Exploration of New Optical Solitons in Magneto-Optical Waveguide with Coupled System of Nonlinear Biswas–Milovic Equation via Kudryashov’s Law Using Extended F-Expansion Method

Abstract

Optical soliton solutions in a magneto-optical waveguide and other exact solutions are investigated for the coupled system of the nonlinear Biswas–Milovic equation with Kudryashov’s law using the extended F-expansion method. Various types of solutions are extracted, such as dark soliton solutions, singular soliton solutions, a dark–singular combo soliton, singular combo soliton solutions, Jacobi elliptic solutions, periodic solutions, combo periodic solutions, hyperbolic solutions, rational solutions, exponential solutions and Weierstrass solutions. The obtained different types of wave solutions help in obtaining nonlinear optical fibers in the future. Furthermore, some selected solutions are described graphically to demonstrate the physical nature of the obtained solutions. The results show that the current method gives effectual and direct mathematical tools for resolving the nonlinear problems in the field of nonlinear wave equations.

1. Introduction

Nonlinear evolution equations play a major role in a variety of scientific and engineering fields. The studies of solitary wave solutions for a nonlinear evolution equation attracted many researchers (see [1,2,3]). The theory of optical solitons is the pioneer area of research in the field of nonlinear optics and the telecommunication industry. Studying optical solitons prorogation through waveguides and optical fibers has been of great interest in the research in the past few years (see [4,5,6,7,8,9,10,11,12,13]). Kudryashov’s law of refractive index is used to explain the soliton transmission dynamics across optical fibers. Nikolay Kudryashov proposed six forms of nonlinear refractive index structures (see [14,15,16,17,18]). Recently, the Biswas–Milovic equation (BME) was a generalized version of the nonlinear Schrödinger’s equation (NLSE) (see [19]). Many authors studied the BME by different methods (see [20,21,22,23,24,25,26,27,28,29,30]). Few authors studied the coupled system of a nonlinear BME (see [31,32]).

In the present study, we consider the coupled system of a nonlinear BME with Kudryashov’s refractive index law as follows [31]:

$$i{\left({\mathit{u}}^m\right)}_t+{\mathit{a}}_1{\left({\mathit{u}}^m\right)}_{\mathrm{xx}}+\left(\frac{{\mathit{b}}_1}{{\left|\mathit{u}\right|}^{2n}}+\frac{{\mathit{c}}_1}{{\left|\mathit{u}\right|}^n}+{\mathit{d}}_1{\left|\mathit{u}\right|}^n+{\mathit{e}}_1{\left|\mathit{u}\right|}^{2n}+\frac{{\mathit{f}}_1}{{\left|\mathit{v}\right|}^{2n}}+\frac{{\mathit{g}}_1}{{\left|\mathit{v}\right|}^n}+{\mathit{h}}_1{\left|\mathit{v}\right|}^n+{\mathit{k}}_1{\left|\mathit{v}\right|}^{2n}\right){\mathit{u}}^m$$

$$={\varrho }_1{\mathit{v}}^m+i\left[{\alpha }_1{\left({\mathit{u}}^m{\left|\mathit{u}\right|}^{2n}\right)}_x+{\theta }_1{\left({\left|\mathit{u}\right|}^{2n}\right)}_x{\mathit{u}}^m+{\mu }_1{\left|\mathit{u}\right|}^{2n}{\left({\mathit{u}}^m\right)}_x+{\xi }_1{\left({\mathit{u}}^m\right)}_x\right],$$

(1)

$$i{\left({\mathit{v}}^m\right)}_t+{\mathit{a}}_2{\left({\mathit{v}}^m\right)}_{\mathrm{xx}}+\left(\frac{{\mathit{b}}_2}{{\left|\mathit{v}\right|}^{2n}}+\frac{{\mathit{c}}_2}{{\left|\mathit{v}\right|}^n}+{\mathit{d}}_2{\left|\mathit{v}\right|}^n+{\mathit{e}}_2{\left|\mathit{v}\right|}^{2n}+\frac{{\mathit{f}}_2}{{\left|\mathit{u}\right|}^{2n}}+\frac{{\mathit{g}}_2}{{\left|\mathit{u}\right|}^n}+{\mathit{h}}_2{\left|\mathit{u}\right|}^n+{\mathit{k}}_2{\left|\mathit{u}\right|}^{2n}\right){\mathit{v}}^m$$

$$={\varrho }_2{\mathit{u}}^m+i\left[{\alpha }_2{\left({\mathit{v}}^m{\left|\mathit{v}\right|}^{2n}\right)}_x+{\theta }_2{\left({\left|\mathit{v}\right|}^{2n}\right)}_x{\mathit{v}}^m+{\mu }_2{\left|\mathit{v}\right|}^{2n}{\left({\mathit{v}}^m\right)}_x+{\xi }_2{\left({\mathit{v}}^m\right)}_x\right],$$

(2)

where $\mathit{u}(x,t)$ and $\mathit{v}(x,t)$ are the profiles of the wave; ${\mathit{a}}_l(l=1,2)$ are the coefficients of the chromatic dispersion; ${\mathit{b}}_l,{\mathit{c}}_l,{\mathit{d}}_l$ and ${\mathit{e}}_l(l=1,2)$ are the coefficients of the self-phase modulation; ${\mathit{f}}_l,{\mathit{g}}_l,{\mathit{h}}_l$ and ${\mathit{k}}_l(l=1,2)$ are the cross-phase modulation (XPM); ${\varrho }_l(l=1,2)$ are the coefficients of a magneto-optic; ${\alpha }_l(l=1,2)$ are the coefficients of self-steepening terms; ${\theta }_l$ and ${\mu }_l(l=1,2)$ are the coefficients of the nonlinear dispersion; ${\xi }_l(l=1,2)$ are the coefficients of the inter-modal dispersion; and n and m are the power nonlinearity and the maximum intensity, respectively.

In this work, the extended F-expansion method scheme is introduced to obtain various and novel solutions for the proposed model. These solutions include dark solitons, singular solitons, dark–singular combo solitons, singular combo solitons, Jacobi elliptic solutions, periodic solutions, combo periodic solutions, hyperbolic solutions, rational solutions, exponential solutions and Weierstrass solutions. Moreover, for the physical illustration, some of the obtained solutions are represented graphically.

2. Extended F-Expansion Method

In this section, a brief description of the extended F-expansion method is introduced (see [33,34]). Assuming the nonlinear partial differential equation (NLPDE) as follows:

$$\mathfrak{F}\left(\mathit{u},{\mathit{u}}_t,{\mathit{u}}_{x_1},{\mathit{u}}_{x_1{x}_2},{\mathit{u}}_{t{x}_1},{\mathit{u}}_{t{x}_2},\dots \right)=0.$$

(3)

Let

$$\mathit{u}(t,x_1,x_2,x_3,\dots ,x_n)=\mathcal{H}\left(ϵ\right),ϵ=\sum _{j=1}^n{x}_j+\gamma t,$$

(4)

then, Equation (3) is reduced to:

$$R(\mathcal{H},{\mathcal{H}}^{\prime },{\mathcal{H}}^{\prime \prime },\dots )=0.$$

(5)

The solution of (5) is written in the following form:

$$\mathcal{H}\left(ϵ\right)={\alpha }_0+\sum _{j=1}^N\left({\alpha }_j{\chi }^j\left(ϵ\right)+\frac{{\beta }_i}{{\chi }^j\left(ϵ\right)}\right),$$

(6)

where ${\alpha }_0,{\alpha }_j$ and ${\beta }_j$ are constants, and $\chi$ achieves the following equation:

$${\chi }^{\prime 2}\left(ϵ\right)=s_0+s_1\chi \left(ϵ\right)+s_2{\chi }^2\left(ϵ\right)+s_3{\chi }^3\left(ϵ\right)+s_4{\chi }^4\left(ϵ\right),$$

(7)

where $s_l,(l=0,1,2,3,4)$ are constants.

