Diverse Variety of Exact Solutions for Nonlinear Gilson–Pickering Equation

Abstract

The purpose of this article is to achieve new soliton solutions of the Gilson–Pickering equation (GPE) with the assistance of Sardar’s subequation method (SSM) and Jacobi elliptic function method (JEFM). The applications of the GPE is wider because we study some valuable and vital equations such as Fornberg–Whitham equation (FWE), Rosenau–Hyman equation (RHE) and Fuchssteiner–Fokas–Camassa–Holm equation (FFCHE) obtained by particular choices of parameters involved in the GPE. Many techniques are available to convert PDEs into ODEs for extracting wave solutions. Most of these techniques are a case of symmetry reduction, known as nonclassical symmetry. In our work, this approach is used to convert a PDE to an ODE and obtain the exact solutions of the NLPDE. The solutions obtained are unique, remarkable, and significant for readers. Mathematica 11 software is used to derive the solutions of the presented model. Moreover, the diagrams of the acquired solutions for distinct values of parameters were demonstrated in two and three dimensions along with contour plots.

1. Introduction

In recent decades, mathematicians and physicists have shown their interest in the field of nonlinear evolution equations (NLEEs) due to their wide applications in every field of science [1,2,3,4,5]. NLEEs have symmetries in different ways. Vinogradov presented four different ideas to study symmetries in nonlinear PDEs [6]. Researchers are referred to these papers to understand symmetry, its different definitions and applications on PDEs, etc. [7,8,9,10,11,12]. NLEEs have uncountable applications in daily life, which is the main reason for mathematicians, engineers, and physicists to study NLEEs. The significance of NLEEs has been increasing in diverse science fields such as optics, plasma physics, mechanical and electromagnetic waves, wave theory, mechanics, fluid dynamics, nonlinear optics, fibers, optical fibers, etc. Many important NLEEs describing the above stated fields are not easy to solve. Therefore, we need different and new techniques to solve such kinds of NLEEs. For this aspect, researchers have developed different, unique, and powerful techniques to solve NLEEs, which include the modified simple equation technique [13,14,15], the variational iteration method [16,17], the variational method [18], the first integral method [19], the perturbation method [20], method of integrability [21], the nonperturbative technique [22], the modified F-expansion method [23,24,25], the exp-function method [26,27], the sine–cosine method [28,29,30], the Riccatti–Bernoulli sub-ODE method [31,32], the Jacobi elliptic function method [33,34], the generalized Kudryashov method [35,36], the functional variable method [37,38], the modified Khater method [39,40], the new extended direct algebraic method [41,42], the Lie symmetry technique [43,44], the $(G^{{}^{\prime }}/G)$-expansion method [45], the tanh–coth method [46,47], the new auxiliary equation method [48,49], the $(G^{{}^{\prime }}/G,1/G)$-expansion method [50], the technique of $(m+1,G^{{}^{\prime }})$ [51], the addendum to Kudryashov’s method [52], and many others [53,54,55,56,57,58].

Here, the third-order nonlinear GPE [59,60] is solved, and we find possible exact solutions with the help of Sardar’s subequation method (SSM) [61,62,63,64], the Jacobi elliptic function method (JEFM) [65,66], and the generalized derivative technique [67]. The GPE can be written as

$$\begin{array}{c}\hfill u_t+2k{u}_x-\zeta u{u}_x-\eta u_x{u}_{xx}-\nu u_{xxt}-\theta u{u}_{xxx}=0.\end{array}$$

(1)

The parameters $\zeta$, k, $\eta$, $\nu$ and $\theta$ are arbitrary constants. When we choose particular values for the parameters, such as $\zeta =-1$, $k=0.5$, $\eta =3$ and $\nu =1$, then Equation (1) becomes an FWE [68,69]

$$\begin{array}{c}\hfill u_t+u_x+u{u}_x-3u_x{u}_{xx}-u_{txx}-\theta u{u}_{xxx}=0.\end{array}$$

(2)

For substituting the values $\zeta =1$, $k=0$, $\eta =3$ and $\nu =0$ in Equation (1), then the RHE [70] is obtained.

