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.

1. 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

$$u_t-\epsilon u_{xxt}+2k{u}_x-u{u}_{xxx}-\alpha u{u}_x-\beta u_x{u}_{xx}=0,$$

(1)

where $\epsilon$, $k$, $\alpha$,$\beta$ 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 $\epsilon =1,\alpha =-1,k=0.5$ and $\beta =3$ the Fornberg–Whitham equation reads [8]. Considered as $\epsilon =0,\alpha =1,k=0$ and $\beta =3$, then, Equation (1) turns into the Rosenau-Hyman equation [7]. If these parameters are taken as $\epsilon =1,\alpha =-3,$ and $\beta =2$, GPE reads as the Fuchssteiner–Fokas–Camassa–Holm equation for the parameters [7]. For $\beta =1$, 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 [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41].

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.

2. 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:

$$P\left(u_x,u_{xx},u_{tt},u{u}_x,\cdots \right)=0,$$

(2)

and the travelling wave transformation is:

$$u\left(x,t\right)=U\left(\xi \right),\xi =mx-nt,$$

(3)

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:

$$N\left(U,U^{\prime },U^{″},\cdots \right)=0,$$

(4)

where $U=U\left(\xi \right),U^{\prime }=\frac{dU}{d\xi },U^{″}=\frac{d^{2}U}{d{\xi }^2},.$

Step 2. Supposing the trial solution of Equation (4) is as following:

$$U={\displaystyle \sum _{i=0}^s{a}_i{F}^i}=a_0+a_{1}F+a_2{F}^2+\cdots +a_s{F}^s,$$

(5)

and

$$F^{\prime }=bF+d{F}^M,$$

(6)

where $F=F\left(\xi \right)$ is the Bernoulli differential polynomial and also $b\ne 0,d\ne 0,M\in ℝ-\left\{0,1,2\right\}.$ Putting Equation (5) along with Equation (6) into Equation (4) produces an equation of polynomial $\Omega \left(F\right)$ of $F$ as following:

$$\Omega \left(F\right)={\rho }_r{F}^r+\cdots +{\rho }_{1}F+{\rho }_0=0.$$

(7)

We can obtain a relationship between $s$ and $M$ under the rules of the balance principle.

Step 3. Setting the coefficients of $\Omega \left(F\right)$ to all be zero gives an algebraic system of equations:

$${\rho }_i=0,i=0,\cdots ,r.$$

(8)

Solving this system, we get the values of $a_0,\dots ,a_s$.

Step 4. When we solve Equation (6), we obtain the following two situations according to $b$ and $d$:

$$\begin{array}{l}F\left(\xi \right)={\left[\frac{-d}{b}+\frac{E}{e^{b\left(M-1\right)\xi }}\right]}^{\frac{1}{1-M}},b\ne d,\\ F\left(\xi \right)={\left[\frac{\left(E-1\right)+\left(E+1\right)\tanh \left(\frac{b\left(1-M\right)\xi }{2}\right)}{1-\tanh \left(\frac{b\left(1-M\right)\xi }{2}\right)}\right]}^{\frac{1}{1-M}},b=d,E\in ℝ.\end{array}$$

(9)

Substituting Equation (5) into Equation (4), we can find the polynomial of $F$. Considering all the coefficients of same power of $F$ to zero gives an algebraic system of equations. By solving this system via several computational programs, we can find some new values of parameters of $a_0,\dots ,a_s$. This process gives many solutions to the model considered. For a better understanding of the solutions obtained in this manner, we can draw 2D, 3D, and contour graphical simulations of solutions for suitable values of parameters.

3. Application of the BSEFM

In this section, BSEFM has been successfully applied to the GPE to find more and new complex soliton solutions.

Example 1.