The integer N in Equation (6) can be estimated from Equation (5) using the balance rule.

Substituting from Equation (6) into Equation (5) with the auxiliary Equation (7), then we obtain a polynomial in $\chi$. In this polynomial, we combine all terms of the same forces and equalize them with zero.

Hence, we obtain a system of algebraic equations that can be solved by Mathematica software to obtain the unknown constants ${\alpha }_0,{\alpha }_j,{\beta }_j(j=1,2,\dots ,N)$, $\gamma$ and $s_l(l=0,1,2,3,4)$. Therefore, various forms of solutions can be raised for Equation (3).

3. Exact Wave Solutions

Assuming the exact traveling wave solutions of the Equations (1) and (2) are:

$$\mathit{u}(x,t)={\mathcal{F}}_1\left(ϵ\right)e^{i(\rho x+\Im t+\vartheta )},$$

(8)

$$\mathit{v}(x,t)={\mathcal{F}}_2\left(ϵ\right)e^{i(\rho x+\Im t+\vartheta )},$$

(9)

where

$$ϵ=x+\gamma t.$$

(10)

The amplitude component of the soliton is ${\mathcal{F}}_j(j=1,2)$. $\gamma ,\rho$ and ℑ are the velocity, frequency and wave number of the soliton, while $\vartheta$ is the phase constant represent.

Applying (8) and (9) in (1) and (2) then splitting the real and imaginary parts, we obtain:

$${\mathit{a}}_{1}m{\mathcal{F}}_1^{m-1}{\mathcal{F}}_1^{\prime \prime }+{\mathit{a}}_{1}m(m-1){\mathcal{F}}_1^{m-2}{\left({\mathcal{F}}_1^{\prime }\right)}^2-m\left({\mathit{a}}_{1}m{\rho }^2-\rho {\xi }_1+\Im \right){\mathcal{F}}_1^m+\left(m\rho \left({\alpha }_1+{\mu }_1\right)+{\mathit{e}}_1\right){\mathcal{F}}_1^{m+2n}$$

$$+{\mathit{b}}_1{\mathcal{F}}_1^{m-2n}+{\mathit{c}}_1{\mathcal{F}}_1^{m-n}+{\mathit{d}}_1{\mathcal{F}}_1^{m+n}+{\mathit{f}}_1{\mathcal{F}}_1^m{\mathcal{F}}_2^{-2n}+{\mathit{g}}_1{\mathcal{F}}_1^m{\mathcal{F}}_2^{-n}+{\mathit{h}}_1{\mathcal{F}}_1^m{\mathcal{F}}_2^n+{\mathit{k}}_1{\mathcal{F}}_1^m{\mathcal{F}}_2^{2n}-{\varrho }_1{\mathcal{F}}_2^m=0,$$

(11)

$${\mathit{a}}_{2}m{\mathcal{F}}_2^{m-1}{\mathcal{F}}_2^{\prime \prime }+{\mathit{a}}_{2}m(m-1){\mathcal{F}}_2^{m-2}{\left({\mathcal{F}}_2^{\prime }\right)}^2-m\left({\mathit{a}}_{2}m{\rho }^2-\rho {\xi }_2+\Im \right){\mathcal{F}}_2^m+\left(m\rho \left({\alpha }_2+{\mu }_2\right)+{\mathit{e}}_2\right){\mathcal{F}}_2^{m+2n}$$

$$+{\mathit{b}}_2{\mathcal{F}}_2^{m-2n}+{\mathit{c}}_2{\mathcal{F}}_2^{m-n}+{\mathit{d}}_2{\mathcal{F}}_2^{m+n}+{\mathit{f}}_2{\mathcal{F}}_2^m{\mathcal{F}}_1^{-2n}+{\mathit{g}}_2{\mathcal{F}}_2^m{\mathcal{F}}_1^{-n}+{\mathit{h}}_2{\mathcal{F}}_2^m{\mathcal{F}}_1^n+{\mathit{k}}_2{\mathcal{F}}_2^m{\mathcal{F}}_1^{2n}-{\varrho }_2{\mathcal{F}}_1^m=0,$$

(12)

and

$$\left[m\left(\gamma +2{\mathit{a}}_{1}m\rho -{\xi }_1\right)-\left(m\left({\alpha }_1+{\mu }_1\right)+2n\left({\alpha }_1+{\theta }_1\right)\right){\mathcal{F}}_1^{2n}\right]{\mathcal{F}}_1^{m-1}{\mathcal{F}}_1^{\prime }=0,$$

(13)

$$\left[m\left(\gamma +2{\mathit{a}}_{2}m\rho -{\xi }_2\right)-\left(m\left({\alpha }_2+{\mu }_2\right)+2n\left({\alpha }_2+{\theta }_2\right)\right){\mathcal{F}}_2^{2n}\right]{\mathcal{F}}_2^{m-1}{\mathcal{F}}_2^{\prime }=0.$$

(14)

When applying the linearly independent principle to Equations (13) and (14), we obtain:

$$m\left({\alpha }_j+{\mu }_j\right)+2n\left({\alpha }_j+{\theta }_j\right)=0,(j=1,2),$$

(15)

$$\gamma ={\xi }_1-2{\mathit{a}}_{1}m\rho ={\xi }_2-2{\mathit{a}}_{2}m\rho .$$

(16)

Through Equation (16), we obtain:

$$\rho =\frac{{\xi }_1-{\xi }_2}{2m\left({\mathit{a}}_1-{\mathit{a}}_2\right)},{\mathit{a}}_1\ne {\mathit{a}}_2.$$

(17)

Setting

$${\mathcal{F}}_1=\mathcal{A}{\mathcal{F}}_2,$$

(18)

where $\mathcal{A}\ne 0,1$, then Equations (11) and (12) are represented as follows:

$${\mathit{a}}_{1}m{\mathcal{F}}_1^{m-1}{\mathcal{F}}_1^{\prime \prime }+{\mathit{a}}_{1}m(m-1){\mathcal{F}}_1^{m-2}{\left({\mathcal{F}}_1^{\prime }\right)}^2+\left({\mathit{c}}_1+{\mathit{g}}_1{\mathcal{A}}^{-n}\right){\mathcal{F}}_1^{m-n}-\left[m\left({\mathit{a}}_{1}m{\rho }^2-{\xi }_1\rho +\Im \right)+{\varrho }_1{\mathcal{A}}^m\right]{\mathcal{F}}_1^m$$

$$+\left({\mathit{b}}_1+{\mathit{f}}_1{\mathcal{A}}^{-2n}\right){\mathcal{F}}_1^{m-2n}+\left({\mathit{d}}_1+{\mathit{h}}_1{\mathcal{A}}^n\right){\mathcal{F}}_1^{m+n}+\left[m\rho \left({\alpha }_1+{\mu }_1\right)+{\mathit{k}}_1{\mathcal{A}}^{2n}+{\mathit{e}}_1\right]{\mathcal{F}}_1^{m+2n}=0,$$