For choosing the values $\zeta =-1$, $\eta =2$ and $\nu =1$ in Equation (1), then the GPE converts into an FFCHE [71,72]

$$\begin{array}{c}\hfill u_t+2k{u}_x+u{u}_x-2u_x{u}_{xx}-u_{txx}-\theta u{u}_{xxx}=0.\end{array}$$

(3)

This paper is presented in the following manner: A brief discussion about the SSM is presented in Section 2. Section 3 and Section 4 illustrate the solutions of the GPE using the SSM and JEFM, respectively. The fifth section is devoted to the result and discussion. The last section consists of a conclusion.

2. Sardar’s Subequation Method (SSM)

In this section, we discuss briefly the SSM in the following steps:

Step 1: Suppose the NLEE

$$\begin{array}{c}\hfill G(u_{,}{u}_t,u_x,u_{tt},u_{xx},\dots )=0.\end{array}$$

(4)

Utilizing the following wave transformation

$$\begin{array}{c}\hfill u(x,t)=U\left(\chi \right),\chi =x-Vt,\end{array}$$

(5)

Equation (4) changes in the following nonlinear ODE

$$\begin{array}{c}\hfill H(U,U^{{}^{\prime }},U^{{}^{″}},U^{{}^{‴}},\dots )=0.\end{array}$$

(6)

Step 2: Consider a solution of Equation (6) as

$$\begin{array}{c}\hfill U\left(\chi \right)=\sum _{i=0}^s{b}_i{F}^i\left(\chi \right),\end{array}$$

(7)

$b_i$, ($i=$ 0, 1, …,s) are constants, and $F\left(\chi \right)$ is the solution of the following equation:

$$\begin{array}{c}\hfill {\left(F^{{}^{\prime }}\left(\chi \right)\right)}^2=\sigma +a{F}^2\left(\chi \right)+F^4\left(\chi \right),\end{array}$$

(8)

where a and $\sigma$ are unknown parameters. Equation (8) provides solutions of the form:

Case I: If $a>0$ and $\sigma =0,$ then

$$\begin{array}{ccc}& & F_1^{\pm }\left(\chi \right)=\pm \sqrt{-pqa}{\mathrm{Sech}}_{pq}\left(\sqrt{a}\chi \right),\hfill \\ & & F_2^{\pm }\left(\chi \right)=\pm \sqrt{pqa}{\mathrm{Csch}}_{pq}\left(\sqrt{a}\chi \right),\hfill \\ & \end{array}$$

where ${\mathrm{Sech}}_{pq}\left(\chi \right)=\frac{2}{p{e}^{\chi }+q{e}^{-\chi }}$, ${\mathrm{Csch}}_{pq}\left(\chi \right)=\frac{2}{p{e}^{\chi }-q{e}^{-\chi }}$.

Case II: If $a<0$ and $\sigma =0,$ then

$$\begin{array}{ccc}& & F_3^{\pm }\left(\chi \right)=\pm \sqrt{-pqa}{\mathrm{Sec}}_{pq}\left(\sqrt{-a}\chi \right),\hfill \\ & & F_4^{\pm }\left(\chi \right)=\pm \sqrt{-pqa}{\mathrm{Csc}}_{pq}\left(\sqrt{-a}\chi \right),\hfill \\ & \end{array}$$

where ${\mathrm{Sec}}_{pq}\left(\chi \right)=\frac{2}{p{e}^{\iota \chi }+q{e}^{-\iota \chi }}$, ${\mathrm{Csc}}_{pq}\left(\chi \right)=\frac{2\iota }{p{e}^{\iota \chi }-q{e}^{-\iota \chi }}$.

