The Jacobi Elliptic Function and Incomplete Elliptic Integral of Second Kind Solutions of the Wazwaz Negative Order Korteweg–de Vries Equation

Abstract

In this research paper, the negative order Korteweg–de Vries equation expressed as nonlinear partial differential equation, firstly introduced by Wazwaz, is solved for the exact Jacobi elliptic function solution. For this purpose, the Jacobi elliptic function scheme, one of the direct algebraic methods, was used. The obtained exact solutions of the negative-order Korteweg–de Vries equation, a symmetry evolution equation, contains the combination of Jacobi elliptic functions and incomplete elliptic integral of second function. The three unique families of exact solutions are classified and presented. The degeneration of the obtained Jacobi elliptic function solutions into various solitons, periodic and rational solutions, is reported using the modulus transformation of Jacobi elliptic function solutions. The necessary condition existence of certain Jacobi elliptic function solutions is presented. The two-dimensional graphs for certain Jacobi elliptic function solutions are drawn to show the variation in wave propogation with respect to velocity and modulus. The non-existence of certain Jacobi elliptic function solutions for negative-order Korteweg–de Vries equations is reported. Finally, the obtained solutions were compared with the previously obtained solutions of negative-order Korteweg–de Vries equation.

1. Introduction

The integrable Korteweg–de Vries equation by the nonlinear partial differential equation is given by [1]:

$$\begin{array}{c}\hfill {\displaystyle \frac{𝜕}{𝜕\mathtt{t}}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)+6\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right){\displaystyle \frac{𝜕}{𝜕\mathtt{x}}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)+{\displaystyle \frac{{𝜕}^3}{𝜕{\mathtt{x}}^3}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)=0.\end{array}$$

The recursion operator Φ symmetry generator of the Korteweg–de Vries equation is defined by [2]:

$$\begin{array}{c}\hfill \mathsf{\Phi }\left(\mathsf{{\rm Y}}\right)=-{𝜕}_x^2+4\mathsf{{\rm Y}}+2{\mathsf{{\rm Y}}}_x{𝜕}_x^{-1}.\end{array}$$

In which ${𝜕}_x$ and ${𝜕}_x^{-1}$ are respective total derivative and integration with respect to x, and the evolution equation hierarchy [3] ${\mathsf{{\rm Y}}}_t=\mathsf{\Phi }{\mathsf{{\rm Y}}}_x$ by which negative-direction evolution equation symmetry hierarchy is defined is given by ${\mathsf{{\rm Y}}}_t={\mathsf{\Phi }}^{-1}{\mathsf{{\rm Y}}}_x$, which may be written as $\mathsf{\Phi }{\mathsf{{\rm Y}}}_t={\mathsf{{\rm Y}}}_x$. Substituting the Korteweg–de Vries equation recursion operator in the negative-direction evolution equation gives the negative-order Korteweg–de Vries equation.

The negative order Korteweg–de Vries (nKdV) equation considered in this research was given by [1], which was based on the recursion operator [2] in the negative direction [3].

$$\begin{array}{c}\hfill {\displaystyle \frac{{𝜕}^4}{𝜕{\mathtt{x}}^3𝜕\mathtt{t}}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)+4{\displaystyle \frac{𝜕}{𝜕\mathtt{x}}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right){\displaystyle \frac{{𝜕}^2}{𝜕\mathtt{x}𝜕\mathtt{t}}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)+2{\displaystyle \frac{𝜕}{𝜕\mathtt{t}}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right){\displaystyle \frac{{𝜕}^2}{𝜕{\mathtt{x}}^2}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)+{\displaystyle \frac{{𝜕}^2}{𝜕{\mathtt{x}}^2}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)=0.\end{array}$$

(1)

In Equation (1), $\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)$ is the unknown function in space and time variables $\mathtt{x}$ and $\mathtt{t}$, ${\displaystyle \frac{{𝜕}^4}{𝜕{\mathtt{x}}^3𝜕\mathtt{t}}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)$ is the dispersion term and the nonlinear part $4{\displaystyle \frac{𝜕}{𝜕\mathtt{x}}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right){\displaystyle \frac{{𝜕}^2}{𝜕\mathtt{x}𝜕\mathtt{t}}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)+2{\displaystyle \frac{𝜕}{𝜕\mathtt{t}}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right){\displaystyle \frac{{𝜕}^2}{𝜕{\mathtt{x}}^2}}\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)$ is advection term. By applying the complex form of simplified Hirota’s method, the single, double and multiple complex solitons of the nKdV Equation (1) are shown in [1]. The one, two and three solutions of nKdV are obtained via the simplified Hirota’s direct method in [4]. The combined KdV and nKdV is solved through the consistent Riccati expansion method in [5] and the exact cnoidal, soliton and their interaction are obtained. Two generalizations of nKdV are formed and their exact solutions are obtained in [6]. The Riemann–Hilbert method for the initial value problem of nKdV is applied in [7] and the square matrix solutions are obtained. Travelling wave solutions of nKdV hierarchy through the dynamical system are shown in [8]. Solitons and kink solutions of one of nKdV hierarchy are studied in [9]. Bilinear Bäcklund transformation is employed to obtain multi solitons and multi-kink solutions in addition to conservation laws in [10]. The different evolution equation nKdV hierarchy is studied in [11] and the traveling wave scheme is applied, showing the non existence of solitary wave solutions.

The extended F expansion method has the equation

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{d}{d\mathsf{\xi }}}\mathsf{F}\left(\mathsf{\xi }\right)\right)}^2={\mathsf{\nu }}_4{\mathsf{F}}^4\left(\mathsf{\xi }\right)+{\mathsf{\nu }}_3{\mathsf{F}}^3\left(\mathsf{\xi }\right)+{\mathsf{\nu }}_2{\mathsf{F}}^2\left(\mathsf{\xi }\right)+{\mathsf{\nu }}_1\mathsf{F}\left(\mathsf{\xi }\right)+{\mathsf{\nu }}_0,\end{array}$$

where ${\mathsf{\nu }}_4$ is non-zero, whose general solution in terms of the Jacobi elliptical function is applied for diverse models in [12]. The twin-core couplers cubic-quartic system in perturbated metamaterial with triple power law nonlinearity is solved for Jacobi elliptic solutions in [12]. The extended F expansion method is applied to the twin-core coupler with perturbation terms and arbitrary Kudryashov sextic law of refraction index is solved for bright, singular and periodic solutions in [13]. The Sasa–Satsuma equation in birefrigent fibers with Kerr law nonlinearity is solved for bright, dark, singular, periodic, rational and Jacobi elliptic solutions in [14].

The generalized Jacobi elliptic function expansion method with different initial seed solution is employed in [15,16] for elliptic function solutions of various models. The birefringent model Biswas–Arshed equation is solved for bright, dark, singular, periodic and Jacobi elliptic solutions in [15], The unstable baroclinic wave propogation in geophysical fluids described by AB system of equations is solved by the extended Jacobi elliptic function expansion method and their exact solutions are obtained in [16]. The system of the Biswas–Milovic equation with Kudryashov law of refractive index is solved for various bright, dark and combo solutions in [17]. The twin and multiple-core couplers with the polynomial law of nonlinearity is solved for bright solitons in [18]. The stochastic Nizhnik–Novikov–Veselov (NNV) system is studied using Jacobi elliptic function and their trigonometric and rational type solutions are obtained in [19]. The $(2+1)$—dimensional Wazwaz–Kaur–Boussinesq equation is solved for Jacobi elliptic solutions in [20].

The modified exponential rational function method with different initially assumed solutions is applied for nonlinear equations in [21]. The ion sound and Langmuir dynamical system of equation is solved using the modified general exponential rational function scheme in [21] and exact exponential solutions are obtained. The Jaulent–Miodek system is solved for exact solutions via the rational exponential function method in [22]. The Fokas–Lenells equation with perturbation term is solved for diverse exact solutions in [23]. The modified Kudryashov method is applied to solve various problems in [24]. The generalized Kuramoto–Sivashinsky equation is studied and the exact solution is obtained using the modified Kudryashov method in [24]. The $\left(3+1\right)$—dimensional gas bubble liquid nonlinear equation is solved for the solitary wave solution using the modified Kudryashov method in [25]. The extended sine Gordon equation expansion method and modified exponential function method are applied for the dispersion equation in [26]. The $(3+1)$—dimension Wazwaz–Kaur–Boussinesq equation [27] is studied.

From the above literature, the Wazwaz negative-order KDV Equation (1) is studied only in two works [1,4]. Hence, the research gap identified is the Jacobi elliptic function solution for Equation (1), which is not studied and reported in the literature so far. Hence, to fill this gap, in this work, the exact Jacobi elliptic function solutions of the nKdV Equation (1) are obtained via the Jacobi elliptic function method. The obtained solutions are the combined Jacobi elliptic function and incomplete elliptic integral of second kind function.

Definition 1.

The incomplete elliptic integral of second kind [28] (Chapter 17, Section 17.4, Page 592) is defined by

$$\begin{array}{c}\hfill EllipticE\left(\mathtt{x},\mathsf{\lambda }\right)\stackrel{\mathrm{def}}{=}{\int }_0^{\mathtt{x}}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\Phi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }.\end{array}$$

This research paper is organized as the Jacobi elliptic function method given in Section 2. The Wazwaz negative order KdV equation in (1) is studied using Section 2 in Section 3. Three unique families of Jacobi elliptic solutions of Equation (1) are given in Section 4. The physical interpretation, the non-existence of certain Jacobi elliptic solutions of Equation (1) and the comparative study of solutions of Equation (1) are given in Section 5. Finally, the conclusion is drawn in Section 6.

2. Description of the Jacobi Elliptic Function Scheme

In this section, the steps of the Jacobi elliptic function method [20,27] are analyzed.