(19)

$${\mathit{a}}_{2}m{\mathcal{A}}^m{\mathcal{F}}_1^{m-1}{\mathcal{F}}_1^{\prime \prime }+{\mathit{a}}_{2}m(m-1){\mathcal{A}}^m{\mathcal{F}}_1^{m-2}{\left({\mathcal{F}}_1^{\prime }\right)}^2+{\mathcal{A}}^m({\mathit{c}}_2{\mathcal{A}}^{-n}+{\mathit{g}}_2){\mathcal{F}}_1^{m-n}-[m{\mathcal{A}}^m({\mathit{a}}_{2}m{\rho }^2-{\xi }_2\rho +\Im )+{\varrho }_2]{\mathcal{F}}_1^m$$

$$+{\mathcal{A}}^m\left({\mathit{f}}_2+{\mathit{b}}_2{\mathcal{A}}^{-2n}\right){\mathcal{F}}_1^{m-2n}+{\mathcal{A}}^m\left({\mathit{h}}_2+{\mathit{d}}_2{\mathcal{A}}^n\right){\mathcal{F}}_1^{m+n}+[{\mathit{k}}_2+\left[m\rho ({\alpha }_2+{\mu }_2)+{\mathit{e}}_2\right]{\mathcal{A}}^{2n}]{\mathcal{A}}^m{\mathcal{F}}_1^{m+2n}=0.$$

(20)

Equations (19) and (20) have the same form under the following constraints:

$${\mathit{a}}_1={\mathit{a}}_2{\mathcal{A}}^m,$$

(21)

$${\mathit{a}}_{1}m(m-1)={\mathit{a}}_{2}m{\mathcal{A}}^m(m-1),$$

(22)

$${\mathit{c}}_1+{\mathit{g}}_1{\mathcal{A}}^{-n}={\mathcal{A}}^m\left({\mathit{c}}_2{\mathcal{A}}^{-n}+{\mathit{g}}_2\right),$$

(23)

$${\mathit{b}}_1+{\mathit{f}}_1{\mathcal{A}}^{-2n}={\mathcal{A}}^m\left({\mathit{f}}_2+{\mathit{b}}_2{\mathcal{A}}^{-2n}\right),$$

(24)

$$\left[m\left({\mathit{a}}_{1}m{\rho }^2-{\xi }_1\rho +\Im \right)+{\varrho }_1{\mathcal{A}}^m\right]=\left[m{\mathcal{A}}^m\left({\mathit{a}}_{2}m{\rho }^2-{\xi }_2\rho +\Im \right)+{\varrho }_2\right],$$

(25)

$${\mathit{d}}_1+{\mathit{h}}_1{\mathcal{A}}^n={\mathcal{A}}^m\left({\mathit{h}}_2+{\mathit{d}}_2{\mathcal{A}}^n\right),$$

(26)

$$\left[m\rho \left({\alpha }_1+{\mu }_1\right)+{\mathit{k}}_1{\mathcal{A}}^{2n}+{\mathit{e}}_1\right]=\left[{\mathit{k}}_2+\left[m\rho \left({\alpha }_2+{\mu }_2\right)+{\mathit{e}}_2\right]{\mathcal{A}}^{2n}\right]{\mathcal{A}}^m.$$

(27)

Through the Equations (17), (21) and (25), we obtain:

$$\Im =\frac{\left({\xi }_1-{\xi }_2\right)\left({\xi }_2{\mathcal{A}}^m-{\xi }_1\right)}{2{\mathit{a}}_{2}m{\left(1-{\mathcal{A}}^m\right)}^2}+\frac{{\varrho }_2-{\varrho }_1{\mathcal{A}}^m}{m\left(1-{\mathcal{A}}^m\right)},{\mathcal{A}}^m\ne 1\mathrm{and}{\mathit{a}}_2\ne 0.$$

(28)

When applying the balancing rule between the terms ${\mathcal{F}}_1^{m+2n}$ and ${\mathcal{F}}_1^{m-1}{\mathcal{F}}_1^{\prime \prime }$ in Equation (19), then $N=\frac{1}{n}$, so we can use the following transformation:

$${\mathcal{F}}_1\left(ϵ\right)={\mathcal{H}}^{\frac{1}{n}}\left(ϵ\right).$$

(29)

Therefore, Equation (19) can be represented as follows:

$$\mathcal{H}{\mathcal{H}}^{\prime \prime }+{\mathcal{L}}_1{\left({\mathcal{H}}^{\prime }\right)}^2+{\mathcal{L}}_2+{\mathcal{L}}_3\mathcal{H}-{\mathcal{L}}_4{\mathcal{H}}^2+{\mathcal{L}}_5{\mathcal{H}}^3+{\mathcal{L}}_6{\mathcal{H}}^4=0,$$

(30)

where ${\mathcal{L}}_1=\frac{m-n}{n},{\mathcal{L}}_2=\frac{n\left({\mathit{b}}_1+{\mathit{f}}_1{\mathcal{A}}^{-2n}\right)}{{\mathit{a}}_{1}m},{\mathcal{L}}_3=\frac{n\left({\mathit{c}}_1+{\mathit{g}}_1{\mathcal{A}}^{-n}\right)}{{\mathit{a}}_{1}m},$${\mathcal{L}}_4=-\frac{n\left[m\left({\mathit{a}}_{1}m{\rho }^2-{\xi }_1\rho +\Im \right)+{\varrho }_1{\mathcal{A}}^m\right]}{{\mathit{a}}_{1}m},$${\mathcal{L}}_5=\frac{n\left({\mathit{d}}_1+{\mathit{h}}_1{\mathcal{A}}^n\right)}{{\mathit{a}}_{1}m}$ and ${\mathcal{L}}_6=\frac{n\left[m\rho \left({\alpha }_1+{\mu }_1\right)+{\mathit{k}}_1{\mathcal{A}}^{2n}+{\mathit{e}}_1\right]}{{\mathit{a}}_{1}m}.$

So, we can now apply the extended F-expansion method to Equation (30) to obtain the traveling wave solutions of this equation as

$$\mathcal{H}\left(ϵ\right)={\alpha }_0+{\alpha }_1\chi \left(ϵ\right)+\frac{{\beta }_1}{\chi \left(ϵ\right)},$$

(31)

where ${\alpha }_1$ or ${\beta }_1\ne 0$.