Case III: If $a<0$ and $\sigma =\frac{a^2}{4},$ then

$$\begin{array}{ccc}& & F_5^{\pm }\left(\chi \right)=\pm \sqrt{\frac{-a}{2}}{\mathrm{Tanh}}_{pq}\left(\sqrt{\frac{-a}{2}}\chi \right),\hfill \\ & & F_6^{\pm }\left(\chi \right)=\pm \sqrt{\frac{-a}{2}}{\mathrm{Coth}}_{pq}\left(\sqrt{\frac{-a}{2}}\chi \right),\hfill \\ & & F_7^{\pm }\left(\chi \right)=\pm \sqrt{\frac{-a}{2}}({\mathrm{Tanh}}_{pq}\left(\sqrt{-2a}\chi \right)\pm \iota \sqrt{pq}{\mathrm{Sech}}_{pq}\left(\sqrt{-2a}\chi \right)),\hfill \\ & & F_8^{\pm }\left(\chi \right)=\pm \sqrt{\frac{-a}{2}}({\mathrm{Coth}}_{pq}\left(\sqrt{-2a}\chi \right)\pm \sqrt{pq}{\mathrm{Csch}}_{pq}\left(\sqrt{-2a}\chi \right)),\hfill \\ & & F_9^{\pm }\left(\chi \right)=\pm \sqrt{\frac{-a}{8}}\left({\mathrm{Tanh}}_{pq}(\sqrt{\frac{-a}{8}}\chi \right)+{\mathrm{Coth}}_{pq}\left(\sqrt{\frac{-a}{8}}\chi \right)),\hfill \\ & \end{array}$$

where ${\mathrm{Tanh}}_{pq}\left(\chi \right)=\frac{p{e}^{\chi }-q{e}^{-\chi }}{p{e}^{\chi }+q{e}^{-\chi }}$, ${\mathrm{Coth}}_{pq}\left(\chi \right)=\frac{p{e}^{\chi }+q{e}^{-\chi }}{p{e}^{\chi }-q{e}^{-\chi }}$.

Case IV: If a > 0 and $\sigma =\frac{a^2}{4},$ then

$$\begin{array}{ccc}& & F_{10}^{\pm }\left(\chi \right)=\pm \sqrt{\frac{a}{2}}{\mathrm{Tan}}_{pq}\left(\sqrt{\frac{a}{2}}\chi \right),\hfill \\ & & F_{11}^{\pm }\left(\chi \right)=\pm \sqrt{\frac{a}{2}}{\mathrm{Cot}}_{pq}\left(\sqrt{\frac{a}{2}}\chi \right),\hfill \\ & & F_{12}^{\pm }\left(\chi \right)=\pm \sqrt{\frac{a}{2}}({\mathrm{Tan}}_{pq}\left(\sqrt{2a}\chi \right)\pm \sqrt{pq}{\mathrm{Sec}}_{pq}\left(\sqrt{2a}\chi \right)),\hfill \\ & & F_{13}^{\pm }\left(\chi \right)=\pm \sqrt{\frac{a}{2}}({\mathrm{Cot}}_{pq}\left(\sqrt{2a}\chi \right)\pm \sqrt{pq}{\mathrm{Csc}}_{pq}\left(\sqrt{2a}\chi \right)),\hfill \\ & & F_{14}^{\pm }\left(\chi \right)=\pm \sqrt{\frac{a}{8}}({\mathrm{Tan}}_{pq}\left(\sqrt{\frac{a}{8}}\chi \right)+{\mathrm{Cot}}_{pq}\left(\sqrt{\frac{a}{8}}\chi \right)),\hfill \\ & \end{array}$$

where ${\mathrm{Tan}}_{pq}\left(\chi \right)=-\iota {\displaystyle \frac{p{e}^{\iota \chi }-q{e}^{-\iota \chi }}{p{e}^{\iota \chi }+q{e}^{-\iota \chi }}}$, ${\mathrm{Cot}}_{pq}\left(\chi \right)=\iota {\displaystyle \frac{p{e}^{\iota \chi }+q{e}^{-\iota \chi }}{p{e}^{\iota \chi }-q{e}^{-\iota \chi }}}.$

The process starts by extracting s with the support of the balancing rule. When s is collected, the predicted solution in Equation (7) is substituted into Equation (6). To obtain a nonzero solution ($F\left(\chi \right)\ne 0$), every coefficient of the same power of $F\left(\chi \right)$ are equated to zero. Then, the resulting system of the algebraic equations is solved to obtain the values of unknown parameters, especially $b_s$ and V.