Taking the travelling wave transformation as$u\left(x,t\right)=U\left(\xi \right),\xi =mx-nt$which$m,n$are real constants and non-zero, respectively, we get the following NODE:

$$2\epsilon n{m}^2{U}^{″}-2m^{3}U{U}^{″}+m^3\left(1-\beta \right){\left(U^{\prime }\right)}^2-\alpha m{U}^2+\left(4km-2n\right)U=0,$$

(10)

in here,$\epsilon ,n,m,\beta ,\alpha ,k$are both real constants and non-zero. Between$U^{″}$and$U^2$, relationship between$s$and$M$can be obtained as following:

$$2M=s+2.$$

(11)

Case 1: If we define $s=4$ and $M=3$ in Equation (11), we can write the following travelling wave solution:

$$U=a_0+a_{1}F+a_2{F}^2+a_3{F}^3+a_4{F}^4,$$

(12)

$$\begin{array}{l}{U}^{\prime }=a_{1}bF+a_{1}d{F}^3+2a_{2}b{F}^2+2a_{2}d{F}^4+3a_{2}b{F}^3++3a_{3}d{F}^5\\ +4a_{4}b{F}^4+4a_{4}d{F}^6,\end{array}$$

(13)

and

$$\begin{array}{l}{U}^{″}=b^{2}F{a}_1+4bd{F}^3{a}_1+3d^2{F}^5{a}_1+4b^2{F}^2{a}_2+12bd{F}^4{a}_2++8d^2{F}^6{a}_2+9b^2{F}^3{a}_3\\ +24bd{F}^5{a}_3+15d^2{F}^7{a}_3+16b^2{F}^4{a}_4++40bd{F}^6{a}_4+24d^2{F}^8{a}_4,\\ ⋮\end{array}$$

(14)

where $a_4\ne 0,b\ne 0,d\ne 0$. 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 $b\ne d,$ the following coefficients can be selected:

$$\begin{array}{l}\beta =-2,a_0=2k\epsilon ,a_1=0,a_3=0,a_4=\frac{-a_2^2}{12k\epsilon },b=-\frac{12dk\epsilon }{a_2},n=\frac{5i{a}_2}{48d{\epsilon }^{3/2}},\\ m=\frac{i{a}_2}{24dk{\epsilon }^{3/2}},\alpha =-\frac{1}{2\epsilon }.\end{array}$$

(15)

If these are entered into the Equation (12), together with Equation (9), the following novel complex solutions are given:

$$u_1\left(x,t\right)=\frac{288c^2{k}^3{\epsilon }^3{e}^{-\frac{i}{\sqrt{\epsilon }}\left(5kt-2x\right)}+192a_{2}c{k}^2{\epsilon }^2{e}^{-\frac{i}{2\sqrt{\epsilon }}\left(5kt-2x\right)}+2k\epsilon a_2^2}{{\left(12ck\epsilon e^{-\frac{i}{2\sqrt{\epsilon }}\left(5kt-2x\right)}+a_2\right)}^2},$$

(16)

here $c,k,\epsilon ,a_2\in ℝ-\left\{0\right\}$. Considering the suitable values of parameters in Equation (16), 2D, 3D, and contour surfaces can be plotted as follows (Figure 1, Figure 2 and Figure 3):

Case 1.2. When

$$\begin{array}{l}\beta =-2,a_1=0,a_3=0,a_0=2k\epsilon ,a_4=-2idm{a}_2\sqrt{\epsilon },b=\frac{i}{2m\sqrt{\epsilon }},\\ n=3km\left(1-\frac{8idkm{\epsilon }^{3/2}}{a_2+24idkm{\epsilon }^{3/2}}\right),\alpha =-\frac{a_2}{\epsilon a_2+24idkm{\epsilon }^{5/2}},\end{array}$$