Suppose the nonlinear partial differential equation is given in the following form:

$$\begin{array}{c}\hfill {\mathsf{{\rm Y}}}_{xx}+{\mathsf{{\rm Y}}}_{tt}+{\mathsf{{\rm Y}}}_{xt}+{\mathsf{{\rm Y}}}_x+{\mathsf{{\rm Y}}}_t+\cdots +\cdots =0.\end{array}$$

(2)

In Equation (2), $\mathsf{{\rm Y}}=\mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)$ and the subscript express the partial derivative. Substituting these gives

$$\begin{array}{c}\hfill \mathsf{{\rm Y}}\left(\mathtt{x},\mathtt{t}\right)=\mathsf{\vartheta }\left(\mathsf{\xi }\right),\mathsf{\xi }=\mathsf{\kappa }\left(x-\mathsf{\tau }t\right).\end{array}$$

(3)

In Equation (3), $\mathsf{\kappa }$ and $\mathsf{\tau }$ are, respectively, the wave number and velocity. Hence, Equation (2) is transformed into following nonlinear ordinary differential equation:

$$\begin{array}{c}\hfill {\mathsf{\kappa }}^2{\mathsf{\vartheta }}^{\prime \prime }+{\mathsf{\kappa }}^2{\mathsf{\tau }}^2{\mathsf{\vartheta }}^{\prime \prime }-{\mathsf{\kappa }}^2\mathsf{\tau }{\mathsf{\vartheta }}^{\prime \prime }+\mathsf{\kappa }{\mathsf{\vartheta }}^{\prime }-\mathsf{\kappa }\mathsf{\tau }{\mathsf{\vartheta }}^{\prime }+\cdots +\cdots =0.\end{array}$$

(4)

In Equation (4), $\mathsf{\vartheta }=\mathsf{\vartheta }\left(\mathsf{\xi }\right)$ and superscript ′ express the derivative with respect to $\mathsf{\xi }$. Let the initial solution of Equation (4) be taken in the following form:

$$\begin{array}{c}\hfill \mathsf{\vartheta }\left(\mathsf{\xi }\right)={\mathsf{\rho }}_0+{\mathsf{\rho }}_1\mathsf{F}\left(\mathsf{\xi }\right)+{\mathsf{\rho }}_2{\mathsf{F}}^2\left(\mathsf{\xi }\right)+\cdots +{\mathsf{\rho }}_N{\mathsf{F}}^N\left(\mathsf{\xi }\right)+{\displaystyle \frac{{\mathsf{\sigma }}_1}{\mathsf{F}\left(\mathsf{\xi }\right)}}+{\displaystyle \frac{{\mathsf{\sigma }}_2}{{\mathsf{F}}^2\left(\mathsf{\xi }\right)}}+\cdots +{\displaystyle \frac{{\mathsf{\sigma }}_N}{{\mathsf{F}}^N\left(\mathsf{\xi }\right)}}.\end{array}$$

(5)

In Equation (5), $\mathsf{N}$ is a positive integer computed by comparing the highest order derivative and highest power nonlinear term in Equation (4) by taking into account Equation (6), and ${\mathsf{\rho }}_0,{\mathsf{\rho }}_1,{\mathsf{\rho }}_2,\cdots ,{\mathsf{\rho }}_N,{\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\cdots ,{\mathsf{\sigma }}_N$ are the unknowns to be found. Next, $\mathsf{F}\left(\mathsf{\xi }\right)$ is the solution of

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{d}{d\mathsf{\xi }}}\mathsf{F}\left(\mathsf{\xi }\right)\right)}^2={\mathsf{\nu }}_1{\mathsf{F}}^4\left(\mathsf{\xi }\right)+{\mathsf{\nu }}_2{\mathsf{F}}^2\left(\mathsf{\xi }\right)+{\mathsf{\nu }}_3.\end{array}$$

(6)

In Equation (6) ${\mathsf{\nu }}_1,{\mathsf{\nu }}_2,{\mathsf{\nu }}_3$ are real parameters with the condition that ${\mathsf{\nu }}_1$ is non-zero and the solution $\mathsf{F}\left(\mathsf{\xi }\right)$ is given in

Table 1. $\mathsf{F}\left(\mathsf{\xi }\right)$ for ${\mathsf{\nu }}_1$, ${\mathsf{\nu }}_2$ and ${\mathsf{\nu }}_3$.

S.No	${\mathsf{\nu }}_1$	${\mathsf{\nu }}_2$	${\mathsf{\nu }}_3$	$\mathsf{F}\left(\mathsf{\xi }\right)$
1	${\mathsf{\lambda }}^2$	$-\left(1+{\mathsf{\lambda }}^2\right)$	1	$\mathrm{sn}\left(\mathsf{\xi },\mathsf{\lambda }\right)$
2	$-1$	$\left(2-{\mathsf{\lambda }}^2\right)$	$\left({\mathsf{\lambda }}^2-1\right)$	$\mathrm{dn}\left(\mathsf{\xi },\mathsf{\lambda }\right)$
3	$-{\mathsf{\lambda }}^2\left(1-{\mathsf{\lambda }}^2\right)$	$\left(2{\mathsf{\lambda }}^2-1\right)$	1	$\mathrm{sd}\left(\mathsf{\xi },\mathsf{\lambda }\right)$
4	1	$\left(2-{\mathsf{\lambda }}^2\right)$	$\left(1-{\mathsf{\lambda }}^2\right)$	$\mathrm{cs}\left(\mathsf{\xi },\mathsf{\lambda }\right)$
5	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{\left(1-2{\mathsf{\lambda }}^2\right)}{2}}$	${\displaystyle \frac{1}{4}}$	$\mathrm{ns}\left(\mathsf{\xi },\mathsf{\lambda }\right)\pm \mathrm{cs}\left(\mathsf{\xi },\mathsf{\lambda }\right)$
6	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{\left({\mathsf{\lambda }}^2-2\right)}{2}}$	${\displaystyle \frac{{\mathsf{\lambda }}^2}{4}}$	$\mathrm{ns}\left(\mathsf{\xi },\mathsf{\lambda }\right)\pm \mathrm{ds}\left(\mathsf{\xi },\mathsf{\lambda }\right)$
7	${\displaystyle \frac{\left({\mathsf{\lambda }}^2-1\right)}{4}}$	${\displaystyle \frac{\left({\mathsf{\lambda }}^2+1\right)}{2}}$	${\displaystyle \frac{\left({\mathsf{\lambda }}^2-1\right)}{4}}$	$\mathsf{\lambda }\mathrm{sd}\left(\mathsf{\xi },\mathsf{\lambda }\right)\pm \mathrm{nd}\left(\mathsf{\xi },\mathsf{\lambda }\right)$
8	1	$2\left(1-2{\mathsf{\lambda }}^2\right)$	1	${\displaystyle \frac{\mathrm{sn}\left(\mathsf{\xi },\mathsf{\lambda }\right)\mathrm{dn}\left(\mathsf{\xi },\mathsf{\lambda }\right)}{\mathrm{cn}\left(\mathsf{\xi },\mathsf{\lambda }\right)}}$
9	${\displaystyle \frac{4}{\mathsf{\lambda }}}$	$\left(-{\mathsf{\lambda }}^2-6\mathsf{\lambda }-1\right)$	${\mathsf{\lambda }}^2\left({\mathsf{\lambda }}^2+2\mathsf{\lambda }+1\right)$	${\displaystyle \frac{\mathsf{\lambda }\mathrm{cn}\left(\mathsf{\xi },\mathsf{\lambda }\right)\mathrm{dn}\left(\mathsf{\xi },\mathsf{\lambda }\right)}{\mathsf{\lambda }{\mathrm{sn}}^2\left(\mathsf{\xi },\mathsf{\lambda }\right)-1}}$
10	${\displaystyle \frac{\left(1-{\mathsf{\lambda }}^2\right)}{4}}$	${\displaystyle \frac{\left(1+{\mathsf{\lambda }}^2\right)}{2}}$	${\displaystyle \frac{\left(1-{\mathsf{\lambda }}^2\right)}{4}}$	${\displaystyle \frac{\mathrm{cn}\left(\mathsf{\xi },\mathsf{\lambda }\right)}{1\pm \mathrm{sn}\left(\mathsf{\xi },\mathsf{\lambda }\right)}}$
11	${\displaystyle \frac{A^2{\left(\mathsf{\lambda }-1\right)}^2}{4}}$	${\displaystyle \frac{\left({\mathsf{\lambda }}^2+1\right)}{2}}+3\mathsf{\lambda }$	${\displaystyle \frac{{\left(\mathsf{\lambda }-1\right)}^2}{4A^2}}$	${\displaystyle \frac{\mathrm{cn}\left(\mathsf{\xi },\mathsf{\lambda }\right)\mathrm{dn}\left(\mathsf{\xi },\mathsf{\lambda }\right)}{A\left(1+\mathrm{sn}\left(\mathsf{\xi },\mathsf{\lambda }\right)\right)\left(1+\mathsf{\lambda }\mathrm{sn}\left(\mathsf{\xi },\mathsf{\lambda }\right)\right)}}$
12	$-B^2\left({\mathsf{\lambda }}^2-2\mathsf{\lambda }+1\right)$	$2\left({\mathsf{\lambda }}^2+1\right)$	${\displaystyle \frac{\left(-{\mathsf{\lambda }}^2+2\mathsf{\lambda }+1\right)}{B^2}}$	${\displaystyle \frac{\mathsf{\lambda }{\mathrm{sn}}^2\left(\mathsf{\xi },\mathsf{\lambda }\right)+1}{B\left(\mathsf{\lambda }{\mathrm{sn}}^2\left(\mathsf{\xi },\mathsf{\lambda }\right)-1\right)}}$

. Next, substituting Equation (5) in Equation (4) and using Equation (6) and its derivatives converts Equation (4) into the polynomial in $\mathsf{F}\left(\mathsf{\xi }\right)$. Then, collecting the coefficient of $\mathsf{F}\left(\mathsf{\xi }\right)$ and its powers from the polynomial leads to the algebraic system. Solving the obtained overdetermined system gives the unknown ${\mathsf{\rho }}_0,{\mathsf{\rho }}_1,{\mathsf{\rho }}_2,\cdots ,{\mathsf{\rho }}_N,{\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\cdots ,{\mathsf{\sigma }}_N,\mathsf{\kappa },\mathsf{\tau }$ values. Substituting the computed unknowns into Equation (5) and using Equation (3) gives the solution of Equation (2).

3. Mathematical Analysis of Wazwaz Negative-Order KdV Equation Using Jacobi Elliptic Function Scheme

In this section, the Wazwaz negative-order KdV Equation (1) is studied with the Jacobi elliptic function scheme.