3. Implementation of the Method

Substituting Equation (5) into Equation (1) and after simplification, we obtain

$$\begin{array}{c}\hfill (2k-\nu )U^{{}^{\prime }}+\nu V{U}^{{}^{‴}}-\theta U{U}^{{}^{‴}}-\zeta U{U}^{{}^{\prime }}-\eta U^{{}^{\prime }}{U}^{{}^{″}}=0.\end{array}$$

(9)

Integrating with respect to $\chi$ once implies

$$\begin{array}{c}\hfill (2k-\nu )U+\nu V{U}^{{}^{″}}-\frac{\zeta }{2}{U}^2+\frac{\theta -\eta }{2}{\left(U^{{}^{\prime }}\right)}^2-\theta U{U}^{{}^{″}}=C,\end{array}$$

(10)

where C is the integration constant.

For finding the value s, we use balancing principle on Equation (10), and get $s=2$.

From Equation (7), we have

$$\begin{array}{c}\hfill U\left(\chi \right)=b_0+b_{1}F\left(\chi \right)+b_2{F}^2\left(\chi \right),b_2\ne 0,\end{array}$$

(11)

where $b_0$, $b_1$, and $b_2$ are arbitrary constants.

Substituting Equation (11) and Equation (8) in Equation (10), we get all the terms having the same power of $F\left(\chi \right)$. Setting these polynomials equal to zero, we obtain

$$\begin{array}{ccc}& & b_0=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta },b_1=0,V=\frac{b_2\left({\zeta }^2-16{\theta }^2\left(a^2-3\sigma \right)\right)+12\theta (2k-v)}{12\nu \zeta },\eta =-2\theta ,\hfill \\ & & C=\frac{9{(v-2k)}^2-b_2^2\left(16a^4{\theta }^2-8a^3\zeta \theta +a^2\left({\zeta }^2-96{\theta }^2\sigma \right)+36a\zeta \theta \sigma +3\sigma \left(48{\theta }^2\sigma -{\zeta }^2\right)\right)}{18\zeta }.\hfill \end{array}$$

(12)

From Equations (10)–(12), we evaluate the following solutions:

Case I: If $a>0$ and $\sigma =0,$ then

$$\begin{array}{c}\hfill U_{1,1}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{-pqa}{\mathrm{Sech}}_{pq}\left(\sqrt{a}\chi \right)\right)}^2,\end{array}$$

(13)

$$\begin{array}{c}\hfill U_{2,2}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{pqa}{\mathrm{Csch}}_{pq}\left(\sqrt{a}\chi \right)\right)}^2.\end{array}$$

(14)

Case II: If $a<0$ and $\sigma =0,$ then

$$\begin{array}{c}\hfill U_{3,3}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{-pqa}{\mathrm{Sec}}_{pq}\left(\sqrt{-a}\chi \right)\right)}^2,\end{array}$$

(15)

$$\begin{array}{c}\hfill U_{4,4}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{-pqa}{\mathrm{Csc}}_{pq}\left(\sqrt{-a}\chi \right)\right)}^2.\end{array}$$

(16)

Case III: If $a<0$ and $\sigma =\frac{a^2}{4}$, then

$$\begin{array}{c}\hfill U_{5,5}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{\frac{-a}{2}}{\mathrm{Tanh}}_{pq}\left(\sqrt{\frac{-a}{2}}\chi \right)\right)}^2,\end{array}$$

(17)

$$\begin{array}{c}\hfill U_{6,6}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{\frac{-a}{2}}{\mathrm{Coth}}_{pq}\left(\sqrt{\frac{-a}{2}}\chi \right)\right)}^2,\end{array}$$

(18)

$$\begin{array}{c}\hfill U_{7,7}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{\frac{-a}{2}}({\mathrm{Tanh}}_{pq}\left(\sqrt{-2a}\chi \right)\pm \iota \sqrt{pq}{\mathrm{Sech}}_{pq}\left(\sqrt{-2a}\chi \right))\right)}^2,\end{array}$$

(19)

$$\begin{array}{c}\hfill U_{8,8}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{\frac{-a}{2}}({\mathrm{Coth}}_{pq}\left(\sqrt{-2a}\chi \right)\pm \sqrt{pq}{\mathrm{Csch}}_{pq}\left(\sqrt{-2a}\chi \right))\right)}^2,\end{array}$$

(20)

$$\begin{array}{c}\hfill U_{9,9}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{\frac{-a}{8}}({\mathrm{Tanh}}_{pq}\left(\sqrt{\frac{-a}{8}}\chi \right)+{\mathrm{Coth}}_{pq}\left(\sqrt{\frac{-a}{8}}\chi \right))\right)}^2.\end{array}$$