(17)

this gives rise to another new complex soliton solution, as following solution:

$$\begin{array}{ll}{u}_2\left(x,t\right)=& 2k\epsilon +\frac{a_2}{c{e}^{-\frac{i}{m\sqrt{\epsilon }}\left(mx-3kmt\left(1-\frac{8idkm{\epsilon }^{3/2}}{a_2+24idkm{\epsilon }^{3/2}}\right)\right)}+2idm\sqrt{\epsilon }}\\ & -2idm\sqrt{\epsilon }{a}_2{\left(c{e}^{-\frac{i}{m\sqrt{\epsilon }}\left(mx-3kmt\left(1-\frac{8idkm{\epsilon }^{3/2}}{a_2+24idkm{\epsilon }^{3/2}}\right)\right)}+2idm\sqrt{\epsilon }\right)}^{-2},\end{array}$$

(18)

where $c,k,\epsilon ,a_2,d,m$ are both real constants and non-zero. With suitable values of parameters entered into Equation (18), the following surfaces can be observed as (Figure 4, Figure 5 and Figure 6):

Case 1.3. For $b\ne d$, if

$$\begin{array}{l}\beta =-2,a_1=0,a_3=0,a_0=2k\epsilon ,a_4=-\frac{a_2^2}{12k\epsilon },b=-\frac{12dk\epsilon }{a_2},\alpha =-\frac{1}{2\epsilon },\\ n=\frac{-5i{a}_2}{48d{\epsilon }^{3/2}},m=\frac{-i{a}_2}{24dk{\epsilon }^{3/2}},\end{array}$$

(19)

then these produce the following new complex soliton solution:

$$u_3\left(x,t\right)=2k\epsilon -\frac{a_2^2}{12k\epsilon }{\left(c{e}^{\frac{-i\kappa a_2}{24dk{\epsilon }^{3/2}}x+\frac{5i\kappa a_2}{48d{\epsilon }^{3/2}}t}+\varpi \right)}^{-2}+a_2{\left(c{e}^{\frac{5i\kappa a_2}{48d{\epsilon }^{3/2}}t-\frac{i\kappa a_2}{24dk{\epsilon }^{3/2}}x}+\varpi \right)}^{-1},$$

(20)

where $\kappa =\frac{24dk\epsilon }{a_2},\varpi =\frac{a_2}{12k\epsilon },$ and $k,\epsilon ,a_2,d,c$ 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 (Figure 7, Figure 8 and Figure 9):

Case 1.4. Other coefficients for $b\ne d$ can be considered as follows:

$$\begin{array}{l}\beta =-2,a_1=0,a_3=0,a_0=2k\epsilon ,a_2=\frac{i{a}_4}{2dm\sqrt{\epsilon }},b=\frac{i}{2m\sqrt{\epsilon }},n=\frac{3km\left(32d^{2}k{m}^2{\epsilon }^2+a_4\right)}{48d^{2}k{m}^2{\epsilon }^2+a_4},\\ \alpha =-\frac{a_4}{48d^{2}k{m}^2{\epsilon }^2+\epsilon a_4}.\end{array}$$

(21)

Taking these coefficients in Equation (12), together with Equations (9) and (21), the following new soliton solution to Equation (1) can be obtained:

$$\begin{array}{ll}{u}_4\left(x,t\right)=& 2k\epsilon +a_4{\left(c{e}^{\frac{-i}{m\sqrt{\epsilon }}\left(mx-\frac{3km}{48d^{2}k{m}^2{\epsilon }^2+a_4}\left(32d^{2}k{m}^2{\epsilon }^2+a_4\right)t\right)}+2idm\sqrt{\epsilon }\right)}^{-2}\\ & +\frac{i{a}_4}{2dm\sqrt{\epsilon }}{\left(c{e}^{\frac{-i}{m\sqrt{\epsilon }}\left(mx-\frac{3km}{48d^{2}k{m}^2{\epsilon }^2+a_4}\left(32d^{2}k{m}^2{\epsilon }^2+a_4\right)t\right)}+2idm\sqrt{\epsilon }\right)}^{-1},\end{array}$$

(22)

where $k,\epsilon ,a_4,c,m,d$ are both real constants and non-zero (Figure 10, Figure 11 and Figure 12).