The application of Equation (3) to Equation (1) gives

$$\begin{array}{c}\hfill -\mathsf{\tau }{\mathsf{\kappa }}^2{\displaystyle \frac{{\mathsf{d}}^4}{\mathsf{d}{\mathsf{\xi }}^4}}\mathsf{\vartheta }\left(\mathsf{\xi }\right)-6\mathsf{\tau }\mathsf{\kappa }{\displaystyle \frac{\mathsf{d}}{\mathsf{d}\mathsf{\xi }}}\mathsf{\vartheta }\left(\mathsf{\xi }\right){\displaystyle \frac{{\mathsf{d}}^2}{\mathsf{d}{\mathsf{\xi }}^2}}\mathsf{\vartheta }\left(\mathsf{\xi }\right)+{\displaystyle \frac{{\mathsf{d}}^2}{\mathsf{d}{\mathsf{\xi }}^2}}\mathsf{\vartheta }\left(\mathsf{\xi }\right)=0.\end{array}$$

(7)

Integrating Equation (7) with respect to $\mathsf{\xi }$ one time and taking the integration constant to zero gives

$$\begin{array}{c}\hfill -\mathsf{\tau }{\mathsf{\kappa }}^2{\displaystyle \frac{{\mathsf{d}}^3}{\mathsf{d}{\mathsf{\xi }}^3}}\mathsf{\vartheta }\left(\mathsf{\xi }\right)-3\mathsf{\tau }\mathsf{\kappa }{\left({\displaystyle \frac{\mathsf{d}}{\mathsf{d}\mathsf{\xi }}}\mathsf{\vartheta }\left(\mathsf{\xi }\right)\right)}^2+{\displaystyle \frac{\mathsf{d}}{\mathsf{d}\mathsf{\xi }}}\mathsf{\vartheta }\left(\mathsf{\xi }\right)=0.\end{array}$$

(8)

Substituting ${\displaystyle \frac{\mathsf{d}}{\mathsf{d}\mathsf{\xi }}}\mathsf{\vartheta }\left(\mathsf{\xi }\right)=\mathsf{\varpi }\left(\mathsf{\xi }\right)$,

$$\begin{array}{c}\hfill -\mathsf{\tau }{\mathsf{\kappa }}^2{\displaystyle \frac{{\mathsf{d}}^2}{\mathsf{d}{\mathsf{\xi }}^2}}\mathsf{\varpi }\left(\mathsf{\xi }\right)-3\mathsf{\tau }\mathsf{\kappa }{\left(\mathsf{\varpi }\left(\mathsf{\xi }\right)\right)}^2+\mathsf{\varpi }\left(\mathsf{\xi }\right)=0.\end{array}$$

(9)

The balancing of ${\displaystyle \frac{{\mathsf{d}}^2}{\mathsf{d}{\mathsf{\xi }}^2}}\mathsf{\varpi }\left(\mathsf{\xi }\right)$ and ${\left(\mathsf{\varpi }\left(\mathsf{\xi }\right)\right)}^2$ in Equation (9) gives the positive integer $\mathsf{N}=2$. So, the initial solution takes the following form:

$$\begin{array}{c}\hfill \mathsf{\varpi }\left(\mathsf{\xi }\right)={\mathsf{\rho }}_0+{\mathsf{\rho }}_1\mathsf{F}\left(\mathsf{\xi }\right)+{\mathsf{\rho }}_2{\mathsf{F}}^2\left(\mathsf{\xi }\right)+{\displaystyle \frac{{\mathsf{\sigma }}_1}{\mathsf{F}\left(\mathsf{\xi }\right)}}+{\displaystyle \frac{{\mathsf{\sigma }}_2}{{\mathsf{F}}^2\left(\mathsf{\xi }\right)}}.\end{array}$$

(10)

In Equation (10), ${\mathsf{\rho }}_2$ and ${\mathsf{\sigma }}_2$ should not be zero simultaneously. Now, substituting Equation (10) in Equation (9) and using Equation (6) and its second derivative leads to the polynomial in ${\mathsf{F}}^k\left(\mathsf{\xi }\right);0,\pm 1,\pm 2,\pm 3,\pm 4$. Extracting each coefficient gives the following overdetermined algebraic system.

$$\begin{array}{cc}\hfill {\mathsf{F}}^4\left(\mathsf{\xi }\right):& \left(-6\mathsf{\tau }{\mathsf{\kappa }}^2{\mathsf{\rho }}_2{\mathsf{\nu }}_1-3\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_2^2\right)=0.\hfill \\ \hfill {\mathsf{F}}^3\left(\mathsf{\xi }\right):& \left(-6\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_1{\mathsf{\rho }}_2-2\mathsf{\tau }{\mathsf{\kappa }}^2{\mathsf{\rho }}_1{\mathsf{\nu }}_1\right)=0.\hfill \\ \hfill {\mathsf{F}}^2\left(\mathsf{\xi }\right):& \left(-4\mathsf{\tau }{\mathsf{\kappa }}^2{\mathsf{\rho }}_2{\mathsf{\nu }}_2-6\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_0{\mathsf{\rho }}_2+{\mathsf{\rho }}_2-3\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_1^2\right)=0.\hfill \\ \hfill \mathsf{F}\left(\mathsf{\xi }\right):& \left(-6\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_0{\mathsf{\rho }}_1+{\mathsf{\rho }}_1-\mathsf{\tau }{\mathsf{\kappa }}^2{\mathsf{\rho }}_1{\mathsf{\nu }}_2-6\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_2{\mathsf{\sigma }}_1\right)=0.\hfill \\ \hfill \mathrm{Constant}:& \left(-2\mathsf{\tau }{\mathsf{\kappa }}^2{\mathsf{\rho }}_2{\mathsf{\nu }}_3-2\mathsf{\tau }{\mathsf{\kappa }}^2{\mathsf{\sigma }}_2{\mathsf{\nu }}_1-3\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_0^2+{\mathsf{\rho }}_0-6\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_1{\mathsf{\sigma }}_1-6\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_2{\mathsf{\sigma }}_2\right)=0.\hfill \\ \hfill {\mathsf{F}}^{-1}\left(\mathsf{\xi }\right):& \left({\mathsf{\sigma }}_1-6\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_0{\mathsf{\sigma }}_1-\mathsf{\tau }{\mathsf{\kappa }}^2{\mathsf{\sigma }}_1{\mathsf{\nu }}_2-6\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_1{\mathsf{\sigma }}_2\right)=0.\hfill \\ \hfill {\mathsf{F}}^{-2}\left(\mathsf{\xi }\right):& \left(-4\mathsf{\tau }{\mathsf{\kappa }}^2{\mathsf{\sigma }}_2{\mathsf{\nu }}_2+{\mathsf{\sigma }}_2-3\mathsf{\tau }\mathsf{\kappa }{\mathsf{\sigma }}_1^2-6\mathsf{\tau }\mathsf{\kappa }{\mathsf{\rho }}_0{\mathsf{\sigma }}_2\right)=0.\hfill \\ \hfill {\mathsf{F}}^{-3}\left(\mathsf{\xi }\right):& \left(-2\mathsf{\tau }{\mathsf{\kappa }}^2{\mathsf{\sigma }}_1{\mathsf{\nu }}_3-6\mathsf{\tau }\mathsf{\kappa }{\mathsf{\sigma }}_1{\mathsf{\sigma }}_2\right)=0.\hfill \\ \hfill {\mathsf{F}}^{-4}\left(\mathsf{\xi }\right):& \left(-6\mathsf{\tau }{\mathsf{\kappa }}^2{\mathsf{\sigma }}_2{\mathsf{\nu }}_3-3\mathsf{\tau }\mathsf{\kappa }{\mathsf{\sigma }}_2^2\right)=0.\hfill \end{array}$$

Solving the above system using computer algebra Maple 13 gives the unknowns of Equation (10), $\mathsf{\kappa }$ and $\mathsf{\tau }$. The solutions of overdetermined system of algebraic equations are given in

Table 2. Algebraic solutions.

Set	${\mathsf{\rho }}_0$	${\mathsf{\rho }}_1$	${\mathsf{\rho }}_2$	${\mathsf{\sigma }}_1$	${\mathsf{\sigma }}_2$	$\mathsf{\tau }$	$\mathsf{\kappa }$
1	$\mp \left[{\displaystyle \frac{\left(-{\mathsf{\nu }}_2+\sqrt{12{\mathsf{\nu }}_1{\mathsf{\nu }}_3+{\mathsf{\nu }}_2^2}\right)}{3\sqrt{\mathsf{\tau }}{\left(12{\mathsf{\nu }}_1{\mathsf{\nu }}_3+{\mathsf{\nu }}_2^2\right)}^{1/4}}}\right]$	0	$\mp \left[{\displaystyle \frac{{\mathsf{\nu }}_1}{\sqrt{\mathsf{\tau }}{\left(12{\mathsf{\nu }}_1{\mathsf{\nu }}_3+{\mathsf{\nu }}_2^2\right)}^{1/4}}}\right]$	0	$\mp \left[{\displaystyle \frac{{\mathsf{\nu }}_3}{\sqrt{\mathsf{\tau }}{\left(12{\mathsf{\nu }}_1{\mathsf{\nu }}_3+{\mathsf{\nu }}_2^2\right)}^{1/4}}}\right]$	$\mathsf{\tau }$	$\pm \left[{\displaystyle \frac{1}{2\sqrt{\mathsf{\tau }}{\left(12{\mathsf{\nu }}_1{\mathsf{\nu }}_3+{\mathsf{\nu }}_2^2\right)}^{1/4}}}\right]$
2	$\mp \left[{\displaystyle \frac{\left(-1+\mathsf{\tau }{\mathsf{\nu }}_2\sqrt{\mathsf{\Lambda }}\right)}{3\mathsf{\tau }{\mathsf{\Lambda }}^{1/4}}}\right]$	0	$\mp \left[{\mathsf{\nu }}_1{\mathsf{\Lambda }}^{1/4}\right]$	0	0	$\mathsf{\tau }$	$\pm \left[{\displaystyle \frac{{\mathsf{\Lambda }}^{1/4}}{2}}\right]$
3	$\mp \left[{\displaystyle \frac{\left(-1+\mathsf{\tau }{\mathsf{\nu }}_2\sqrt{\mathsf{\Lambda }}\right)}{3\mathsf{\tau }{\mathsf{\Lambda }}^{1/4}}}\right]$	0	0	0	$\pm \left[{\displaystyle \frac{{\mathsf{\nu }}_3}{{\mathsf{\Lambda }}^{3/4}{\mathsf{\tau }}^2\left(3{\mathsf{\nu }}_1{\mathsf{\nu }}_3-{\mathsf{\nu }}_2^2\right)}}\right]$	$\mathsf{\tau }$	$\pm \left[{\displaystyle \frac{{\mathsf{\Lambda }}^{1/4}}{2}}\right]$

.

In Table 2,

$$\begin{array}{c}\hfill \mathsf{\Lambda }=-\left[{\displaystyle \frac{1}{{\mathsf{\tau }}^2\left(3{\mathsf{\nu }}_1{\mathsf{\nu }}_3-{\mathsf{\nu }}_2^2\right)}}\right].\end{array}$$

Replacing the obtained values in Equation (10) and integrating once with respect to $\mathsf{\xi }$ then using Equation (3) leads to the exact solution of Equation (1). The proposed scheme is given in Algorithm 1.

Algorithm 1 Description of proposed scheme

Step 1. Convert the given Equation (2) to Equation (4) using Equation (3). Step 2. Assume the initial solution of form Equation (5) to Equation (4) through balancing of Equation (4). Step 3. Substitute Step 2 into Step 1 and using Equation (6) form the polynomial. Step 4. Substitution of initial assumed solution into Equation (8) does not solve the algebraic system. Step 5. Convert the Equation (8) to Equation (9) through ${\displaystyle \frac{\mathsf{d}}{\mathsf{d}\mathsf{\xi }}}\mathsf{\vartheta }\left(\mathsf{\xi }\right)=\mathsf{\varpi }\left(\mathsf{\xi }\right)$. Step 6. Again assume the initial solution of Equation (9) as Equation (10) and proceed Steps 2 and 3. Step 7. Makeshift the computed unknowns in Equation (10) and integrate one time with respect to $\mathsf{\xi }$. Step 8. Apply Equation (3) in Step 7. Step 9. Exact solution of Equation (2) is obtained after simplification.