Compensation by Equation (31) with Equation (7) in Equation (30), then we extract exact solutions of (1) and (2) as follows:

Case: 1 $s_1=s_3=0$.

$${\mathcal{L}}_3=-\frac{\left(2{\mathcal{L}}_1^2+5{\mathcal{L}}_1+2\right){\mathcal{L}}_5\left({\mathcal{L}}_4{\mathcal{L}}_6{\left(2{\mathcal{L}}_1+3\right)}^2+\left({\mathcal{L}}_1^2+3{\mathcal{L}}_1+2\right){\mathcal{L}}_5^2\right)}{2\left({\mathcal{L}}_1+1\right){\left(2{\mathcal{L}}_1+3\right)}^3{\mathcal{L}}_6^2},{\alpha }_0=-\frac{\left({\mathcal{L}}_1+2\right){\mathcal{L}}_5}{2\left(2{\mathcal{L}}_1+3\right){\mathcal{L}}_6},$$

(1.1) 

${\alpha }_1=\frac{\sqrt{-s_4\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}},{\beta }_1=0,s_2=\frac{2{\mathcal{L}}_4{\mathcal{L}}_6{\left(2{\mathcal{L}}_1+3\right)}^2+3\left({\mathcal{L}}_1^2+3{\mathcal{L}}_1+2\right){\mathcal{L}}_5^2}{2\left({\mathcal{L}}_1+1\right){\left(2{\mathcal{L}}_1+3\right)}^2{\mathcal{L}}_6},$

$${\mathcal{L}}_2=\frac{{\mathcal{L}}_1\left({\mathcal{L}}_1+2\right)\left(16s_0{s}_4\left({\mathcal{L}}_1+1\right){\mathcal{L}}_6^2{\left(2{\mathcal{L}}_1+3\right)}^4-4\left({\mathcal{L}}_1+2\right){\mathcal{L}}_4{\mathcal{L}}_5^2{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right){}^2-5\left({\mathcal{L}}_1+1\right)\left({\mathcal{L}}_1+2\right){}^2{\mathcal{L}}_5^4\right)}{16\left({\mathcal{L}}_1+1\right){\left(2{\mathcal{L}}_1+3\right)}^4{\mathcal{L}}_6^3}.$$

(1.2) 

${\beta }_1=\frac{\sqrt{-s_0\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}},{\alpha }_1=0s_2=\frac{2{\mathcal{L}}_4{\mathcal{L}}_6{\left(2{\mathcal{L}}_1+3\right)}^2+3\left({\mathcal{L}}_1^2+3{\mathcal{L}}_1+2\right){\mathcal{L}}_5^2}{2\left({\mathcal{L}}_1+1\right){\left(2{\mathcal{L}}_1+3\right)}^2{\mathcal{L}}_6},$

$${\mathcal{L}}_2=\frac{{\mathcal{L}}_1\left({\mathcal{L}}_1+2\right)\left(16s_0{s}_4\left({\mathcal{L}}_1+1\right){\mathcal{L}}_6^2{\left(2{\mathcal{L}}_1+3\right)}^4-4\left({\mathcal{L}}_1+2\right){\mathcal{L}}_4{\mathcal{L}}_5^2{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right){}^2-5\left({\mathcal{L}}_1+1\right)\left({\mathcal{L}}_1+2\right){}^2{\mathcal{L}}_5^4\right)}{16\left({\mathcal{L}}_1+1\right){\left(2{\mathcal{L}}_1+3\right)}^4{\mathcal{L}}_6^3}.$$

(1.3) 

${\alpha }_1=\frac{\sqrt{-s_4\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}},{\beta }_1=\frac{\sqrt{-s_0\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}},s_2=\frac{2{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right){}^2\left(6\sqrt{s_0}\sqrt{s_4}\left({\mathcal{L}}_1+1\right)+{\mathcal{L}}_4\right)+3\left({\mathcal{L}}_1^2+3{\mathcal{L}}_1+2\right){\mathcal{L}}_5^2}{2\left({\mathcal{L}}_1+1\right)\left(2{\mathcal{L}}_1+3\right){}^2{\mathcal{L}}_6},$

${\mathcal{L}}_2=-\frac{{\mathcal{L}}_1\left({\mathcal{L}}_1+2\right)\left(16\sqrt{s_0}\sqrt{s_4}{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right){}^2+\left({\mathcal{L}}_1+2\right){\mathcal{L}}_5^2\right)\left(4{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right){}^2\left(4\sqrt{s_0}\sqrt{s_4}\left({\mathcal{L}}_1+1\right)+{\mathcal{L}}_4\right)+5\left({\mathcal{L}}_1^2+3{\mathcal{L}}_1+2\right){\mathcal{L}}_5^2\right)}{16\left({\mathcal{L}}_1+1\right)\left(2{\mathcal{L}}_1+3\right){}^4{\mathcal{L}}_6^3}.$

From the case(1.1), under the conditions ${\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)<0\mathrm{and}{\mathcal{L}}_5\left(2{\mathcal{L}}_1+3\right)>0$, the exact solutions of (1) and (2) are obtained in the following forms:

(1.1,1) 

If $s_0=1,s_2=-\left(r^2+1\right),s_4=r^2,$ and $0<r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,1}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{sn}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(32)

$${\mathit{v}}_{1.1,1}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{sn}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(33)

or

$${\mathit{u}}_{1.1,2}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{cd}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(34)

$${\mathit{v}}_{1.1,2}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{cd}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(35)

Special case. We extract the dark solitons when $r=1$, as

$${\mathit{u}}_{1.1,3}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tanh[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(36)

$${\mathit{v}}_{1.1,3}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tanh[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(37)

(1.1,2) 

If $s_0=r^2,s_2=-\left(r^2+1\right),s_4=1,$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,4}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{ns}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(38)

$${\mathit{v}}_{1.1,4}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{ns}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(39)

or

$${\mathit{u}}_{1.1,5}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{dc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(40)

$${\mathit{v}}_{1.1,5}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{dc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(41)

Special case. We derive singular solitons or singular periodic solutions when $r=1$ or $r=0$, as

$${\mathit{u}}_{1.1,6}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+coth[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(42)

$${\mathit{v}}_{1.1,6}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+coth[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(43)

or

$${\mathit{u}}_{1.1,7}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csc[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(44)

$${\mathit{v}}_{1.1,7}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csc[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(45)

and

$${\mathit{u}}_{1.1,8}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+sec[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(46)

$${\mathit{v}}_{1.1,8}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+sec[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(47)

(1.1,3) 

If $s_0=-r^2,s_2=2r^2-1,s_4=1-r^2,$ and $0\le r<1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,9}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}\mathrm{nc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(48)

$${\mathit{v}}_{1.1,9}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}\mathrm{nc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(49)

(1.1,4) 

If $s_0=1,s_2=2-r^2,s_4=1-r^2,$ and $0\le r<1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,10}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}\mathrm{sc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(50)

$${\mathit{v}}_{1.1,10}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}\mathrm{sc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(51)

Special case. We obtain periodic solutions when $r=0$, as

$${\mathit{u}}_{1.1,11}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tan[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(52)

$${\mathit{v}}_{1.1,11}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tan[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(53)

(1.1,5) 

If $s_0=1-r^2,s_2=2-r^2,s_4=1,$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,12}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{cs}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(54)

$${\mathit{v}}_{1.1,12}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{cs}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(55)

Special case. When $r=1$ or $r=0$, we derive singular solitons or periodic solutions as

$${\mathit{u}}_{1.1,13}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\csc \mathrm{h}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(56)

$${\mathit{v}}_{1.1,13}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\csc \mathrm{h}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(57)

or

$${\mathit{u}}_{1.1,14}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+cot[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(58)

$${\mathit{v}}_{1.1,14}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+cot[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(59)

(1.1,6) 

If $s_0=-r^2(1-r^2),s_2=2r^2-1,s_4=1,$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,15}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{ds}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(60)

$${\mathit{v}}_{1.1,15}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{ds}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(61)