(21)

Case IV: If a > 0 and $\sigma =\frac{a^2}{4}$, then

$$\begin{array}{c}\hfill U_{10,10}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{\frac{a}{2}}{\mathrm{Tan}}_{pq}\left(\sqrt{\frac{a}{2}}\chi \right)\right)}^2,\end{array}$$

(22)

$$\begin{array}{c}\hfill U_{11,11}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{\frac{a}{2}}{\mathrm{Cot}}_{pq}\left(\sqrt{\frac{a}{2}}\chi \right)\right)}^2,\end{array}$$

(23)

$$\begin{array}{c}\hfill U_{12,12}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{\frac{a}{2}}({\mathrm{Tan}}_{pq}\left(\sqrt{2a}\chi \right)\pm \sqrt{pq}{\mathrm{Sec}}_{pq}\left(\sqrt{2a}\chi \right))\right)}^2,\end{array}$$

(24)

$$\begin{array}{c}\hfill U_{13,13}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{\frac{a}{2}}({\mathrm{Cot}}_{pq}\left(\sqrt{2a}\chi \right)\pm \sqrt{pq}{\mathrm{Csc}}_{pq}\left(\sqrt{2a}\chi \right))\right)}^2,\end{array}$$

(25)

$$\begin{array}{c}\hfill U_{14,14}^{\pm }\left(\chi \right)=\frac{a{b}_2(\zeta -a\theta )+6k-3v}{3\zeta }+b_2{\left(\sqrt{\frac{a}{8}}({\mathrm{Tan}}_{pq}\left(\sqrt{\frac{a}{8}}\chi \right)+{\mathrm{Cot}}_{pq}\left(\sqrt{\frac{a}{8}}\chi \right))\right)}^2.\end{array}$$

(26)

4. Implementation of Jacobi Elliptic Function Method Depending on ${\mathit{F}}^{\prime }=\sqrt{\frac{\mathbf{1}}{\mathbf{2}}\mathit{p}{\mathit{F}}^{\mathbf{4}}+\mathit{q}{\mathit{F}}^{\mathbf{2}}+\mathit{r}}$

To find out the solutions of Equation (10), the JEFM is utilized [65,66]. As a consequence, we are capable of acquiring the solutions of Equation (1).

Suppose the solution of Equation (10) has the form

$$\begin{array}{c}\hfill U\left(\chi \right)=\sum _{i=0}^s{b}_i{F}^i\left(\chi \right)(b_s\ne 0),\end{array}$$

(27)

where F satisfies the following ODE

$$\begin{array}{c}\hfill F^{\prime }=\sqrt{\frac{1}{2}p{F}^4+q{F}^2+r},\end{array}$$

(28)

where r, q, and p are parameters. Equation (28) has different solutions depending upon the values of r, q, and p.

$sn\left(\chi \right)=sn(\chi ,m)$, $cn\left(\chi \right)=cn(\chi ,m)$, $dn\left(\chi \right)=dn(\chi ,m)$, and $0<m<1$ are Jacobi elliptic functions (JEFs). When $m\to 1$, then the JEFs are changed to the following hyperbolic functions:

$$\begin{array}{ccc}& & cn\left(\chi \right)\to \mathrm{sech}\left(\chi \right),\mathrm{sn}\left(\chi \right)\to \tanh \left(\chi \right),\mathrm{cs}\left(\chi \right)\to \mathrm{csch}\left(\chi \right),\hfill \\ & & \mathrm{ds}\left(\chi \right)\to \mathrm{csch}\left(\chi \right),\mathrm{dn}\left(\chi \right)\to \mathrm{sech}\left(\chi \right).\hfill \end{array}$$

(29)

After applying the balancing principle, we obtain the solution of Equation (7) as

$$\begin{array}{c}\hfill U\left(\chi \right)=b_0+b_{1}F\left(\chi \right)+b_2{F}^2\left(\chi \right),b_2\ne 0,\end{array}$$

(30)

where $b_0$, $b_1$, and $b_2$ are arbitrary constants.