4. The Jacobi Elliptic Solutions of the Wazwaz Negative-Order KDV Equation

In this section, the exact Jacobi elliptic solutions of Equation (1) are classified based on sets of Table 2 and reported uniquely.

Set 1:

From Set 1 of Table 2, the initial solution of Equation (9) is given by

$$\begin{array}{c}\hfill \mathsf{\varpi }\left(\mathsf{\xi }\right)={\mathsf{\rho }}_0++{\mathsf{\rho }}_2{\mathsf{F}}^2\left(\mathsf{\xi }\right)+{\displaystyle \frac{{\mathsf{\sigma }}_2}{{\mathsf{F}}^2\left(\mathsf{\xi }\right)}}.\end{array}$$

(11)

Using entries 1–12 from Table 1 in Equation (11) captures the following first set of exact solutions of Equation (1).

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,1\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{1}{3\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_1\right)}^{1/4}\mathrm{sn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)}}\right]\left\{\left({\mathsf{\xi }}_1\sqrt{{\mathsf{\Omega }}_1}+6{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_1\left({\mathsf{\lambda }}^2-5\right)\right)\right.\hfill \\ \hfill & \left.\times \mathrm{sn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)+3\mathrm{cn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,2\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{1}{3\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_2\right)}^{1/4}\mathrm{dn}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)}}\right]\left\{\left({\mathsf{\xi }}_2\sqrt{{\mathsf{\Omega }}_2}+6{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_2\left({\mathsf{\lambda }}^2-2\right)\right)\right.\hfill \\ \hfill & \left.\times \mathrm{dn}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)-3{\mathsf{\lambda }}^2\mathrm{sn}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,3\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{4}{3\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_3\right)}^{1/4}\mathrm{nd}\left({\mathsf{\xi }}_3,\mathsf{\lambda }\right)\mathrm{sd}\left({\mathsf{\xi }}_3,\mathsf{\lambda }\right)}}\right]\left\{\left({\displaystyle \frac{{\mathsf{\xi }}_3}{4}}\sqrt{{\mathsf{\Omega }}_3}+{\displaystyle \frac{3}{2}}{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_3,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_3\left({\mathsf{\lambda }}^2-{\displaystyle \frac{5}{4}}\right)\right)\right.\hfill \\ \hfill & \left.\times \mathrm{nd}\left({\mathsf{\xi }}_3,\mathsf{\lambda }\right)\mathrm{sd}\left({\mathsf{\xi }}_3,\mathsf{\lambda }\right)+{\displaystyle \frac{3}{2}}\left({\displaystyle \frac{1}{2}}{\mathrm{nd}}^2\left({\mathsf{\xi }}_3,\mathsf{\lambda }\right)-{\mathsf{\lambda }}^2{\mathrm{sd}}^2\left({\mathsf{\xi }}_3,\mathsf{\lambda }\right)\right)\mathrm{cd}\left({\mathsf{\xi }}_3,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,4\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{1}{3\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_2\right)}^{1/4}\mathrm{cs}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)\mathrm{ns}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)}}\right]\left\{\left({\mathsf{\xi }}_2\sqrt{{\mathsf{\Omega }}_2}+6{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_2\left({\mathsf{\lambda }}^2-2\right)\right)\right.\hfill \\ \hfill & \left.\times \mathrm{cs}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)\mathrm{ns}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)+3\left(2{\mathrm{cs}}^2\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)-{\mathrm{ns}}^2\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)\right)\mathrm{ds}\left({\mathsf{\xi }}_2,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,5\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{1}{3\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_4\right)}^{1/4}\mathrm{ns}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)}}\right]\left\{\left({\mathsf{\xi }}_4\sqrt{{\mathsf{\Omega }}_4}+3{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_4\left({\mathsf{\lambda }}^2-2\right)\right)\right.\hfill \\ \hfill & \left.\times \mathrm{ns}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)+3\mathrm{cs}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\mathrm{ds}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,6\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{1}{12\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_5\right)}^{1/4}{\mathsf{\lambda }}^2\mathrm{ns}\left({\mathsf{\xi }}_5,\mathsf{\lambda }\right)}}\right]\left\{\left(2{\mathsf{\lambda }}^2{\mathsf{\xi }}_5\sqrt{{\mathsf{\Omega }}_5}+6\left({\mathsf{\lambda }}^2+1\right){\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_5,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right.\right.\hfill \\ \hfill & \left.\left.+6\left(\pm {\mathsf{\lambda }}^2\mp 1\right)\mathrm{cs}\left({\mathsf{\xi }}_5,\mathsf{\lambda }\right)+{\mathsf{\xi }}_5\left({\mathsf{\lambda }}^4+{\mathsf{\lambda }}^2-6\right)\right)\mathrm{ns}\left({\mathsf{\xi }}_5,\mathsf{\lambda }\right)+6\left({\mathsf{\lambda }}^2+1\right)\mathrm{cs}\left({\mathsf{\xi }}_5,\mathsf{\lambda }\right)\mathrm{ds}\left({\mathsf{\xi }}_5,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,7\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{1}{3\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_4\right)}^{1/4}\mathrm{nd}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)}}\right]\left\{\left({\mathsf{\xi }}_4\sqrt{{\mathsf{\Omega }}_4}+3{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_4\left({\mathsf{\lambda }}^2-2\right)\right)\right.\hfill \\ \hfill & \left.\times \mathrm{nd}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)-3{\mathsf{\lambda }}^2\mathrm{cd}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\mathrm{sd}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,8\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{1}{2\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_4\right)}^{1/4}}}\right]\left\{\pm \left({\displaystyle \frac{{\mathsf{\xi }}_4\left(2\left(2{\mathsf{\lambda }}^2-1\right)+4\sqrt{{\mathsf{\Omega }}_4}\right)}{3}}\right)\mp \left({\displaystyle \frac{1}{\mathrm{cn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)}}\right)\right.\hfill \\ \hfill & \times \left[\left({\mathsf{\xi }}_4-2{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right)\mathrm{cn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)+\mathrm{sn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\right]\hfill \\ \hfill & \mp \left[\left({\displaystyle \frac{\left(2{\mathsf{\lambda }}^2{\mathrm{cn}}^2\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)-2{\mathsf{\lambda }}^2+1\right)\mathrm{sn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)}{{\mathsf{\lambda }}^2{\mathrm{cn}}^4\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)-\left(2{\mathsf{\lambda }}^2-1\right){\mathrm{cn}}^2\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)+{\mathsf{\lambda }}^2-1}}\right)+\left(2{\mathsf{\lambda }}^2-1\right){\mathsf{\xi }}_4+2\left(1-{\mathsf{\lambda }}^2\right){\mathsf{\xi }}_4\right.\hfill \\ \hfill & \left.\left.-2{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right]\right\}.\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,9\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{1}{3\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_6\right)}^{1/4}\left({\mathsf{\lambda }}^2{\mathrm{cn}}^2\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)-{\mathsf{\lambda }}^2+1\right)\left(\mathsf{\lambda }{\mathrm{cn}}^2\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)-\mathsf{\lambda }+1\right)\mathrm{cn}\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)}}\right]\hfill \\ \hfill & \times \left\{\left[{\mathsf{\xi }}_6\sqrt{{\mathsf{\Omega }}_6}+12{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right]\right.\hfill \\ \hfill & \times \left({\mathsf{\lambda }}^2{\mathrm{cn}}^2\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)-{\mathsf{\lambda }}^2+1\right)\left(\mathsf{\lambda }{\mathrm{cn}}^2\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)-\mathsf{\lambda }+1\right)\mathrm{cn}\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)+{\mathsf{\xi }}_6{\mathsf{\lambda }}^3\left({\mathsf{\lambda }}^2-6\mathsf{\lambda }-11\right){\mathrm{cn}}^5\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)\hfill \\ \hfill & -12{\mathsf{\lambda }}^3\mathrm{sn}\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right){\mathrm{cn}}^4\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)-2{\mathsf{\xi }}_6\mathsf{\lambda }\left(\mathsf{\lambda }-1\right)\left({\mathsf{\lambda }}^2-6\mathsf{\lambda }-11\right)\left(\mathsf{\lambda }+{\displaystyle \frac{1}{2}}\right){\mathrm{cn}}^3\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)\hfill \\ \hfill & +15\mathsf{\lambda }\left(\mathsf{\lambda }-1\right)\left(\mathsf{\lambda }+{\displaystyle \frac{3}{5}}\right)\mathrm{sn}\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right){\mathrm{cn}}^2\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)+{\mathsf{\xi }}_6{\left(\mathsf{\lambda }-1\right)}^2\left(\mathsf{\lambda }+1\right)\left({\mathsf{\lambda }}^2-6\mathsf{\lambda }-11\right)\mathrm{cn}\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)\hfill \\ \hfill & \left.-3{\left(\mathsf{\lambda }-1\right)}^2\left(\mathsf{\lambda }+1\right)\mathrm{sn}\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_6,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,10\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{1}{3\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_4\right)}^{1/4}\left(\mathrm{sn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)+1\right)\left(\mathrm{sn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)-1\right)}}\right]\left\{\left({\mathsf{\xi }}_4{\mathrm{sn}}^2\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)-{\mathsf{\xi }}_4\right)\sqrt{{\mathsf{\Omega }}_4}+\left(3\left({\mathrm{sn}}^2\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)-1\right)\right)\right.\hfill \\ \hfill & \left.\times {\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_4\left({\mathsf{\lambda }}^2-2\right){\mathrm{sn}}^2\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)+{\mathsf{\xi }}_4\left(2-{\mathsf{\lambda }}^2\right)+3\mathrm{sn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_4,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,11\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{1}{3\sqrt{\mathsf{\tau }}{\left({\mathsf{\Omega }}_1\right)}^{1/4}\left({\mathsf{\lambda }}^2{\mathrm{sn}}^2\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)-1\right)\left({\mathrm{sn}}^2\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)-1\right)}}\right]\left\{{\mathsf{\xi }}_1\left({\mathsf{\lambda }}^2{\mathrm{sn}}^4\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)-\left({\mathsf{\lambda }}^2+1\right){\mathrm{sn}}^2\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)+1\right)\right.\hfill \\ \hfill & \times \sqrt{{\mathsf{\Omega }}_1}+6\left({\mathsf{\lambda }}^2{\mathrm{sn}}^4\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)-\left({\mathsf{\lambda }}^2+1\right){\mathrm{sn}}^2\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)+1\right){\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\hfill \\ \hfill & +{\mathsf{\xi }}_1{\mathsf{\lambda }}^2\left({\mathsf{\lambda }}^2-5\right){\mathrm{sn}}^4\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)+6{\mathsf{\lambda }}^2{\mathrm{sn}}^3\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)+{\mathsf{\xi }}_1\left(-{\mathsf{\lambda }}^4+4{\mathsf{\lambda }}^2+5\right){\mathrm{sn}}^2\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)\hfill \\ \hfill & \left.-3\left({\mathsf{\lambda }}^2+1\right)\mathrm{sn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_1,\mathsf{\lambda }\right)+{\mathsf{\xi }}_1\left({\mathsf{\lambda }}^2-5\right)\right\}.\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[1,12\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{4}{3\sqrt{\mathsf{\tau }}{\left(\mathsf{\lambda }+1\right)}^2{\left({\mathsf{\Omega }}_7\right)}^{1/4}\left({\mathsf{\lambda }}^2{\mathrm{sn}}^4\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)-1\right)}}\right]\left\{{\displaystyle \frac{{\mathsf{\xi }}_7}{2}}{\left(\mathsf{\lambda }+1\right)}^2\left({\mathsf{\lambda }}^2{\mathrm{sn}}^4\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)-1\right)\sqrt{{\mathsf{\Omega }}_7}\right.\hfill \\ \hfill & +3{\mathsf{\lambda }}^2\left({\mathsf{\lambda }}^2{\mathrm{sn}}^4\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)-1\right){\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_7{\mathsf{\lambda }}^2\left(\mathsf{\lambda }+1\right)\left({\mathsf{\lambda }}^3-2{\mathsf{\lambda }}^2-{\displaystyle \frac{1}{2}}\mathsf{\lambda }-{\displaystyle \frac{1}{2}}\right){\mathrm{sn}}^4\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)\hfill \\ \hfill & +3{\mathsf{\lambda }}^4{\mathrm{sn}}^3\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)+6\mathsf{\lambda }\left(\mathsf{\lambda }+{\displaystyle \frac{1}{2}}\right)\mathrm{sn}\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_7,\mathsf{\lambda }\right)\hfill \\ \hfill & \left.-{\mathsf{\xi }}_7\left(\mathsf{\lambda }+1\right)\left({\mathsf{\lambda }}^3-2{\mathsf{\lambda }}^2-{\displaystyle \frac{1}{2}}\mathsf{\lambda }-{\displaystyle \frac{1}{2}}\right)\right\}.\hfill \end{array}$$