Case 2: Taking $s=6$ and $M=4$ in Equation (11), we can write the following travelling wave solution:

$$U=a_0+a_{1}F+a_2{F}^2+a_3{F}^3+a_4{F}^4+a_5{F}^5+a_6{F}^6,$$

(23)

$$\begin{array}{l}{U}^{\prime }=a_1{F}^{\prime }+2a_2{F}^{\prime }F+3a_3{F}^{\prime }{F}^2+4a_4{F}^{\prime }{F}^3+5a_5{F}^{\prime }{F}^4+6a_6{F}^{\prime }{F}^5,\\ ⋮\end{array}$$

(24)

where $a_6\ne 0,F^{\prime }=bF+d{F}^4,b\ne 0,d\ne 0,$ 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 $b\ne d,$ the following coefficients can be chosen as:

$$\begin{array}{l}\beta =-2,a_0=2k\epsilon ,a_1=a_2=a_4=a_5=0,a_6=-3idm{a}_3\sqrt{\epsilon },b=\frac{i}{3m\sqrt{\epsilon }},\\ n=3km\left(1-\frac{12idkm{\epsilon }^{3/2}}{36idkm{\epsilon }^{3/2}+a_3}\right),\alpha =-\frac{a_3}{\epsilon a_3+36idkm{\epsilon }^{5/2}}.\end{array}$$

(25)

Considering Equation (25) along with Equation (9) into Equation (23) produces another novel complex soliton solution:

$$u_5\left(x,t\right)=2k\epsilon +\frac{a_3}{c{e}^{\frac{-i}{\sqrt{\epsilon }}\left(x-3kt+i\omega t\right)}+3idm\sqrt{\epsilon }}-\frac{3idm\sqrt{\epsilon }{a}_3}{{\left(c{e}^{\frac{-i}{\sqrt{\epsilon }}\left(x-3kt+i\omega t\right)}+3idm\sqrt{\epsilon }\right)}^2},$$

(26)

where $\omega =\frac{36d{k}^{2}m{\epsilon }^{3/2}}{36idkm{\epsilon }^{3/2}+a_3}$ and $k,\epsilon ,a_3,c,d,m$ are both real constants and non-zero. With suitable values of parameters in Equation (26), 2D, 3D, and contour surfaces can be seen as (Figure 13, Figure 14 and Figure 15):

Case 2.2. Other coefficients for Equation (1) and for $b\ne d$ can be considered as follows:

$$\begin{array}{l}\beta =-2,a_0=2k\epsilon ,a_1=a_2=a_4=a_5=0,a_6=\frac{d{a}_3}{b},n=\frac{ik\left(-8dk\epsilon +b{a}_3\right)}{b\sqrt{\epsilon }\left(-12dk\epsilon +b{a}_3\right)},\\ m=\frac{i}{3b\sqrt{\epsilon }},\alpha =\frac{b{a}_3}{12dk{\epsilon }^2-b\epsilon a_3}.\end{array}$$

(27)

Regulating Equation (23) under the terms of Equations (9) and (27), another new complex solution for Equation (1) can be found as following:

$$u_6\left(x,t\right)=2k\epsilon +\frac{d{a}_3}{b{\left(-\frac{d}{b}+c{e}^{\frac{-ix}{\sqrt{\epsilon }}+\frac{3ikt\left(-8dk\epsilon +b{a}_3\right)}{\sqrt{\epsilon }\left(-12dk\epsilon +b{a}_3\right)}}\right)}^2}+\frac{a_3}{-\frac{d}{b}+c{e}^{\frac{-ix}{\sqrt{\epsilon }}+\frac{3ikt\left(-8dk\epsilon +b{a}_3\right)}{\sqrt{\epsilon }\left(-12dk\epsilon +b{a}_3\right)}}},$$

(28)

in which $k,\epsilon ,b,d,c,a_3$ are both real constants and non-zero (Figure 16, Figure 17 and Figure 18).

4. 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 Figure 1, Figure 4, Figure 7, Figure 10, Figure 13, Figure 16 and Figure 2, 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.