Substituting Equation (30) and Equation (28) into Equation (10), we obtain the system of equations in the powers of $U\left(\chi \right)$. By taking each coefficient of $U^k\left(\chi \right)$ equal to zero, the following system of equations is obtained

$$\begin{array}{ccc}& & -\frac{1}{2}{b}_0^2\zeta +2b_{0}k-\frac{1}{2}{b}_1^2\eta r-2b_2{b}_0\theta r+\frac{1}{2}{b}_1^2\theta r+2b_{2}rvV-b_{0}v-C=0,\hfill \\ & & -\frac{1}{2}{b}_1^2\zeta -b_0{b}_2\zeta +2b_{2}k-\frac{1}{2}{b}_1^2\eta q-\frac{1}{2}{b}_1^2\theta q-4b_0{b}_2\theta q+4b_{2}qvV-2b_2^2\eta r-b_{2}v=0,\hfill \\ & & -\frac{b_2^2\zeta }{2}-\frac{1}{4}{b}_1^2\eta p-\frac{3}{4}{b}_1^2\theta p-3b_0{b}_2\theta p+3b_{2}pvV-2b_2^2\eta q-2b_2^2\theta q=0,\hfill \\ & & -b_2{b}_1\zeta -b_0{b}_1\theta p+b_{1}pvV-2b_2{b}_1\eta q-3b_2{b}_1\theta q=0,\hfill \\ & & -b_0{b}_1\zeta +2b_{1}k-b_0{b}_1\theta q+b_{1}qvV-2b_2{b}_1\eta r-b_{1}v=0,\hfill \\ & & b_1{b}_2\eta (-p)-3b_1{b}_2\theta p=0,\hfill \\ & & b_2^2\eta (-p)-2b_2^2\theta p=0.\hfill \end{array}$$

(31)

Solving system (31) for the involved constants, we get

$$\begin{array}{ccc}& & b_0=b_0,b_1=0,b_2=b_2,\eta =-2\theta ,k=\frac{3pv+3b_0\zeta p+2(-6\theta pr+4\theta q^2-\zeta q)b_2}{6p},\hfill \\ & & V=\frac{6b_0\theta p+(\zeta -4\theta q)b_2}{6pv},C=\frac{3b_0^2\zeta p+4b_0{b}_2+2r{b}_2^2\left((\zeta -4\theta q)\right)}{6p}.\hfill \end{array}$$

(32)

Hence, the solution in Equation (30) becomes

$$\begin{array}{c}\hfill U\left(\chi \right)=b_0+b_2{F}^2\left(\chi \right).\end{array}$$

(33)

The solutions for Equation (28) are given in

Table 1. All the solutions of $F^{\prime }=\sqrt{\frac{1}{2}p{F}^4+q{F}^2+r}$.

Case	p	q	r	F
1	$2m^2$	$-(1+m^2)$	1	$sn\left(\chi \right)$
2	2	$2m^2-1$	$-m^2(1-m^2)$	$ds\left(\chi \right)$
3	2	$2-m^2$	$1-m^2$	$cs\left(\chi \right)$
4	$-2m^2$	$2m^2-1$	$1-m^2$	$cn\left(\chi \right)$
5	-2	$2-m^2$	$m^2-1$	$dn\left(\chi \right)$
6	$\frac{m^2}{2}$	$\frac{(m^2-2)}{2}$	$\frac{1}{4}$	$\frac{sn\left(\chi \right)}{1\pm dn\left(\chi \right)}$
7	$\frac{m^2}{2}$	$\frac{m^2-2}{2}$	$\frac{m^2}{4}$	$\frac{sn\left(\chi \right)}{1\pm dn\left(\chi \right)}$
8	$\frac{-1}{2}$	$\frac{m^2+1}{2}$	$\frac{-{(1-m^2)}^2}{4}$	$mcn\left(\chi \right)\pm dn\left(\chi \right)$
9	$\frac{m^2-1}{2}$	$\frac{m^2+1}{2}$	$\frac{m^2-1}{4}$	$\frac{dn\left(\chi \right)}{1\pm sn\left(\chi \right)}$
10	$\frac{1-m^2}{2}$	$\frac{1-m^2}{2}$	$\frac{1-m^2}{4}$	$\frac{cn\left(\chi \right)}{1\pm sn\left(\chi \right)}$
11	$\frac{{(1-m^2)}^2}{2}$	$\frac{{(1-m^2)}^2}{2}$	$\frac{1}{4}$	$\frac{sn\left(\chi \right)}{dn\pm cn\left(\chi \right)}$
12	2	0	0	$\frac{c}{\chi }$
13	0	1	0	$c{e}^{\chi }$

.