(23)

In Equations (12) and (22), ${\mathsf{\Omega }}_1=\left({\mathsf{\lambda }}^4+14{\mathsf{\lambda }}^2+1\right)$; in Equations (13) and (15), ${\mathsf{\Omega }}_2=\left({\mathsf{\lambda }}^4-16{\mathsf{\lambda }}^2+16\right)$; in Equation (14), ${\mathsf{\Omega }}_3=\left(16{\mathsf{\lambda }}^4-16{\mathsf{\lambda }}^2+1\right)$; in Equations (16), (18), (19) and (22), ${\mathsf{\Omega }}_4=\left({\mathsf{\lambda }}^4-{\mathsf{\lambda }}^2+1\right)$; in Equation (17), ${\mathsf{\Omega }}_5=\left({\mathsf{\lambda }}^4-{\mathsf{\lambda }}^2+4\right)$; in Equation (20), ${\mathsf{\Omega }}_6=\left({\mathsf{\lambda }}^4+60{\mathsf{\lambda }}^3+134{\mathsf{\lambda }}^2+60\mathsf{\lambda }+1\right)$; and in Equation (23), ${\mathsf{\Omega }}_7=\left(16{\mathsf{\lambda }}^4-48{\mathsf{\lambda }}^3+56{\mathsf{\lambda }}^2-8\right)$. Solutions ${\mathsf{{\rm Y}}}_{\left[1,1\right]}\left(\mathtt{x},\mathtt{t}\right)-{\mathsf{{\rm Y}}}_{\left[1,12\right]}\left(\mathtt{x},\mathtt{t}\right)$ Equations (12)–(23), ${\mathsf{\xi }}_i;i=1,2,\cdots ,7$ are defined in

Table 3. ${\mathsf{\xi }}_i$ for $i=1,2,\cdots ,7$.

i	${\mathsf{\xi }}_{\mathit{i}}$
1	$\pm \left[{\displaystyle \frac{\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)}{2\sqrt{\mathsf{\tau }}{\left({\mathsf{\lambda }}^4+14{\mathsf{\lambda }}^2+1\right)}^{1/4}}}\right]$
2	$\pm \left[{\displaystyle \frac{\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)}{2\sqrt{\mathsf{\tau }}{\left({\mathsf{\lambda }}^4-16{\mathsf{\lambda }}^2+16\right)}^{1/4}}}\right]$
3	$\pm \left[{\displaystyle \frac{\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)}{2\sqrt{\mathsf{\tau }}{\left(16{\mathsf{\lambda }}^4-16{\mathsf{\lambda }}^2+1\right)}^{1/4}}}\right]$
4	$\pm \left[{\displaystyle \frac{\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)}{2\sqrt{\mathsf{\tau }}{\left({\mathsf{\lambda }}^4-{\mathsf{\lambda }}^2+1\right)}^{1/4}}}\right]$
5	$\pm \left[{\displaystyle \frac{\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)}{2\sqrt{\mathsf{\tau }}{\left(\frac{1}{4}{\mathsf{\lambda }}^4-\frac{1}{4}{\mathsf{\lambda }}^2+1\right)}^{1/4}}}\right]$
6	$\pm \left[{\displaystyle \frac{\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)}{2\sqrt{\mathsf{\tau }}{\left({\mathsf{\lambda }}^4+60{\mathsf{\lambda }}^3+134{\mathsf{\lambda }}^2+60\mathsf{\lambda }+1\right)}^{1/4}}}\right]$
7	$\pm \left[{\displaystyle \frac{\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)}{2\sqrt{\mathsf{\tau }}{\left(16{\mathsf{\lambda }}^4-48{\mathsf{\lambda }}^3+56{\mathsf{\lambda }}^2-8\right)}^{1/4}}}\right]$

.

When the Jacobi elliptic modulus $\mathsf{\lambda }\to 0$,

• Solutions Equations (12), (14)–(16), (19)–(22) degenerate into periodic functions solutions of Equation (1).
• Solutions Equations (2) and (7) degenerate into rational function solutions of Equation (1).
• Solution Equation (23) degenerates into complex rational function solution of Equation (1).
• Solution Equation (17) is undefined due to division by zero; hence, the existence condition of ${\mathsf{{\rm Y}}}_{\left[1,6\right]}\left(\mathtt{x},\mathtt{t}\right)$ is $\mathsf{\lambda }\ne 0$.

When the Jacobi modulus $\mathsf{\lambda }\to 1$,

• Solutions Equations (12)–(17), (19) and (23) degenerate into singular solutions.
• Solutions Equations (18) and (20)–(22) vanish to zero. Hence, $\mathsf{\lambda }\ne 1$ for the solutions ${\mathsf{{\rm Y}}}_{\left[1,7\right]}\left(\mathtt{x},\mathtt{t}\right)$, ${\mathsf{{\rm Y}}}_{\left[1,9\right]}\left(\mathtt{x},\mathtt{t}\right)$, ${\mathsf{{\rm Y}}}_{\left[1,10\right]}\left(\mathtt{x},\mathtt{t}\right)$, ${\mathsf{{\rm Y}}}_{\left[1,11\right]}\left(\mathtt{x},\mathtt{t}\right)$ exists.

Set 2:

From Set 2 of Table 2, the initial solution of Equation (9) is given by

$$\begin{array}{c}\hfill \mathsf{\varpi }\left(\mathsf{\xi }\right)={\mathsf{\rho }}_0++{\mathsf{\rho }}_2{\mathsf{F}}^2\left(\mathsf{\xi }\right).\end{array}$$

(24)

Applying entries 1–12 in Table 1 in Equation (24) gives the following second set of exact solutions of Equation (1).

$$\begin{array}{c}\hfill {\mathsf{{\rm Y}}}_{\left[2,1\right]}\left(\mathtt{x},\mathtt{t}\right)=\pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_1\right)}^{1/4}}{3\mathsf{\tau }}}\right]\left\{\mathsf{\tau }\left(3{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_8\left({\mathsf{\lambda }}^2-2\right)\right)\sqrt{{\mathsf{\Lambda }}_1}+{\mathsf{\xi }}_8\right\}.\end{array}$$

(25)

$$\begin{array}{c}\hfill {\mathsf{{\rm Y}}}_{\left[2,2\right]}\left(\mathtt{x},\mathtt{t}\right)=\pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_1\right)}^{1/4}}{3\mathsf{\tau }}}\right]\left\{\mathsf{\tau }\left(3{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_8\left({\mathsf{\lambda }}^2-2\right)\right)\sqrt{{\mathsf{\Lambda }}_1}+{\mathsf{\xi }}_8\right\}.\end{array}$$

