On the New Wave Behaviors of the Gilson-Pickering Equation

In this article, we study the fully non-linear third-order partial differential equation, namely the Gilson-Pickering equation. The (1/G′)-expansion method, and the generalized exponential rational function method are used to construct various exact solitary wave solutions for a given equation. These methods are based on a homogeneous balance technique that provides an order for the estimation of a polynomial-type solution. In order to convert the governing equation into a nonlinear ordinary differential equation, a traveling wave transformation has been implemented. As a result, we have constructed a variety of solitary wave solutions, such as singular solutions, compound singular solutions, complex solutions, and topological and non-topological solutions. Besides, the 2D, 3D, and contour surfaces are plotted for all obtained solutions by choosing appropriate parameter values.

The third-order nonlinear partial differential equation (NLPDE) was introduced in [25] by Gilson and Pickering as where ε, α, κ, and β are non-zero real numbers. Recently, the Gilson-Pickering equation has been investigated using a variety of methods, such as the (G ′ /G)-expansion method [26], the anstaz method [27], the (G ′ /G)-expansion method to tanh, the coth, cot, and the logical forms under certain conditions [28], the Bernoulli sub-equation model [29], a not a knot meshless method [30], and the symmetry method [31].
The core of this paper is to investigate the Gilson-Pickering equation using the (1/G ′ )-expansion method and the generalized exponential rational function method (GERF).

THE BASIC CONCEPTS OF THE (1/G ′ )-EXPANSION METHOD
In this section, the fundamental steps of the (1/G ′ )-expansion method are presented [38,39]: Step 1. Let us consider the general form of a two-variable nonlinear partial differential equation (NPDE) as follows: where p = p(x, t), and Q is a partial differential equation.
Step 2. To convert Equation (2) to a nonlinear ordinary differential equation (NODE), we employ the following wave transformation where h is a scalar. After some procedures, Equation (2) reduces to the following NODE: where W is an ordinary differential equation.
Step 3. Assume that Equation (4) has a solution of the form where a 0 , a 1 , a 2 , ..., a m are scalars to be determined, m is a balance term, and G = G(η) satisfies the following secondorder linear ODE: where λ and µ are scalars.
The solution of Equation (6) is given by If we convert the algebraic expression given by Equation (7) to a trigonometric function, we can write it as the following: Inserting Equation (6) and its necessary derivatives along with Equation (5) into Equation (4) returns the polynomial of 1 Summing the 1 G ′ i coefficients with the same power and then setting every summation to zero, we get a system of algebraic equations for a i , i ≥ 0. Eventually, solving this system simply gives the value of the variables. Putting these values of variables with the value of the balance term m into Equation (4), we can get solutions for Equation (2).

THE BASIC CONCEPTS OF THE GERF
In this section, the basic steps of the GERF are presented.
Step1. Let us consider that the general form of a nonlinear partial differential equation is given by: where Q is a partial differential equation. Suppose that the wave transformation takes the form: where h is a scalar. Using Equation (10) in Equation (9), we get the nonlinear ordinary differential equation where W is an ordinary differential equation.
Step 2. Suppose that the solitary wave solutions of Equation (11) are given by: where where r m , s m (1 ≤ n ≤ 4) are real/complex constants, A 0 , A K , B K are constants to be determined, and m will be determined by the balance principle.
Step 3. Substituting Equation (12) into Equation (11), we get the polynomials that are dependent on Equation (12). By equating the same order terms, we obtain an algebraic system of equations. With the help of computational programs such as Mathematica, Matlab, and Maple, we solve this system and determine the values of A 0 , A K , B K . Finally one can easily obtain the nontrivial exact solutions of Equation (11).