For $m\to 1$ in Table 1, we obtain the following solutions of Equation (28) given in

Table 2. All the solutions of $F^{\prime }=\sqrt{\frac{1}{2}p{F}^4+q{F}^2+r}$ when $m\to 1$.

Case	p	q	r	F
1	2	$-2$	1	$\mathrm{Tanh}\left(\chi \right)$
2	2	1	0	$\mathrm{Csch}\left(\chi \right)$
3	2	1	0	$\mathrm{Csch}\left(\chi \right)$
4	$-2$	1	0	$\mathrm{Sech}\left(\chi \right)$
5	-2	1	0	$\mathrm{Sech}\left(\chi \right)$
6	$\frac{1}{2}$	$\frac{-1}{2}$	$\frac{1}{4}$	$\frac{\mathrm{Tanh}\left(\chi \right)}{1\pm \mathrm{Sech}\left(\chi \right)}$
7	$\frac{1}{2}$	$\frac{-1}{2}$	$\frac{1}{4}$	$\frac{\mathrm{Tanh}\left(\chi \right)}{1\pm \mathrm{Sech}\left(\chi \right)}$
8	$\frac{-1}{2}$	1	0	$\mathrm{Sech}\left(\chi \right)\pm \mathrm{Sech}\left(\chi \right)$
9	0	1	0	$\frac{\mathrm{Sech}\left(\chi \right)}{1\pm \mathrm{Tanh}\left(\chi \right)}$
10	0	0	0	$\frac{\mathrm{Sech}\left(\chi \right)}{1\pm \mathrm{Tanh}\left(\chi \right)}$
11	0	0	$\frac{1}{4}$	$\frac{\mathrm{Tanh}\left(\chi \right)}{\mathrm{Sech}\pm \mathrm{Sech}\left(\chi \right)}$
12	2	0	0	$\frac{c}{\chi }$
13	0	1	0	$c{e}^{\chi }$

.

F can be collected from the last column of Table 2.

Here, we discuss some solutions of Equation (10) for the graphical illustration using Table 2:

Case I: For $p=2$, $q=-2$, and $r=1$, we have

$$\begin{array}{c}\hfill U_1\left(\chi \right)=b_0+b_2{\mathrm{Tanh}}^2\left(\chi \right).\end{array}$$

(34)

Case II: For $p=2$, $q=1$, and $r=0$, we have

$$\begin{array}{c}\hfill U_2\left(\chi \right)=b_0+b_2{\mathrm{Csch}}^2\left(\chi \right).\end{array}$$

(35)

Case III: For $p=-2$, $q=1$, and $r=0$, we have

$$\begin{array}{c}\hfill U_3\left(\chi \right)=b_0+b_2{\mathrm{Sech}}^2\left(\chi \right).\end{array}$$

(36)

5. Results and Discussion

A graphical representation plays a fundamental role in the article because graphs show the physical activities of solutions. Here, we explore the behavior and physical interpretation of the solutions of the Gilson–Pickering equation that we have obtained with the assistance of the SSM and JEFM. It is noted that the results provided in [60] have only the form of a periodic singular soliton, bright soliton, and shock waves. In the present method, we give the solutions in form of dark, singular, bright, periodic singular, dark–bright-combined and dark–singular-combined solitons. Using the SSM, the solutions $|U_{1,1}^{\pm }|$ and $|U_{5,5}^{\pm }|$ provide the bright and dark solitons, respectively. $|U_{3,3}^{\pm }|$, $|U_{4,4}^{\pm }|$, $|U_{10,10}^{\pm }|$, $|U_{11,11}^{\pm }|$, $|U_{12,12}^{\pm }|$, $|U_{13,13}^{\pm }|$, and $|U_{14,14}^{\pm }|$ illustrate the singular solitons while $|U_{2,2}^{\pm }|$, $|U_{6,6}^{\pm }|$, and $|U_{8,8}^{\pm }|$ represent periodic singular solitons. Dark–bright-combined, dark–singular-combined solitons are elaborated by $|U_{7,7}^{\pm }|$ and $|U_{9,9}^{\pm }|$, respectively. The solutions $|U_1\left(\chi \right)|$, $|U_2\left(\chi \right)|$, and $|U_3\left(\chi \right)|$ obtained by the JEFM provide dark, singular, and bright solitons, respectively. We depict 2D, 3D, and contour plots of some of the obtained solutions with the appropriate values of constants obtained by the SSM and JEFM. We can say that graphs are nonlinear data structures that illustrate the properties and behavior which actually demonstrate the solutions.