(26)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[2,3\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_1\right)}^{1/4}}{3\mathsf{\tau }\mathrm{nd}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}}\right]\left\{\left[\left({\mathsf{\xi }}_8\left({\mathsf{\lambda }}^2-2\right)+3{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right)\mathrm{nd}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right.\right.\hfill \\ \hfill & \left.\left.-3{\mathsf{\lambda }}^2\mathrm{cd}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\mathrm{sd}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right]\mathsf{\tau }\sqrt{{\mathsf{\Lambda }}_1}+{\mathsf{\xi }}_8\mathrm{nd}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[2,4\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_1\right)}^{1/4}}{3\mathsf{\tau }\mathrm{ns}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}}\right]\left\{\left[\left({\mathsf{\xi }}_8\left({\mathsf{\lambda }}^2-2\right)+3{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right)\mathrm{ns}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right.\right.\hfill \\ \hfill & \left.\left.+3\mathrm{cs}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\mathrm{ds}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right]\mathsf{\tau }\sqrt{{\mathsf{\Lambda }}_1}+{\mathsf{\xi }}_8\mathrm{ns}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[2,5\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{16}^{3/4}{\left({\mathsf{\Lambda }}_2\right)}^{1/4}}{48\mathsf{\tau }\mathrm{ns}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)}}\right]\left\{4\mathsf{\tau }\left[\left({\mathsf{\xi }}_9\left({\mathsf{\lambda }}^2-{\displaystyle \frac{5}{4}}\right)\pm {\displaystyle \frac{3}{2}}\mathrm{ds}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)+{\displaystyle \frac{3}{2}}{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right)\mathrm{ns}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\right.\right.\hfill \\ \hfill & \left.\left.+{\displaystyle \frac{3}{2}}\mathrm{cs}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\mathrm{ds}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\right]\sqrt{{\mathsf{\Lambda }}_2}+{\mathsf{\xi }}_9\mathrm{ns}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[2,6\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{16}^{3/4}{\left({\mathsf{\Lambda }}_3\right)}^{1/4}}{192\mathsf{\tau }\mathrm{ns}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)}}\right]\left\{4\mathsf{\tau }\left[\left({\mathsf{\xi }}_{10}\left({\mathsf{\lambda }}^2-2\right)\pm 6\mathrm{cs}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)+6{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right)\mathrm{ns}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)\right.\right.\hfill \\ \hfill & \left.\left.+6\mathrm{cs}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)\mathrm{ds}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)\right]\sqrt{{\mathsf{\Lambda }}_3}+4{\mathsf{\xi }}_{10}\mathrm{ns}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[2,7\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{16}^{3/4}{\left({\mathsf{\Lambda }}_4\right)}^{1/4}}{192\mathsf{\tau }\mathrm{nd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)}}\right]\left\{4\mathsf{\tau }\left[\left({\mathsf{\xi }}_{11}\left({\mathsf{\lambda }}^2-5\right)\mp 6\mathrm{cd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)+6{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right)\mathrm{nd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right.\right.\hfill \\ \hfill & \left.\left.-6{\mathsf{\lambda }}^2\mathrm{cd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\mathrm{sd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right]\sqrt{{\mathsf{\Lambda }}_4}+4{\mathsf{\xi }}_{11}\mathrm{nd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[2,8\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{4{\left({\mathsf{\Lambda }}_2\right)}^{1/4}}{3\mathsf{\tau }\mathrm{cn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)}}\right]\left\{\mathsf{\tau }\left[\left({\mathsf{\xi }}_9\left({\mathsf{\lambda }}^2-{\displaystyle \frac{5}{4}}\right)+{\displaystyle \frac{3}{2}}{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right)\mathrm{cn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\right.\right.\hfill \\ \hfill & \left.\left.-{\displaystyle \frac{3}{4}}\mathrm{sn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\right]\sqrt{{\mathsf{\Lambda }}_2}+{\displaystyle \frac{{\mathsf{\xi }}_9}{4}}\mathrm{cn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[2,9\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_4\right)}^{1/4}}{3\mathsf{\tau }\left(\mathsf{\lambda }{\mathrm{cn}}^2\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)-\mathsf{\lambda }+1\right)}}\right]\left\{\mathsf{\tau }\left[{\mathsf{\xi }}_{11}\mathsf{\lambda }\left({\mathsf{\lambda }}^2-5\right){\mathrm{cn}}^2\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)+6\left(\mathsf{\lambda }{\mathrm{cn}}^2\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)-\mathsf{\lambda }+1\right)\right.\right.\hfill \\ \hfill & \left.\times {\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }-6\mathsf{\lambda }\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)-{\mathsf{\xi }}_{11}\left({\mathsf{\lambda }}^2-5\right)\left(\mathsf{\lambda }-1\right)\right]\hfill \\ \hfill & \left.\times \sqrt{{\mathsf{\Lambda }}_4}+{\mathsf{\xi }}_{11}\left(\mathsf{\lambda }{\mathrm{cn}}^2\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)-\mathsf{\lambda }+1\right)\right\}.\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[2,10\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{16}^{3/4}{\left({\mathsf{\Lambda }}_4\right)}^{1/4}}{192\mathsf{\tau }\left(\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\pm 1\right)}}\right]\left\{4\mathsf{\tau }\left[{\mathsf{\xi }}_{11}\left({\mathsf{\lambda }}^2-5\right)\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)+6\left(\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\pm 1\right)\right.\right.\hfill \\ \hfill & \left.\times {\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+6\mathsf{\lambda }\mathrm{cn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\pm {\mathsf{\xi }}_{11}\left({\mathsf{\lambda }}^2-5\right)\right]\hfill \\ \hfill & \left.\times \sqrt{{\mathsf{\Lambda }}_4}+4{\mathsf{\xi }}_{11}\left(\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\pm 1\right)\right\}.\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[2,11\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{16}^{3/4}{\left({\mathsf{\Lambda }}_5\right)}^{1/4}}{192\mathsf{\tau }\left(\mathsf{\lambda }\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)+1\right)\left(\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)+1\right)}}\right]\left\{48\mathsf{\tau }\left(\mathsf{\lambda }\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)+1\right)\left(\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)+1\right)\right.\hfill \\ \hfill & \times {\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+4\mathsf{\tau }{\mathsf{\xi }}_{12}\mathsf{\lambda }\left({\mathsf{\lambda }}^2-6\mathsf{\lambda }-11\right){\mathrm{sn}}^2\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)\hfill \\ \hfill & +4\mathsf{\tau }\left(12\mathsf{\lambda }\mathrm{cn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)+{\mathsf{\xi }}_{12}\left(\mathsf{\lambda }+1\right)\left({\mathsf{\lambda }}^2-6\mathsf{\lambda }-11\right)\right)\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)\hfill \\ \hfill & +24\mathsf{\tau }\left(\mathsf{\lambda }+1\right)\mathrm{cn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)+4{\mathsf{\xi }}_{12}\mathsf{\tau }\left({\mathsf{\lambda }}^2-6\mathsf{\lambda }-11\right)\sqrt{{\mathsf{\Lambda }}_5}\hfill \\ \hfill & \left.+4{\mathsf{\xi }}_{12}\left(\mathsf{\lambda }\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)+1\right)\left(\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)+1\right)\right\}.\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[2,12\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_6\right)}^{1/4}}{3\mathsf{\tau }\left(\mathsf{\lambda }{\mathrm{sn}}^2\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)-1\right)}}\right]\left\{\mathsf{\tau }\left[{\mathsf{\xi }}_{13}\mathsf{\lambda }\left({\mathsf{\lambda }}^2-5\right){\mathrm{sn}}^2\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)+6\left(\mathsf{\lambda }{\mathrm{sn}}^2\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)-1\right)\right.\right.\hfill \\ \hfill & \left.\times {\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+6\mathsf{\lambda }\mathrm{sn}\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)-{\mathsf{\xi }}_{13}\left({\mathsf{\lambda }}^2-5\right)\right]\hfill \\ \hfill & \left.\times \sqrt{{\mathsf{\Lambda }}_6}+{\mathsf{\xi }}_{13}\left(\mathsf{\lambda }{\mathrm{sn}}^2\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)-1\right)\right\}.\hfill \end{array}$$

(36)

In solutions ${\mathsf{{\rm Y}}}_{\left[2,1\right]}\left(\mathtt{x},\mathtt{t}\right)-{\mathsf{{\rm Y}}}_{\left[2,12\right]}\left(\mathtt{x},\mathtt{t}\right)$, Equations (25)–(36), ${\mathsf{\xi }}_i;i=8,9,\cdots ,13$ are defined in

Table 4. ${\mathsf{\xi }}_i$ for $i=8,9,\cdots ,13$.

i	${\mathsf{\xi }}_{\mathit{i}}$
8	$\pm \left[{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left({\mathsf{\lambda }}^4-{\mathsf{\lambda }}^2+1\right)}}\right)}^{1/4}\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)\right]$
9	$\pm \left[{\displaystyle \frac{{16}^{1/4}}{2}}{\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left(16{\mathsf{\lambda }}^4-16{\mathsf{\lambda }}^2+1\right)}}\right)}^{1/4}\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)\right]$
10	$\pm \left[{\displaystyle \frac{{16}^{1/4}}{2}}{\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left(4{\mathsf{\lambda }}^4-19{\mathsf{\lambda }}^2+16\right)}}\right)}^{1/4}\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)\right]$
11	$\pm \left[{\displaystyle \frac{{16}^{1/4}}{2}}{\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left({\mathsf{\lambda }}^4+14{\mathsf{\lambda }}^2+1\right)}}\right)}^{1/4}\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)\right]$
12	$\pm \left[{\displaystyle \frac{{16}^{1/4}}{2}}{\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left({\mathsf{\lambda }}^4+60{\mathsf{\lambda }}^3+134{\mathsf{\lambda }}^2+60\mathsf{\lambda }+1\right)}}\right)}^{1/4}\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)\right]$
13	$\pm \left[{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left({\mathsf{\lambda }}^4+12{\mathsf{\lambda }}^3-4{\mathsf{\lambda }}^2+7\right)}}\right)}^{1/4}\left(\mathtt{x}-\mathsf{\lambda }\mathtt{t}\right)\right]$

.

When the Jacobi modulus $\mathsf{\lambda }\to 0$,

• Solutions Equations (25)–(27), (31), (33) and (36) reduce to rational functions.
• Solutions Equations (28)–(30), (32), (34) and (35) reduce to periodic functions.

When the Jacobi modulus $\mathsf{\lambda }\to 1$,

• Solutions Equations (25)–(26), (28)–(30) and (32) reduce to solitons.
• Solutions Equations (27), (33) and (36) vanish to zero, so the existence conditions of ${\mathsf{{\rm Y}}}_{\left[2,3\right]}\left(\mathtt{x},\mathtt{t}\right),{\mathsf{{\rm Y}}}_{\left[2,9\right]}\left(\mathtt{x},\mathtt{t}\right),{\mathsf{{\rm Y}}}_{\left[2,12\right]}\left(\mathtt{x},\mathtt{t}\right)$ are $\mathsf{\lambda }\ne 1$.
• Solutions Equations (31), (34) and (35) reduce to rational functions.

Set 3:

From Set 3 of Table 2, the initial solution of Equation (9) is given by

$$\begin{array}{c}\hfill \mathsf{\varpi }\left(\mathsf{\xi }\right)={\mathsf{\rho }}_0+{\displaystyle \frac{{\mathsf{\sigma }}_2}{{\mathsf{F}}^2\left(\mathsf{\xi }\right)}}.\end{array}$$

(37)

Using entries 1–12 from Table 1 in the Equation (37) yields the following third set of exact solutions of Equation (1).