Special case. When $r=1$ or $r=0$, we obtain singular solitons or periodic solutions as

$${\mathit{u}}_{1.1,16}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csch[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(62)

$${\mathit{v}}_{1.1,16}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csch[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(63)

or

$${\mathit{u}}_{1.1,17}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csc[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(64)

$${\mathit{v}}_{1.1,17}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csc[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(65)

(1.1,7) 

If $s_0=\frac{1}{4},s_2=\frac{1-2r^2}{2},s_4=\frac{1}{4},$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,18}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+(\mathrm{ns}[x+\gamma t]\pm \mathrm{cs}[x+\gamma t])\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(66)

$${\mathit{v}}_{1.1,18}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+(\mathrm{ns}[x+\gamma t]\pm \mathrm{cs}[x+\gamma t])\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(67)

Special case. In the case of $r=1$ or $r=0$, we obtain singular solitons and dark solitons or periodic solutions as

$${\mathit{u}}_{1.1,19}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+coth\left[\frac{x+\gamma t}{2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(68)

$${\mathit{v}}_{1.1,19}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+coth\left[\frac{x+\gamma t}{2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(69)

and

$${\mathit{u}}_{1.1,20}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tanh\left[\frac{x+\gamma t}{2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(70)

$${\mathit{v}}_{1.1,20}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tanh\left[\frac{x+\gamma t}{2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(71)

or

$${\mathit{u}}_{1.1,21}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+cot\left[\frac{x+\gamma t}{2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(72)

$${\mathit{v}}_{1.1,21}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+cot\left[\frac{x+\gamma t}{2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(73)

and

$${\mathit{u}}_{1.1,22}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tan\left[\frac{x+\gamma t}{2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(74)

$${\mathit{v}}_{1.1,22}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tan\left[\frac{x+\gamma t}{2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(75)

(1.1,8) 

If $s_0=s_4=\frac{1-r^2}{4},s_2=\frac{1+r^2}{2}$ and $0\le r<1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,23}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}(\mathrm{nc}[x+\gamma t]+\mathrm{sc}[x+\gamma t])\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(76)

$${\mathit{v}}_{1.1,23}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}(\mathrm{nc}[x+\gamma t]+\mathrm{sc}[x+\gamma t])\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(77)

Special case. In the case of $r=0$, the combo periodic solutions are obtained as follows:

$${\mathit{u}}_{1.1,24}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+(sec[x+\gamma t]+tan[x+\gamma t])\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(78)

$${\mathit{v}}_{1.1,24}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+(sec[x+\gamma t]+tan[x+\gamma t])\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(79)

(1.1,9) 

If $s_0=\frac{r^2}{4},s_2=\frac{r^2-2}{2},s_4=\frac{1}{4}$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,25}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+(\mathrm{ns}[x+\gamma t]\pm \mathrm{ds}[x+\gamma t])\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(80)

$${\mathit{v}}_{1.1,25}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+(\mathrm{ns}[x+\gamma t]\pm \mathrm{ds}[x+\gamma t])\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(81)

Special case. In the case of $r=0$, we obtain singular periodic solutions as follows:

$${\mathit{u}}_{1.1,26}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2csc[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(82)

$${\mathit{v}}_{1.1,26}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2csc[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(83)

(1.1,10) 

If $s_0=\frac{s_2^2}{4s_4},s_2<0$ and $s_4>0$; thus, the dark soliton solutions are obtained as follows:

$${\mathit{u}}_{1.1,27}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{-\frac{s_2}{2}}tanh\left[(x+\gamma t)\sqrt{\frac{-s_2}{2}}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(84)

$${\mathit{v}}_{1.1,27}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{\frac{-s_2}{2}}tanh\left[(x+\gamma t)\sqrt{\frac{-s_2}{2}}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(85)

(1.1,11) 

If $s_0=\frac{s_2^2}{4s_4},s_2>0$ and $s_4>0$; thus, the periodic solutions are obtained as follows:

$${\mathit{u}}_{1.1,28}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{\frac{s_2}{2}}tan\left[(x+\gamma t)\sqrt{\frac{s_2}{2}}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(86)

$${\mathit{v}}_{1.1,28}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{\frac{s_2}{2}}tan\left[(x+\gamma t)\sqrt{\frac{s_2}{2}}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(87)

(1.1,12) 

If $s_0=\frac{r^2{s}_2^2}{s_4{\left(r^2+1\right)}^2},s_2<0,s_4>0$ and $0<r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,29}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{\frac{-r^2{s}_2}{r^2+1}}\mathrm{sn}\left[(x+\gamma t)\sqrt{\frac{-s_2}{r^2+1}}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(88)

$${\mathit{v}}_{1.1,29}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{\frac{-r^2{s}_2}{r^2+1}}\mathrm{sn}\left[(x+\gamma t)\sqrt{\frac{-s_2}{r^2+1}}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(89)

(1.1,13) 

If $s_0=s_4=1,s_2=2-4r^2$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,30}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\mathrm{dn}[x+\gamma t]\mathrm{sn}[x+\gamma t]}{\mathrm{cn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(90)

$${\mathit{v}}_{1.1,30}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\mathrm{dn}[x+\gamma t]\mathrm{sn}[x+\gamma t]}{\mathrm{cn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(91)

(1.1,14) 

If $s_0=\frac{{(r+1)}^2}{4D_1^2},s_2=\frac{r^2-6r+1}{2},s_4=\frac{D_1^2{(r+1)}^2}{4}$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,31}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{(r+1)\mathrm{cn}[x+\gamma t]\mathrm{dn}[x+\gamma t]}{2\left(\mathrm{sn}\right[x+\gamma t]+1)(1-r\mathrm{sn}[x+\gamma t\left]\right)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(92)

$${\mathit{v}}_{1.1,31}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{(r+1)\mathrm{cn}[x+\gamma t]\mathrm{dn}[x+\gamma t]}{2\left(\mathrm{sn}\right[x+\gamma t]+1)(1-r\mathrm{sn}[x+\gamma t\left]\right)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(93)

Special case. In the case of $r=0$, we thus obtain combo periodic solutions as follows:

$${\mathit{u}}_{1.1,32}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{cos[x+\gamma t]}{sin[x+\gamma t]+1}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(94)

$${\mathit{v}}_{1.1,32}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{cos[x+\gamma t]}{sin[x+\gamma t]+1}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(95)

where $D_1$ is a constant.

(1.1,15) 

If $s_0=r^4+2r^3+r^2,s_2=-\left(r^2+6r+1\right),s_4=\frac{4}{r}$ and $0<r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,33}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{2\sqrt{r}\mathrm{cn}[x+\gamma t]\mathrm{dn}[x+\gamma t]}{r{\mathrm{sn}}^2[x+\gamma t]-1}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(96)

$${\mathit{v}}_{1.1,33}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{2\sqrt{r}\mathrm{cn}[x+\gamma t]\mathrm{dn}[x+\gamma t]}{r{\mathrm{sn}}^2[x+\gamma t]-1}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(97)

(1.1,16) 

If $s_0=\frac{r^4}{4\left(D_2^2+D_3^2\right)},s_2=\frac{r^2-2}{2},s_4=\frac{D_2^2+D_3^2}{4}$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,34}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\sqrt{D_2^2+D_3^2}\mathrm{dn}[x+\gamma t]+\sqrt{-D_3^2{r}^2+D_2^2+D_3^2}}{D_3\mathrm{cn}[x+\gamma t]+D_2\mathrm{sn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(98)

$${\mathit{v}}_{1.1,34}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\sqrt{D_2^2+D_3^2}\mathrm{dn}[x+\gamma t]+\sqrt{-D_3^2{r}^2+D_2^2+D_3^2}}{D_3\mathrm{cn}[x+\gamma t]+D_2\mathrm{sn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(99)

where $D_2$ and $D_3$ are constant.

