dξ 1 and U 2 = dU dξ 2 .
For the given vcCDG equation with two independent variables ξ 1 , ξ 2 and dependent variable U, the solutions of Equation (5) are written where p i,j (t), b r (t) and c r (t) are functions that contain t. Then, by balancing condition, we need to balance U 11111 and UU 111 in Equation (5), then we have n = 2(k − 1), k = 2, 3, . . . . Next, we will use the unified method to find the travelling wave solutions of Equation (5) in the case of n = 2.

The solitary solutions
In this instance, the solutions of Equation (5) are By taking Equations (7) and (8) into Equation (5), we can set the coefficients of φ j (ξ j ) to 0. Through symbolic computation, and making all the coefficients of φ j (ξ j )(j = 1, 2) equal to zero, we get , where We can easily work out the solutions of Equation (8) are Then, by taking Equations (9) and (10) into Equations (7) and (8), we can receive where . Given the parameters of travelling wave solutions in Equation (11), Figure 1. (a) and (b) illustrate the interaction between two-soliton waves with two different velocities. And they converge into one wave at the origin and gradually split into two waves, which travel in the other direction. Moreover, we can see that except x = 0 and near the origin, the values of other solutions are all greater than zero in (c). Similarly, in (d), except t = 0 and near the origin, the values of other solutions are all greater than zero.

The elliptic solutions
In the light of the method, we take into account the solutions of Equation (5) in the form By a similar way, we obtain , , Obviously, if the coefficients of Equation (18) The solutions of the auxiliary equations [35] in Equation (18) can be obtained where 0 < m 1 < 1 and 0 < m 2 < 1.

The modified Kudryashov method
In this part, we will employ the modified Kudryashov method to study and analyse the vcCDG equation. To solve Equation (1), we need to make the following assumptions: where α 1 and α 2 are arbitrary constants. By above transformation, Equation (1) can be transformed into ordinary differential equation where U = dU dξ .
On the basis of the method, the layout of the solutions of Equation (24) can be set where the auxiliary equation with regard to x is Therefore, the first thing we need to do is to calculate the value of N. After that we can determine the specific form of the solution of Equation (24). Similarly, N can be obtained by balancing U (5) and U 2 U in Equation (24), we compute Therefore, we can get the exact form of Equation (25) is According to this method, we need to bring Equations (26) and (28) together into Equation (24). And by combining the same terms, we can get a series of equations of φ(ξ ) and make each coefficient of φ(ξ ) equal to zero. Finally, according to the above steps and symbolic calculation, we can acquire the following four solutions: Case 1: Then, putting Equation (29) into Equation (28), the solution of Equation (24) can be found Finally, we assign values to the parameters in the above solution. When the parameters α 1 = 1 − 2 cos(3t), q 0 (t) = 1 2 + sin(5t), σ 2 (t) = −5 + sin(4t), σ 3 (t) = −1 and σ 4 (t) = 1 + 3 cos(t), we can get Figure 3. They show the interaction between multiple bright soliton waves in (a) and (b). Obviously, the solutions in (c) change greatly. In (d), the value of u changes greatly, and then it turns to be gradual after a sharp change.

Conclusions
Here, the vcCDG equation is studied and analysed by the unified method and the modified Kudryashov method. Compared with other methods, we receive more different travelling wave solutions and classify some solutions like solitary wave solutions, soliton wave solutions and elliptic wave solutions. These solutions contribute to a better understanding of physics phenomena in different branches of engineering science, mathematical physics, quantum mechanics and nonlinear optics, and other technical fields. By choosing appropriate parameters, 3D and 2D plots of some solutions are plotted, from which we are able to obtain periodic waves, rogue waves and kink solitary waves. Through these, we can get more physical meaning of the cvCDG equation. The unified method and the modified Kudryashov method are significant in the study of solutions of nonlinear partial differential equations, have good research prospect and also provide an idea and direction for solving other problems.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by the Natural Science Foundation of Shanxi [grant number 202103021224068].