The graphs in Figure 1 and Figure 2 show a singular soliton and a dark–bright soliton, respectively, obtained by the SSM. Moreover, some solutions acquired from the JEFM are plotted in Figure 3, Figure 4 and Figure 5, representing the dark, singular, and bright solutions. Maple 18 software was used to draw these graphs.

In Figure 1, the graph of $|U_{2,2}^{\pm }|$ is drawn for the values: $\theta =0.9$, $\zeta =0.8$, $v=1.7$, $a=-1$, $p=0.85$, $q=0.98$, $k=0.2$, and $b_2=0.05$, which demonstrates the shape of a singular soliton where (a) and (b) represent the 3D and contour graphs for $-10\le x,\le t\le 10$, while (c) shows the 2D graph for $-10\le x\le 10$ with $t=0.0$. The graph of $|U_{7,7}^{\pm }|$ presented in Figure 2 illustrates the combined dark–bright soliton for the values: $\theta =0.9$, $\zeta =0.8$, $v=1.7$, $a=-1$, $p=0.85$, $q=0.98$, $k=0.2$, and $b_2=0.05$, where (a) and (b) demonstrate the 3D and contour graphs for $-10\le x,\le t\le 10$, while (c) depicts the 2D graph for $-10\le x\le 10$ with $t=0.0$. Figure 3 represents the graph of $|U_1\left(\chi \right)|$ for the values: $\zeta =1.2$, $\theta =0.95$, $b_0=0.3$, $b_2=0.1$, $\nu =0.5$, $p=0.95$, and $q=0.8$. It illustrates the shape of a dark soliton where (a) and (b) are the 3D and contour graphs for $-10\le x,\le t\le 10$, while (c) is the 2D graph for $-10\le x\le 10$ with $t=0.0$. The graph of solution $|U_2\left(\chi \right)|$ provides the singular soliton for the values: $\zeta =0.2$, $\theta =0.1$, $b_0=0.05$, $b_2=-0.1$, $\nu =0.01$, $p=0.5$, and $q=0.85$; it is illustrated in Figure 4, where (a) and (b) represent the 3D and contour graphs for $-10\le x,\le t\le 10$, respectively, while (c) depicts the 2D graph for $-10\le x\le 10$ with $t=0.0$. The graph of $|U_3\left(\chi \right)|$ for $\zeta =0.2$, $\theta =0.5$, $b_0=0.3$, $b_2=0.3$, $\nu =0.3$, $p=0.75$, and $q=1.1$ is drawn in Figure 5, which elaborates the bright soliton, where (a) and (b) represent the 3D and contour graphs for $-10\le x,\le t\le 10$, while (c) demonstrates the 2D graph for $-10\le x\le 10$ with $t=0.0$.

6. Conclusions

In this paper, the SSM and JEFM were used effectively to investigate the behavior of the GPE, which has applications in different fields of applied sciences. The affluent families of solutions were explored in single and combined forms. These families of solutions in the single form were singular soliton, singular periodic soliton, dark soliton, and bright soliton, while in combined form, the combined bright–dark soliton and combined dark–singular solitons were obtained. Moreover, some rational solutions also occurred through derivation. The solutions obtained from the proposed methods were distinct, unique, fruitful, and had a specific physical structure. Graphical illustrations of some solutions were also presented in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5. Mathematica 11 and Maple 18 were used to derive the solutions of the presented model and for plotting graphs, respectively. Finally, we can say that the Sardar’s subequation method and the Jacobi elliptic function method are helpful and powerful techniques for solving NLEEs.