Special case. In the case of $r=1$ or $r=0$, we extract singular combo solitons or combo periodic solutions as follows:

$${\mathit{u}}_{1.1,35}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{D_{2}cosh[x+\gamma t]+\sqrt{D_2^2+D_3^2}}{D_{2}sinh[x+\gamma t]+D_3}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(100)

$${\mathit{v}}_{1.1,35}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{D_{2}cosh[x+\gamma t]+\sqrt{D_2^2+D_3^2}}{D_{2}sinh[x+\gamma t]+D_3}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(101)

or

$${\mathit{u}}_{1.1,36}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{2\sqrt{D_2^2+D_3^2}}{D_{2}sin[x+\gamma t]+D_{3}cos[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(102)

$${\mathit{v}}_{1.1,36}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{2\sqrt{D_2^2+D_3^2}}{D_{2}sin[x+\gamma t]+D_{3}cos[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(103)

(1.1,17) 

If $s_0=s_4=\frac{1-r^2}{4},s_2=\frac{1+r^2}{2}$ and $0\le r<1$; thus, the Jacobi elliptic solutions are obtained as

$${\mathit{u}}_{1.1,37}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\sqrt{1-r^2}\mathrm{cn}[x+\gamma t]}{1+\mathrm{sn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(104)

$${\mathit{v}}_{1.1,37}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\sqrt{1-r^2}\mathrm{cn}[x+\gamma t]}{1+\mathrm{sn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(105)

or

$${\mathit{u}}_{1.1,38}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}(\mathrm{nc}[x+\gamma t]+\mathrm{sc}[x+\gamma t])\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(106)

$${\mathit{v}}_{1.1,38}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}(\mathrm{nc}[x+\gamma t]+\mathrm{sc}[x+\gamma t])\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(107)

(1.1,18) 

If $s_0=\frac{1}{4},s_2=\frac{1+r^2}{2},s_4=\frac{{(1-r^2)}^2}{4}$ and $0\le r<1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.1,39}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\left(1-r^2\right)\mathrm{sn}[x+\gamma t]}{\mathrm{dn}[x+\gamma t]\pm \mathrm{cn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(108)

$${\mathit{v}}_{1.1,39}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\left(1-r^2\right)\mathrm{sn}[x+\gamma t]}{\mathrm{dn}[x+\gamma t]\pm \mathrm{cn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(109)

(1.1,19) 

If $s_0=\frac{1}{4},s_2=\frac{r^2-2}{2},s_4=\frac{r^4}{4}$ and $0<r\le 1$, we derive the Jacobi elliptic solutions as follows:

$${\mathit{u}}_{1.1,40}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{r^2\mathrm{cn}[x+\gamma t]}{\mathrm{dn}[x+\gamma t]+\sqrt{1-r^2}}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(110)

$${\mathit{v}}_{1.1,40}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{r^2\mathrm{cn}[x+\gamma t]}{\mathrm{dn}[x+\gamma t]+\sqrt{1-r^2}}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(111)

or

$${\mathit{u}}_{1.1,41}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{r^2\mathrm{sn}[x+\gamma t]}{\mathrm{dn}[x+\gamma t]+1}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(112)

$${\mathit{v}}_{1.1,41}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{r^2\mathrm{sn}[x+\gamma t]}{\mathrm{dn}[x+\gamma t]+1}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(113)

(1.1,20) 

If $s_0=0,s_2<0,s_4>0$ and $0\le r\le 1$, we extract the periodic solutions as follows:

$${\mathit{u}}_{1.1,42}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{-s_2}sec\left[(x+\gamma t)\sqrt{-s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(114)

$${\mathit{v}}_{1.1,42}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{-s_2}sec\left[(x+\gamma t)\sqrt{-s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(115)

or

$${\mathit{u}}_{1.1,43}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{-s_2}csc\left[(x+\gamma t)\sqrt{-s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(116)

$${\mathit{v}}_{1.1,43}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{-s_2}csc\left[(x+\gamma t)\sqrt{-s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(117)

From the case(1.2), under the conditions ${\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)<0$ and ${\mathcal{L}}_5\left(2{\mathcal{L}}_1+3\right)>0$, the exact solutions of (1) and (2) are obtained in the following forms:

(1.2,1) 

If $s_0=1,s_2=-\left(r^2+1\right),s_4=r^2,$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.2,1}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{ns}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(118)

$${\mathit{v}}_{1.2,1}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{ns}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(119)

or

$${\mathit{u}}_{1.2,2}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{dc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(120)

$${\mathit{v}}_{1.2,2}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{dc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(121)

Special case. In the case of $r=1$ or $r=0$, we extract singular solitons or periodic solutions as

$${\mathit{u}}_{1.2,3}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+coth[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(122)

$${\mathit{v}}_{1.2,3}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+coth[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(123)

or

$${\mathit{u}}_{1.2,4}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csc[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(124)

$${\mathit{v}}_{1.2,4}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csc[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(125)

and

$${\mathit{u}}_{1.2,5}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+sec[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(126)

$${\mathit{v}}_{1.2,5}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+sec[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(127)

(1.2,2) 

If $s_0=1-r^2,s_2=2r^2-1,s_4=-r^2,$ and $0\le r<1$, we establish Jacobi elliptic solutions as

$${\mathit{u}}_{1.2,6}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}\mathrm{nc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(128)

$${\mathit{v}}_{1.2,6}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}\mathrm{nc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(129)

(1.2,3) 

If $s_0=r^2,s_2=-\left(r^2+1\right),s_4=1,$ and $0<r\le 1$, we obtain Jacobi elliptic solutions as

$${\mathit{u}}_{1.2,7}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{sn}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(130)

$${\mathit{v}}_{1.2,7}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{sn}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(131)

or

$${\mathit{u}}_{1.2,8}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{cd}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(132)

$${\mathit{v}}_{1.2,8}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{cd}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(133)

Special case. In the case of $r=1$, we derive dark solitons as

$${\mathit{u}}_{1.2,9}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tanh[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(134)

$${\mathit{v}}_{1.2,9}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tanh[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(135)

(1.2,4) 

If $s_0=1,s_2=2-r^2,s_4=1-r^2,$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.2,10}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{cs}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(136)

$${\mathit{v}}_{1.2,10}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{cs}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(137)

Special case. In the case of $r=1$ or $r=0$, we extract singular solitons or periodic solutions as follows:

$${\mathit{u}}_{1.2,11}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csch[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(138)

$${\mathit{v}}_{1.2,11}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csch[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(139)

or

$${\mathit{u}}_{1.2,12}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+cot[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(140)

$${\mathit{v}}_{1.2,12}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+cot[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(141)

(1.2,5) 

If $s_0=1,s_2=2r^2-1,s_4=-r^2(1-r^2),$ and $0\le r\le 1$, we obtain Jacobi elliptic solutions as follows:

$${\mathit{u}}_{1.2,13}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{ds}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(142)

$${\mathit{v}}_{1.2,13}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\mathrm{ds}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(143)