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,1\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_1\right)}^{1/4}}{3\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}}\right]\left\{\left[{\mathsf{\xi }}_8\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_1}}}+3{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_8\left({\mathsf{\lambda }}^2-2\right)\right]\right.\hfill \\ \hfill & \times \left.\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)+3\mathrm{cn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,2\right]}\left(\mathtt{x},\mathtt{t}\right)=& \mp \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_1\right)}^{1/4}}{3\mathrm{dn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}}\right]\left\{\left[{\mathsf{\xi }}_8\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_1}}}+3{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_8\left({\mathsf{\lambda }}^2-2\right)\right]\right.\hfill \\ \hfill & \times \left.\mathrm{dn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)-3{\mathsf{\lambda }}^2\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,3\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_1\right)}^{1/4}}{3\mathrm{nd}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\mathrm{sd}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}}\right]\left\{\left[{\mathsf{\xi }}_8\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_1}}}+3{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_8\left({\mathsf{\lambda }}^2-2\right)\right]\right.\hfill \\ \hfill & \times \left.\mathrm{nd}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\mathrm{sd}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)-3\left({\mathsf{\lambda }}^2{\mathrm{sd}}^2\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)-{\mathrm{nd}}^2\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right)\mathrm{cd}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,4\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_1\right)}^{1/4}}{3\mathrm{cs}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\mathrm{ns}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}}\right]\left\{\left[{\mathsf{\xi }}_8\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_1}}}+3{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\xi }}_8\left({\mathsf{\lambda }}^2-2\right)\right]\right.\hfill \\ \hfill & \times \left.\mathrm{cs}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\mathrm{ns}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)-3\left({\mathrm{ns}}^2\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)-{\mathrm{cs}}^2\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right)\mathrm{ds}\left({\mathsf{\xi }}_8,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,5\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{16}^{1/4}{\left({\mathsf{\Lambda }}_2\right)}^{1/4}}{3\mathrm{ns}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)}}\right]\left\{\left[4{\mathsf{\xi }}_9\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_2}}}+{\displaystyle \frac{3}{2}}{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right.\right.\hfill \\ \hfill & \left.\left.+{\mathsf{\xi }}_9\left({\mathsf{\lambda }}^2-{\displaystyle \frac{5}{4}}\right)\mp {\displaystyle \frac{3}{2}}\mathrm{ds}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\right]\mathrm{ns}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)+{\displaystyle \frac{3}{2}}\mathrm{cs}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\mathrm{ds}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,6\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{16}^{1/4}{\left({\mathsf{\Lambda }}_3\right)}^{1/4}}{12{\mathsf{\lambda }}^2\mathrm{ns}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)}}\right]\left\{\left[{\mathsf{\lambda }}^2{\mathsf{\xi }}_{10}\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_3}}}+6{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right.\right.\hfill \\ \hfill & \left.\left.-{\mathsf{\xi }}_{10}\left(2{\mathsf{\lambda }}^4-7{\mathsf{\lambda }}^2+6\right)\mp 6\mathrm{cs}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)\right]\mathrm{ns}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)+6\mathrm{cs}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)\mathrm{ds}\left({\mathsf{\xi }}_{10},\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,7\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{16}^{1/4}{\left({\mathsf{\Lambda }}_4\right)}^{1/4}}{48\mathrm{nd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)}}\right]\left\{\left[4{\mathsf{\xi }}_{11}\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_4}}}+24{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right.\right.\hfill \\ \hfill & \left.\left.+4{\mathsf{\xi }}_{11}\left({\mathsf{\lambda }}^2-5\right)\pm 24\mathrm{cd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right]\mathrm{nd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)-24{\mathsf{\lambda }}^2\mathrm{cd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\mathrm{sd}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,8\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\left({\mathsf{\Lambda }}_2\right)}^{1/4}\right]\left\{\left[{\displaystyle \frac{\left(2{\mathsf{\lambda }}^2{\mathrm{cn}}^2\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)-2{\mathsf{\lambda }}^2+1\right)\mathrm{sn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)}{{\mathsf{\lambda }}^2{\mathrm{cn}}^4\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)+\left(1-2{\mathsf{\lambda }}^2\right){\mathrm{cn}}^2\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)+{\mathsf{\lambda }}^2-1}}\right]\right.\hfill \\ \hfill & \left.+{\mathsf{\xi }}_9\left(\left(2{\mathsf{\lambda }}^2-1\right)+2\left(1-{\mathsf{\lambda }}^2\right)\right)-2{\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_9,\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }\right\}\pm \left[{\displaystyle \frac{{\mathsf{\xi }}_9}{3\mathsf{\tau }}}\left(-1+2\mathsf{\tau }\left(1-2{\mathsf{\lambda }}^2\right)\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_2}}}\right)\right].\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,9\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_4\right)}^{1/4}}{3\left(1-{\mathsf{\lambda }}^2{\mathrm{sn}}^2\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right)\mathrm{cn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)}}\right]\left\{{\mathsf{\xi }}_{11}\left(1-{\mathsf{\lambda }}^2{\mathrm{sn}}^2\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right)\mathrm{cn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_4}}}\right.\hfill \\ \hfill & +6\left({\mathsf{\lambda }}^2{\mathrm{cn}}^3\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)-\left({\mathsf{\lambda }}^2+1\right)\mathrm{cn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right){\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+{\mathsf{\lambda }}^2{\mathsf{\xi }}_{11}\left({\mathsf{\lambda }}^2-5\right){\mathrm{cn}}^3\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\hfill \\ \hfill & \left.-6{\mathsf{\lambda }}^2\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right){\mathrm{cn}}^2\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)-{\mathsf{\xi }}_{11}\left({\mathsf{\lambda }}^4-6{\mathsf{\lambda }}^2+5\right)\mathrm{cn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)-3\left(1-{\mathsf{\lambda }}^2\right)\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right\}.\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,10\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{16}^{1/4}{\left({\mathsf{\Lambda }}_4\right)}^{1/4}}{48{\mathsf{\tau }}^2\left(\mp 1+\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right)}}\right]\left\{4{\mathsf{\xi }}_{11}\left(\mp 1+\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right)\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_4}}}+24\left(\mp 1+\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\right)\right.\hfill \\ \hfill & \left.\times {\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+4{\mathsf{\xi }}_{11}\left({\mathsf{\lambda }}^2-5\right)\mathrm{sn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)+24\mathrm{cn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{11},\mathsf{\lambda }\right)\mp 4{\mathsf{\xi }}_{11}\left({\mathsf{\lambda }}^2-5\right)\right\}.\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,11\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{16}^{1/4}{\left({\mathsf{\Lambda }}_5\right)}^{1/4}}{48\left(\mathsf{\lambda }\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)-1\right)\left(\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)-1\right)}}\right]\left\{4\mathsf{\xi }\left(\mathsf{\lambda }\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)-1\right)\left(\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)-1\right)\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_5}}}\right.\hfill \\ \hfill & +48\left(\mathsf{\lambda }\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)-1\right)\left(\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)-1\right){\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+4{\mathsf{\xi }}_{12}\mathsf{\lambda }\left({\mathsf{\lambda }}^2-6\mathsf{\lambda }-11\right){\mathrm{sn}}^2\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)\hfill \\ \hfill & +\left(48\mathsf{\lambda }\mathrm{cn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)-4{\mathsf{\xi }}_{12}\left(\mathsf{\lambda }+1\right)\left({\mathsf{\lambda }}^2-6\mathsf{\lambda }-11\right)\right)\mathrm{sn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)\hfill \\ \hfill & \left.-24\left(\mathsf{\lambda }+1\right)\mathrm{cn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{12},\mathsf{\lambda }\right)+4{\mathsf{\xi }}_{12}\left({\mathsf{\lambda }}^2-6\mathsf{\lambda }-11\right)\right\}.\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_{\left[3,12\right]}\left(\mathtt{x},\mathtt{t}\right)=& \pm \left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}_6\right)}^{1/4}}{3{\left(\mathsf{\lambda }+1\right)}^2\left(\mathsf{\lambda }{\mathrm{sn}}^2\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)+1\right)}}\right]\left\{{\mathsf{\xi }}_{13}{\left(\mathsf{\lambda }+1\right)}^2\left(\mathsf{\lambda }{\mathrm{sn}}^2\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)+1\right)\sqrt{{\displaystyle \frac{1}{{\mathsf{\Lambda }}_6}}}\right.\hfill \\ \hfill & +6\left({\mathsf{\lambda }}^2-2\mathsf{\lambda }-1\right)\left(\mathsf{\lambda }{\mathrm{sn}}^2\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)+1\right){\int }_0^{\mathrm{sn}\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)}{\displaystyle \frac{\sqrt{1-{\mathsf{\lambda }}^2{\mathsf{\varphi }}^2}}{\sqrt{1-{\mathsf{\Phi }}^2}}}d\mathsf{\Phi }+\mathsf{\lambda }{\mathsf{\xi }}_{13}\left(\mathsf{\lambda }+1\right)\left({\mathsf{\lambda }}^3-11{\mathsf{\lambda }}^2+\mathsf{\lambda }+1\right)\hfill \\ \hfill & \left.\times {\mathrm{sn}}^2\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)+6\mathsf{\lambda }\left({\mathsf{\lambda }}^2-2\mathsf{\lambda }-1\right)\mathrm{sn}\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)\mathrm{cn}\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)\mathrm{dn}\left({\mathsf{\xi }}_{13},\mathsf{\lambda }\right)+{\mathsf{\xi }}_{13}\left(\mathsf{\lambda }+1\right)\left({\mathsf{\lambda }}^3-11{\mathsf{\lambda }}^2+\mathsf{\lambda }+1\right)\right\}.\hfill \end{array}$$

(49)

In Equations (25)–(28), (38)–(41) ${\mathsf{\Lambda }}_1=\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left({\mathsf{\lambda }}^4-{\mathsf{\lambda }}^2+1\right)}}\right)$, Equations (29), (32), (42) and (45) ${\mathsf{\Lambda }}_2=\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left(16{\mathsf{\lambda }}^4-16{\mathsf{\lambda }}^2+1\right)}}\right)$, Equations (30) and (43) ${\mathsf{\Lambda }}_3=\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left(4{\mathsf{\lambda }}^4-19{\mathsf{\lambda }}^2+16\right)}}\right)$, Equations (31), (33)–(34), (44), (46) and (47) ${\mathsf{\Lambda }}_4=\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left({\mathsf{\lambda }}^4-14{\mathsf{\lambda }}^2+1\right)}}\right)$, Equations (35) and (48) ${\mathsf{\Lambda }}_5=\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left({\mathsf{\lambda }}^4+60{\mathsf{\lambda }}^3+134{\mathsf{\lambda }}^2+60\mathsf{\lambda }+1\right)}}\right)$ and Equations (36) and (49) ${\mathsf{\Lambda }}_6=\left({\displaystyle \frac{1}{{\mathsf{\tau }}^2\left({\mathsf{\lambda }}^4+12{\mathsf{\lambda }}^3-4{\mathsf{\lambda }}^2+7\right)}}\right)$. Solutions ${\mathsf{{\rm Y}}}_{\left[3,1\right]}\left(\mathtt{x},\mathtt{t}\right)-{\mathsf{{\rm Y}}}_{\left[3,12\right]}\left(\mathtt{x},\mathtt{t}\right)$ Equations (38)–(49), ${\mathsf{\xi }}_i;i=8,9,\cdots ,13$ are defined in Table 4. When the Jacobi modulus $\mathsf{\lambda }\to 0$,

• Solutions Equations (38), (40)–(42), (45)–(48) downgrade to periodic functions.
• Solutions Equations (39), (44) and (49) downgrade to rational functions.
• Solution Equation (43) is undefined due to division by zero; hence, $\mathsf{\lambda }\ne 0$ in ${\mathsf{{\rm Y}}}_{\left[3,6\right]}\left(\mathtt{x},\mathtt{t}\right)$ exists.

