Complex Soliton Solutions to the Gilson – Pickering Model

Abstract: In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D graphs and contour simulations to the complex soliton solutions are plotted with the help of computational programs. Finally, at the end of the manuscript a conclusion about new complex soliton solutions is given.


Introduction
In the last several decades, many scientists have focused on new findings to the nonlinear differential equation, such as travelling wave solutions, complex, trigonometric, Jacobi elliptic functions, and so on.More recently, some methods have been considered as powerful tools for finding new solutions to the new models.In this sense, Claire Gilson and Andrew Pickering have introduced a model, after named the Gilson-Pickering equation (GPE), in 1995 [1].Many other properties of this model defined as where ε, k, α, β are real constants have been investigated [2].They have investigated this model in Painleve analysis by giving travelling wave solutions.This model includes many other nonlinear models, such as the Camassa-Holm equation, the Fornberg-Whitham equation, and the Rosenau-Hyman equation [3,4].A. Chen et al. have studied Bifurcation of travelling wave solutions for Equation (1) [5].The invariance and multiplier approach to the GPE have been introduced by Ghodrat Ebadi et al. [6].C. Li, S. Tang, W. Huang, and A. Chen have presented the solitary patterns and periodic solutions for the generalized GPE in 2009 under some conditions [7].Considering the special values of the parameters in Equation (1) as ε = 1, α = −1, k = 0.5 and β = 3 the Fornberg-Whitham equation reads [8].Considered as ε = 0, α = 1, k = 0 and β = 3, then, Equation (1) turns into the Rosenau-Hyman equation [7].If these parameters are taken as ε = 1, α = −3, and β = 2, GPE reads as the Fuchssteiner-Fokas-Camassa-Holm equation for the parameters [7].For β = 1, some new types of exponential function solutions, by using the first integral method, have been presented [9]. A. Irshad et al. have employed the Tanh-Coth method to seek the solutions of Equation (1), and some hyperbolic and exponential function solutions have been introduced to the literature [9].Some important nonlinear models including such properties have been presented .
This manuscript is composed of the following sections.In Section 2, we present the Bernoulli sub-equation function method (BSEFM) [42][43][44][45][46]. W.X. Ma et al. have presented the BSEFM in a detailed manner and used it to find explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation in [47].Similarly, many different solutions, such as complex, Lump, and mixed Lump-soliton solutions to the various models such as generalized Hirota-Satsuma-Ito, BKP and (2+1)-dimensional Ito equations have been obtained in [48][49][50].We apply BSEFM to the GPE to find new contour simulations along with new solutions in Section 3. In the last section of the paper, we present a comprehensive conclusion by mentioning new findings.