(1.2,6) 

If $s_0=1-r^2,s_2=2-r^2,s_4=1,$ and $0\le r<1$, we obtain Jacobi elliptic solutions as follows:

$${\mathit{u}}_{1.2,14}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}\mathrm{sc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(144)

$${\mathit{v}}_{1.2,14}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}\mathrm{sc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(145)

Special case. In the case of $r=0$, we obtain periodic solutions as follows:

$${\mathit{u}}_{1.2,15}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tan[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(146)

$${\mathit{v}}_{1.2,15}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+tan[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(147)

(1.2,7) 

If $s_0=s_4=\frac{1-r^2}{4},s_2=\frac{1+r^2}{2}$ and $0\le r<1$; thus, the Jacobi elliptic solutions are obtained as

$${\mathit{u}}_{1.2,16}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\sqrt{1-r^2}}{\mathrm{nc}[x+\gamma t]+\mathrm{sc}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(148)

$${\mathit{v}}_{1.2,16}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\sqrt{1-r^2}}{\mathrm{nc}[x+\gamma t]+\mathrm{sc}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(149)

Special case. In the case of $r=0$, we establish periodic solutions as

$${\mathit{u}}_{1.2,17}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{1}{tan[x+\gamma t]+sec[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(150)

$${\mathit{v}}_{1.2,17}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{1}{tan[x+\gamma t]+sec[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(151)

(1.2,8) 

If $s_0=\frac{s_2^2}{4s_4},s_2>0$ and $s_4>0$, we establish periodic solutions as

$${\mathit{u}}_{1.2,18}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{\frac{s_2}{2}}cot\left[(x+\gamma t)\sqrt{\frac{s_2}{2}}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(152)

$${\mathit{v}}_{1.2,18}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{\frac{s_2}{2}}cot\left[(x+\gamma t)\sqrt{\frac{s_2}{2}}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(153)

(1.2,9) 

If $s_0=\frac{s_2^2}{4s_4},s_2<0$ and $s_4>0$, we establish singular solitons as

$${\mathit{u}}_{1.2,19}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{-\frac{s_2}{2}}coth\left[(x+\gamma t)\sqrt{\frac{-s_2}{2}}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(154)

$${\mathit{v}}_{1.2,19}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{\frac{-s_2}{2}}coth\left[(x+\gamma t)\sqrt{\frac{-s_2}{2}}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(155)

(1.2,10) 

If $s_0=s_4=1,s_2=2-4r^2$ and $0\le r\le 1$, we establish Jacobi elliptic solutions as

$${\mathit{u}}_{1.2,20}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\mathrm{cn}[x+\gamma t]}{\mathrm{dn}[x+\gamma t]\mathrm{sn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(156)

$${\mathit{v}}_{1.2,20}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\mathrm{cn}[x+\gamma t]}{\mathrm{dn}[x+\gamma t]\mathrm{sn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(157)

(1.2,11) 

If $s_0=\frac{r^4}{4\left(D_2^2+D_3^2\right)},s_2=\frac{r^2-2}{2},s_4=\frac{D_2^2+D_3^2}{4}$ and $0<r\le 1$, we obtain Jacobi elliptic solutions as

$${\mathit{u}}_{1.2,21}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{r^2\left(D_3\mathrm{cn}[x+\gamma t]+D_2\mathrm{sn}[x+\gamma t]\right)}{\sqrt{D_2^2+D_3^2}\mathrm{dn}[x+\gamma t]+\sqrt{-D_3^2{r}^2+D_2^2+D_3^2}}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(158)

$${\mathit{v}}_{1.2,21}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{r^2\left(D_3\mathrm{cn}[x+\gamma t]+D_2\mathrm{sn}[x+\gamma t]\right)}{\sqrt{D_2^2+D_3^2}\mathrm{dn}[x+\gamma t]+\sqrt{-D_3^2{r}^2+D_2^2+D_3^2}}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(159)

where $D_2$ and $D_3$ are constant.

Special case. In the case of $r=1$, we establish hyperbolic solutions as

$${\mathit{u}}_{1.2,22}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{D_{2}sinh[x+\gamma t]+D_3}{D_{2}cosh[x+\gamma t]+\sqrt{D_2^2+D_3^2}}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(160)

$${\mathit{v}}_{1.2,22}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{D_{2}sinh[x+\gamma t]+D_3}{D_{2}cosh[x+\gamma t]+\sqrt{D_2^2+D_3^2}}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(161)

(1.2,12) 

If $s_0=\frac{1}{4},s_2=\frac{1+r^2}{2},s_4=\frac{{(1-r^2)}^2}{4}$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as

$${\mathit{u}}_{1.2,23}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\mathrm{dn}[x+\gamma t]+\mathrm{cn}[x+\gamma t]}{\mathrm{sn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta },$$

(162)

$${\mathit{v}}_{1.2,23}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\mathrm{dn}[x+\gamma t]+\mathrm{cn}[x+\gamma t]}{\mathrm{sn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta }.$$

(163)

(1.2,13) 

If $s_0=\frac{1}{4},s_2=\frac{r^2-2}{2},s_4=\frac{r^4}{4}$ and $0\le r\le 1$, we extract Jacobi elliptic solutions as

$${\mathit{u}}_{1.2,24}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\mathrm{dn}[x+\gamma t]+\sqrt{1-r^2}}{\mathrm{cn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(164)

$${\mathit{v}}_{1.2,24}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\mathrm{dn}[x+\gamma t]+\sqrt{1-r^2}}{\mathrm{cn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(165)

or

$${\mathit{u}}_{1.2,25}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\mathrm{dn}[x+\gamma t]+1}{\mathrm{sn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(166)

$${\mathit{v}}_{1.2,25}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\mathrm{dn}[x+\gamma t]+1}{\mathrm{sn}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(167)

From the case(1.3), if ${\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)<0\mathrm{and}{\mathcal{L}}_5\left(2{\mathcal{L}}_1+3\right)>0$, the exact solutions of (1) and (2) are obtained in the following forms:

(1.3,1) 

If $s_0=1,s_2=-\left(r^2+1\right),s_4=r^2$ or $s_0=r^2,s_2=-\left(r^2+1\right),s_4=1$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as follows:

$${\mathit{u}}_{1.3,1}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{sn}[x+\gamma t]+\mathrm{ns}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(168)

$${\mathit{v}}_{1.3,1}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{sn}[x+\gamma t]+\mathrm{ns}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(169)

$${\mathit{u}}_{1.3,2}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{cd}[x+\gamma t]+\frac{1}{\mathrm{cd}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(170)

$${\mathit{v}}_{1.3,2}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+r\mathrm{cd}[x+\gamma t]+\mathrm{dc}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(171)

Special case. In the case of $r=1$ or $r=0$, we derive singular solitons or periodic solutions as

$${\mathit{u}}_{1.3,3}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2coth\left[2(x+\gamma t)\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(172)

$${\mathit{v}}_{1.3,3}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2coth\left[2(x+\gamma t)\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(173)

or

$${\mathit{u}}_{1.3,4}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csc[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(174)

$${\mathit{v}}_{1.3,4}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csc[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(175)

$${\mathit{u}}_{1.3,5}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+sec[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(176)

$${\mathit{v}}_{1.3,5}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+sec[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(177)