When the Jacobi modulus $\mathsf{\lambda }\to 1$,

• Solutions Equations (38), (40), (42)–(43), (45) and (49) downgrade to singular solution.
• Solutions Equations (44) and (47)–(48) downgrade to rational functions.
• Solutions Equations (39), (41) and (46) vanish to zero; hence, $\mathsf{\lambda }\ne 1$ in ${\mathsf{{\rm Y}}}_{\left[3,2\right]}\left(\mathtt{x},\mathtt{t}\right)$, ${\mathsf{{\rm Y}}}_{\left[3,4\right]}\left(\mathtt{x},\mathtt{t}\right)$, ${\mathsf{{\rm Y}}}_{\left[3,9\right]}\left(\mathtt{x},\mathtt{t}\right)$ exists.

5. Interpretation of Results and Discussion

In this section, the obtained results are analyzed and interpreted. The two-dimensional plots for the Jacobi elliptic solutions ${\mathsf{{\rm Y}}}_{\left[1,2\right]}\left(\mathtt{x},\mathtt{t}\right),{\mathsf{{\rm Y}}}_{\left[1,12\right]}\left(\mathtt{x},\mathtt{t}\right),{\mathsf{{\rm Y}}}_{\left[2,3\right]}\left(\mathtt{x},\mathtt{t}\right),{\mathsf{{\rm Y}}}_{\left[2,9\right]}\left(\mathtt{x},\mathtt{t}\right),{\mathsf{{\rm Y}}}_{\left[3,7\right]}\left(\mathtt{x},\mathtt{t}\right)$$\mathrm{and}{\mathsf{{\rm Y}}}_{\left[3,12\right]}\left(\mathtt{x},\mathtt{t}\right)$ in the Equations (13), (23), (27), (33), (44) and (49) are drawn. The 2D plots are drawn by taking the spatial axis $\mathtt{x}=1$ and Jacobi elliptic function modulus $\mathsf{\lambda }=0.5$ in the time domain $\mathtt{t}\in \left[-50,50\right]$ for different values of velocity $\mathsf{\tau }=0.1,0.2,0.3,0.4\mathrm{and}0.5$. Also, the 2D plots are drawn by taking the velocity $\mathsf{\tau }=0.5$ in the time domain $\mathtt{t}\in \left[-50,50\right]$ for different values of the Jacobi elliptic function modulus $\mathsf{\lambda }=0.1,0.2,0.3,0.4\mathrm{and}0.5$ in the respective Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6. The following observations are made from the 2D graphs.

• For the modulus $\mathsf{\lambda }=0.5$ when the velocity $\mathsf{\tau }$ increases, the wave propogation ${\mathsf{{\rm Y}}}_{\left[1,2\right]}\left(\mathtt{x},\mathtt{t}\right)$ and ${\mathsf{{\rm Y}}}_{\left[3,12\right]}\left(\mathtt{x},\mathtt{t}\right)$ decreases.
• For the modulus $\mathsf{\lambda }=0.5$ when the velocity $\mathsf{\tau }$ increases, the wave propogation ${\mathsf{{\rm Y}}}_{\left[1,12\right]}\left(\mathtt{x},\mathtt{t}\right),{\mathsf{{\rm Y}}}_{\left[2,3\right]}\left(\mathtt{x},\mathtt{t}\right),{\mathsf{{\rm Y}}}_{\left[2,9\right]}\left(\mathtt{x},\mathtt{t}\right)$ and positive part of ${\mathsf{{\rm Y}}}_{\left[3,7\right]}\left(\mathtt{x},\mathtt{t}\right)$ increases.
• For the velocity $\mathsf{\tau }=0.5$ when the modulus $\mathsf{\lambda }$ increases, the wave propogation ${\mathsf{{\rm Y}}}_{\left[1,2\right]}\left(\mathtt{x},\mathtt{t}\right)$ and ${\mathsf{{\rm Y}}}_{\left[2,3\right]}\left(\mathtt{x},\mathtt{t}\right)$ do not have much variation.
• For the velocity $\mathsf{\tau }=0.5$ when the modulus $\mathsf{\lambda }$ increases, the wave propogation ${\mathsf{{\rm Y}}}_{\left[1,12\right]}\left(\mathtt{x},\mathtt{t}\right),{\mathsf{{\rm Y}}}_{\left[2,9\right]}\left(\mathtt{x},\mathtt{t}\right)$ and positive part of ${\mathsf{{\rm Y}}}_{\left[3,7\right]}\left(\mathtt{x},\mathtt{t}\right)$ increases.
• For the velocity $\mathsf{\tau }=0.5$ when the modulus $\mathsf{\lambda }$ increases, the wave propogation ${\mathsf{{\rm Y}}}_{\left[3,12\right]}\left(\mathtt{x},\mathtt{t}\right)$ decreases.

The above observations of 2D graphs may vary when the Jacobi elliptic function modulus $\mathsf{\lambda }$ and velocity $\mathsf{\tau }$ have different numerical values. This variation also holds while the space $\mathtt{x}$ and time $\mathtt{t}$ take different domains.

The following Jacobi elliptic solutions of Equation (6) do not exist for Wazwaz negative-order KDV Equation (1) since the function $\mathsf{F}\left(\mathsf{\xi }\right)$ is non-integrable.

• For ${\mathsf{\nu }}_1>0$, ${\mathsf{\nu }}_2<0$ and ${\mathsf{\nu }}_3={\displaystyle \frac{{\mathsf{\lambda }}^2{\mathsf{\nu }}_2^2}{{\left(1+{\mathsf{\lambda }}^2\right)}^2{\mathsf{\nu }}_1}}$, $\mathsf{F}\left(\mathsf{\xi }\right)={\left(-{\displaystyle \frac{{\mathsf{\lambda }}^2{\mathsf{\nu }}_2}{\left(1+{\mathsf{\lambda }}^2\right){\mathsf{\nu }}_1}}\right)}^{1/2}\mathrm{sn}\left({\left(-{\displaystyle \frac{{\mathsf{\nu }}_2}{\left(1+{\mathsf{\lambda }}^2\right)}}\right)}^{1/2}\mathsf{\xi },\mathsf{\lambda }\right)$.
• For ${\mathsf{\nu }}_1>0$, ${\mathsf{\nu }}_2<0$ and ${\mathsf{\nu }}_3={\displaystyle \frac{{\mathsf{\lambda }}^2\left({\mathsf{\lambda }}^2-1\right){\mathsf{\nu }}_2^2}{{\left(2{\mathsf{\lambda }}^2-1\right)}^2{\mathsf{\nu }}_1}}$, $\mathsf{F}\left(\mathsf{\xi }\right)={\left(-{\displaystyle \frac{{\mathsf{\lambda }}^2{\mathsf{\nu }}_2}{\left(2{\mathsf{\lambda }}^2-1\right){\mathsf{\nu }}_1}}\right)}^{1/2}\mathrm{cn}\left({\left({\displaystyle \frac{{\mathsf{\nu }}_2}{\left(2{\mathsf{\lambda }}^2-1\right)}}\right)}^{1/2}\mathsf{\xi },\mathsf{\lambda }\right)$.
• For ${\mathsf{\nu }}_1>0$, ${\mathsf{\nu }}_2<0$ and ${\mathsf{\nu }}_3={\displaystyle \frac{\left(1-{\mathsf{\lambda }}^2\right){\mathsf{\nu }}_2^2}{{\left({\mathsf{\lambda }}^2-2\right)}^2{\mathsf{\nu }}_1}}$, $\mathsf{F}\left(\mathsf{\xi }\right)={\left(-{\displaystyle \frac{{\mathsf{\nu }}_2}{\left(2-{\mathsf{\lambda }}^2\right){\mathsf{\nu }}_1}}\right)}^{1/2}\mathrm{dn}\left({\left({\displaystyle \frac{{\mathsf{\nu }}_2}{\left(2-{\mathsf{\lambda }}^2\right)}}\right)}^{1/2}\mathsf{\xi },\mathsf{\lambda }\right)$.

6. Concluding Remarks

Using the recursion operator, the celebrated Korteweg–de Vries (KDV) equation was derived for the opposite direction and named negative-order Korteweg–de Vries equation (nKdV) by Wazwaz. In our present work, the Jacobi elliptic function scheme is employed and the Jacobi elliptic solutions of Wazwaz negative order KdV Equation (1) are obtained. Processing the method described in Section 2 to Equation (1) leads to Equation (8); however, the method given in Section 2 cannot be processed as the algebraic system of equations includes $\sqrt{{\mathsf{\nu }}_1{\mathsf{F}}^4\left(\mathsf{\xi }\right)+{\mathsf{\nu }}_2{\mathsf{F}}^2\left(\mathsf{\xi }\right)+{\mathsf{\nu }}_3}$. So, using the transformation ${\displaystyle \frac{\mathsf{d}}{\mathsf{d}\mathsf{\xi }}}\mathsf{\vartheta }\left(\mathsf{\xi }\right)=\mathsf{\varpi }\left(\mathsf{\xi }\right)$ results in Equation (9) where the Section 2 method is applied. After the implementation, the obtained solution of Equation (9) needs to be integrated once with respect to $\mathsf{\xi }$ and the integral constant is ignored. Since the solution of Equation (9) is in the Jacobi elliptic function, once it is integrated, this results in the incomplete elliptic integral of the second function defined in Definition 1. The detailed steps are given in the Algorithm 1.

Three groups for $\left({\mathsf{\rho }}_0,{\mathsf{\rho }}_2,{\mathsf{\sigma }}_2,\mathsf{\tau },\mathsf{\kappa }\right)$, $\left({\mathsf{\rho }}_0,{\mathsf{\rho }}_2,\mathsf{\tau },\mathsf{\kappa }\right)$ and $\left({\mathsf{\rho }}_0,{\mathsf{\sigma }}_2,\mathsf{\tau },\mathsf{\kappa }\right)$ are reported and the unique Jacobi elliptic function solutions of Equation (1) are obtained. The modular transformations $\mathsf{\lambda }\to 0$ and $\mathsf{\lambda }\to 1$ for each set are given, followed by the two-dimensional plot for selective Jacobi solutions being drawn to show their surfaces. Certain Jacobi elliptic solutions vanish to zero and certain solutions are undefined due to division by zero during the modular transformation process, which is highlighted, and the necessary condition for their existence is given after each set. All the obtained exact solutions are verified and it is ensured that they satisfy the original Wazwaz nKdV Equation (1) by substituting the obtained exact solutions in Equation (1) in Maple. The comparitive study of solutions of Equation (1) is given in

Table 5. Comparision of Wazwaz negative-order KDV solutions.

Ref	Method	Solution
[1]	Complex simplified Hirota	One, two and three complex soliton
[4]	Simplified Hirota direct	One, two and three soliton
This work	Jacobi elliptic function	Jacobi elliptic solution

. To the best of our knowledge, the combined incomplete elliptic integral of second kind and Jacobi elliptic function solutions of Equation (1) are appearing for the first time in our contribution to the literature. In the future, the other Jacobi elliptic function methods like the extended F-expansion method may be applied for Equation (1). The generalized Jacobi elliptic function method given may be applied for Equation (1), which can lead to other forms of Jacobi elliptic solutions.