MATHEMATICAL CALCULATION
In this section, the mathematical calculation of the Gilson-Pickering equation is presented. Consider the Gilson-Pickering equation (Equation 1) stated in section 1. Inserting the wave transformation into Equation (1), the following NODE can be obtained where ǫ, β, α, h, and k are non-zero real numbers.
Integrating Equation (15) once with respect to η and assuming that the integration constant is zero, we have.
In this section, the application of the (1/G ′ )-expansion method to the Gilson-Pickering equation is presented.
Applying the balance principle, by taking the nonlinear term P 2 and the highest derivative P ′′ in Equation (16) gives m = 2. With m = 2, Equation (5) takes the form Inserting Equation (17) and its necessary derivatives into Equation (16), returns the polynomial of 1 i coefficients with the likely power and then setting every summation to zero, we get a system of algebraic equations. Solving this system simply gives the following families of solutions: we get where M = −h + 2k α, L = √ 2h − 4k + hαǫ.

Family 2. When
we get

Family 3. When
gives

Family 4. When
we get

Family 5. When
we get Family 6. When we have

IMPLEMENTATION OF THE GERF METHOD
In this section, the application of the GERF method to the Gilson-Pickering equation is presented.
Applying the balance principle, by taking the nonlinear term P 2 and the highest derivative P ′′ in Equation (16) gives m = 2. With m = 2, Equation (12) takes the form where ϕ (η) is given by Equation (13). Following the methodology described above in section 4, we obtain the following nontrivial solutions of Equation (1):

Case 1.
we get

Case 2. When
we get

Case 1. When
we get where .

Case 2.
we get

Case 1. When
we have  − sinh (D)

Case 2.
we get

RESULT AND DISCUSSION
The  Frontiers in Physics | www.frontiersin.org [28] used (G ′ /G) and the ansatz method and found the solitary wave solutions to Equation (1). Baskonus [29] investigated the Gilson-Pickering equation by using the first integral method. Zabihi and Saffarian [30] implemented the simplified (G ′ /G) expansion method to reveal the hyperbolic, trigonometric function, and rational function solutions. Singla and Gupta [31] reported some new complex soliton solutions to Equation (1) with the aid of the Bernoulli sub-equation function method. Camsssa et al. [32] used a not a knot meshless method to obtain numerical solutions to Equation (1). Fuchssteiner and Fokas [33] performed Lie symmetry analysis and found conservation laws for the space-time fractional Gilson-Pickering equation.
In this article, we obtained the singular, compound singular, complex, topological, and non-topological wave solutions to the studied equation. It is known that non-topological solutions detect waves with an intensity lower than the background, topological solutions with such a maximum intensity higher than the background, and singular solutions that are waves with discontinuous derivatives.

CONCLUSION
In this study, we have successfully applied the (1/G ′ ) expansion method and the generalized exponential rational function method to find new exact solutions for the Gilson-Pickering equation. In order to convert the governing equation into a NODE, a traveling wave transformation has been implemented. Various analytical solutions of the proposed model have been constructed such as singular solutions, as shown in Figures 1, 2, 3, compound singular solution, as seen in Figure 4, complex solution, as seen in Figure 5, as well as a singular solution, can be shown in Figure 6. The non-topological solution, as shown in Figure 7, topological solutions, as shown in Figure 8, and compound singular solutions, as seen in Figures 9, 10. Also, topological solution and non-topological solution as seen in Figures 11, 12, respectively. Compared with the results reported in Fan et al. [28], Baskonus [29], Zabihi and Saffarian [30], Singla and Gupta [31], Camsssa et al. [32], and Fuchssteiner and Fokas [33], the solutions obtained are novel. Both methods are efficient for solving complex nonlinear partial differential equations, but, by using the generalized exponential rational function method, we can get more solutions than with the (1/G ′ ) expansion method. Furthermore, the 2D, 3D, and contour surfaces are plotted for all obtained solutions by selecting suitable values for the parameters.

DATA AVAILABILITY STATEMENT
All datasets generated for this study are included in the article/supplementary material.

AUTHOR CONTRIBUTIONS
RY and SN suggested the problem first. KA drafted the first version of the problem statement with the help of HD. All authors made several suggestions to make improvements in the problem statement and contributed to the development of solution in their best possible ways.