General Properties of BSEFM
The general steps of the BSEFM are given as follows: Step 1.It can be considered that the following is a nonlinear partial differential equation: and the travelling wave transformation is: in which m, n are real constants and non-zero, respectively.Substituting Equation (3) into Equation ( 2) yields a nonlinear ordinary differential equation (NODE) as follows: where Step 2. Supposing the trial solution of Equation ( 4) is as following: and where F = F(ξ) is the Bernoulli differential polynomial and also b = 0, d = 0, M ∈ R − {0, 1, 2}.Putting Equation ( 5) along with Equation (6) into Equation ( 4) produces an equation of polynomial Ω(F) of F as following: We can obtain a relationship between s and M under the rules of the balance principle.
Step 3. Setting the coefficients of Ω(F) to all be zero gives an algebraic system of equations: Solving this system, we get the values of a 0 , . . ., a s .
Step 4. When we solve Equation (6), we obtain the following two situations according to b and d: Example 1. Taking the travelling wave transformation asu(x, t) = U(ξ), ξ = mx − nt which m, n are real constants and non-zero, respectively, we get the following NODE: in here, ε, n, m, β, α, k are both real constants and non-zero.Between U and U 2 , relationship between s and M can be obtained as following: Case 1: If we define s = 4 and M = 3 in Equation ( 11), we can write the following travelling wave solution: and where a 4 = 0, b = 0, d = 0. Putting Equations ( 12)-( 14) into Equation (10), a system of algebraic equations, including various powers of F, can be obtained.Solving this system by using various computer programs, such as Mathematica, Maple, and Matlap, gives new complex soliton solutions, as follows: Case 1.1.For b = d, the following coefficients can be selected: If these are entered into the Equation ( 12), together with Equation (9), the following novel complex solutions are given: Considering the suitable values of parameters in Equation ( 16), 2D, 3D, and contour surfaces can be plotted as follows (Figures 1-3): ( ) here Considering the suitable values of parameters in Equation ( 16), 2D, 3D, and contour surfaces can be plotted as follows (Figures 1-3):    16).
Case 1.2.When this gives rise to another new complex soliton solution, as following solution: where c, k, ε, a 2 , d, m are both real constants and non-zero.With suitable values of parameters entered into Equation ( 18), the following surfaces can be observed as (Figures 4-6): Figure 3.The contour simulations of Equation ( 16) Case 1.2.When this gives rise to another new complex soliton solution, as following solution: ( )  18), the following surfaces can be observed as (Figures 4-6): , , 48 24 then these produce the following new complex soliton solution: ( ) where Considering the suitable values of parameters in Equation ( 20), the 2D, 3D, and contour surfaces can be observed as following (Figures 7-9): then these produce the following new complex soliton solution: where κ = 24dkε a 2 , = a 2 12kε , and k, ε, a 2 , d, c are both real constants and non-zero.Considering the suitable values of parameters in Equation ( 20), the 2D, 3D, and contour surfaces can be observed as following (Figures 7-9 Taking these coefficients in Equation ( 12), together with Equations ( 9) and ( 21), the following new soliton solution to Equation (1) can be obtained: where k, ε, a 4 , c, m, d are both real constants and non-zero (Figures 10-12).( )     11), we can write the following travelling wave solution: 23) and ( 24) into Equation ( 10), Case 2: Taking s = 6 and M = 4 in Equation ( 11), we can write the following travelling wave solution: Axioms 2019, 8, 18 where a 6 = 0, F = bF + dF 4 , b = 0, d = 0, When we put Equations ( 23) and ( 24) into Equation (10), we can find a system including various powers of F. Solving this system with the help of using different computer programs, such as Mathematica, Maple, and Matlap, gives the following coefficients and soliton solutions: Case 2.1.For b = d, the following coefficients can be chosen as: Considering Equation ( 25) along with Equation ( 9) into Equation ( 23) produces another novel complex soliton solution: where ω = 36dk 2 mε 3/2       Case 2.2.Other coefficients for Equation ( 1) and for b d ≠ can be considered as follows: Case 2.2.Other coefficients for Equation (1) and for b = d can be considered as follows: Regulating Equation ( 23) under the terms of Equations ( 9) and ( 27), another new complex solution for Equation (1) can be found as following: in which k, ε, b, d, c, a 3 are both real constants and non-zero (Figures 16-18).
Regulating Equation ( 23) under the terms of Equations ( 9) and ( 27), another new complex solution for Equation (1) can be found as following:

Conclusions
This manuscript presents a set of complex soliton solutions to the GPE with the help of BSEFM.New complex soliton solutions, such as exponential, rational, and the complex exponential are obtained.To give better understanding of the physical importance of the solutions found in this paper, the process of choosing the suitable values for the parameters is shown, and the three-and two-dimensional graphs and contour simulations of the these solutions are drawn.The alternative points of view to the solutions (i.e., Equations 16, 18, 20, 22, 26, and 28) can be seen in the 3D and 2D graphs in Figures 1,4,7,10,13,16 and Figures 2,5,8,11,14,17, respectively.The contour patterns on which the wave propagates along the x-axis for complex solutions obtained in this paper can be also viewed from Figures 3,6,9,12,15,18, separately as imaginary and real parts of the solutions.The contour surfaces are another and new alternative simulation to the 3D graph, giving more detailed information in terms of physical properties of the constructed solutions.Comparing such results, produced in this manuscript, with the papers which have been published in [1,4], it can be seen that these complex soliton solutions are entirely new for the GPE.The calculations show that this method is a reliable and efficient scheme which yields many complex results to the other nonlinear partial differential equations.

Conclusions
This manuscript presents a set of complex soliton solutions to the GPE with the help of BSEFM.New complex soliton solutions, such as exponential, rational, and the complex exponential are obtained.To give better understanding of the physical importance of the solutions found in this paper, the process of choosing the suitable values for the parameters is shown, and the three-and two-dimensional graphs and contour simulations of the these solutions are drawn.The alternative points of view to the solutions (i.e., Equations ( 16), ( 18), ( 20), ( 22), (26), and ( 28)) can be seen in the 3D and 2D graphs in Figures 1, 2, 4, 7, 10, 13 and 16, Figure 5, Figure 8, Figure 11, Figure 14, Figure 17, respectively.The contour patterns on which the wave propagates along the x-axis for complex solutions obtained in this paper can be also viewed from Figure 3, Figure 6, Figure 9, Figure 12, Figure 15, Figure 18, separately as imaginary and real parts of the solutions.The contour surfaces are another and new alternative simulation to the 3D graph, giving more detailed information in terms of physical properties of the constructed solutions.Comparing such results, produced in this manuscript, with the papers which have been published in [1,4], it can be seen that these complex soliton solutions are entirely new for the GPE.The calculations show that this method is a reliable and efficient scheme which yields many complex results to the other nonlinear partial differential equations.

18 ) where 2 ,
, , , , c k a d m ε are both real constants and non-zero.With suitable values of parameters entered into Equation (
constants and non-zero.