(1.3,2) 

If $s_0=1,s_2=2-r^2,s_4=1-r^2$ or $s_0=1-r^2,s_2=2-r^2,s_4=1$ and $0\le r\le 1$, we obtain Jacobi elliptic solutions as

$${\mathit{u}}_{1.3,6}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}\mathrm{sc}[x+\gamma t]+\mathrm{cs}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(178)

$${\mathit{v}}_{1.3,6}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{1-r^2}\mathrm{sc}[x+\gamma t]+\mathrm{cs}[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(179)

Special case. In the case of $r=1$ or $r=0$, we establish singular solitons or periodic solutions as

$${\mathit{u}}_{1.3,7}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csch[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(180)

$${\mathit{v}}_{1.3,7}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csch[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(181)

or

$${\mathit{u}}_{1.3,8}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csc\left[2(x+\gamma t)\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(182)

$${\mathit{v}}_{1.3,8}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+csc\left[2(x+\gamma t)\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(183)

(1.3,3) 

If $s_0=s_4=\frac{1}{4},s_2=\frac{1-2r^2}{2}$ and $0\le r\le 1$, we derive Jacobi elliptic solutions as

$${\mathit{u}}_{1.3,9}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{{(\mathrm{ns}[x+\gamma t]+\mathrm{cs}[x+\gamma t])}^2+1}{\mathrm{ns}[x+\gamma t]+\mathrm{cs}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(184)

$${\mathit{v}}_{1.3,9}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{{(\mathrm{ns}[x+\gamma t]+\mathrm{cs}[x+\gamma t])}^2+1}{\mathrm{ns}[x+\gamma t]+\mathrm{cs}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(185)

Special case. In the case of either $r=1$ or $r=0$, we extract singular solitons or periodic solutions as

$${\mathit{u}}_{1.3,10}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2coth[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(186)

$${\mathit{v}}_{1.3,10}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2coth[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(187)

or

$${\mathit{u}}_{1.3,11}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2cot[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(188)

$${\mathit{v}}_{1.3,11}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2cot[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(189)

(1.3,4) 

If $s_0=s_4=\frac{1-r^2}{4},s_2=\frac{1+r^2}{2}$ and $0\le r\le 1$, we obtain Jacobi elliptic solutions as

$${\mathit{u}}_{1.3,12}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\sqrt{1-r^2}{(\mathrm{nc}[x+\gamma t]+\mathrm{sc}[x+\gamma t])}^2+1}{\mathrm{nc}[x+\gamma t]+\mathrm{sc}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(190)

$${\mathit{v}}_{1.3,12}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{\sqrt{1-r^2}{(\mathrm{nc}[x+\gamma t]+\mathrm{sc}[x+\gamma t])}^2+1}{\mathrm{nc}[x+\gamma t]+\mathrm{sc}[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(191)

Special case. In the case of $r=0$, we derive combo periodic solutions as

$${\mathit{u}}_{1.3,13}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{{(sec[x+\gamma t]+tan[x+\gamma t])}^2+1}{sec[x+\gamma t]+tan[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(192)

$${\mathit{v}}_{1.3,13}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{{(sec[x+\gamma t]+tan[x+\gamma t])}^2+1}{sec[x+\gamma t]+tan[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(193)

(1.3,5) 

If $s_0=\frac{s_2^2}{4s_4},s_2<0$ and $s_4>0$, we derive singular solitons as

$${\mathit{u}}_{1.2,14}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{-2s_2}csch\left[(x+\gamma t)\sqrt{-2s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(194)

$${\mathit{v}}_{1.3,14}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{-2s_2}csch\left[(x+\gamma t)\sqrt{-2s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(195)

(1.3,6) 

If $s_0=\frac{s_2^2}{4s_4},s_2>0$ and $s_4>0$, we obtain periodic solutions as

$${\mathit{u}}_{1.2,15}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{2s_2}csc\left[(x+\gamma t)\sqrt{2s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(196)

$${\mathit{v}}_{1.3,15}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\sqrt{2s_2}csc\left[(x+\gamma t)\sqrt{2s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(197)

(1.3,7) 

If $s_0=\frac{{(r\pm 1)}^2}{4D_1^2},s_2=\frac{r^2\mp 6r+1}{2},s_4=\frac{D_1^2{(r\pm 1)}^2}{4}$ and $0\le r<1$, we establish Jacobi elliptic solutions as

$${\mathit{u}}_{1.3,16}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\left|r\pm 1\right|\mathcal{W}(x,t)+\frac{\left|r\pm 1\right|}{\mathcal{W}(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(198)

$${\mathit{v}}_{1.3,16}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\left|r\pm 1\right|\mathcal{W}(x,t)+\frac{\left|r\pm 1\right|}{\mathcal{W}(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(199)

where $\mathcal{W}(x,t)=\frac{\mathrm{cn}[x+\gamma t]\mathrm{dn}[x+\gamma t]}{\left(\mathrm{sn}\right[x+\gamma t]+1)(\pm r\mathrm{sn}[x+\gamma t]+1)}$ and $D_1$ is a constant.

Special case. In the case of $r=0$, we derive periodic solutions as

$${\mathit{u}}_{1.3,17}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2sec[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(200)

$${\mathit{v}}_{1.3,17}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2sec[x+\gamma t]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(201)

(1.3,8) 

If $s_0=r^4+2r^3+r^2,s_2=-\left(r^2+6r+1\right),s_4=\frac{4}{r}$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as

$${\mathit{u}}_{1.3,18}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2\sqrt{r}{\mathcal{S}}_1(x,t)+\frac{r+1}{{\mathcal{S}}_1(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(202)

$${\mathit{v}}_{1.3,18}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+2\sqrt{r}{\mathcal{S}}_1(x,t)+\frac{r+1}{{\mathcal{S}}_1(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(203)

where ${\mathcal{S}}_1(x,t)=\frac{\mathrm{cn}[x+\gamma t]\mathrm{dn}[x+\gamma t]}{r{\mathrm{sn}}^2[x+\gamma t]-1}$.

(1.3,9) 

If $s_0=\frac{r^4}{4\left(D_2^2+D_3^2\right)},s_2=\frac{r^2-2}{2},s_4=\frac{D_2^2+D_3^2}{4}$ and $0\le r\le 1$; thus, the Jacobi elliptic solutions are obtained as

$${\mathit{u}}_{1.3,19}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+{\mathcal{S}}_2(x,t)+\frac{r^2}{{\mathcal{S}}_2(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(204)

$${\mathit{v}}_{1.3,19}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+{\mathcal{S}}_2(x,t)+\frac{r^2}{{\mathcal{S}}_2(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(205)

where ${\mathcal{S}}_2(x,t)=\frac{\sqrt{D_2^2+D_3^2}\mathrm{dn}[x+\gamma t]+\sqrt{-D_3^2{r}^2+D_2^2+D_3^2}}{D_3\mathrm{cn}[x+\gamma t]+D_2\mathrm{sn}[x+\gamma t]}$, while $D_2$ and $D_3$ are constant.

Special case. In the case of either $r=1$ or $r=0$, the singular–bright combo solitons or combo periodic solutions are thus obtained as

$${\mathit{u}}_{1.3,20}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+{\mathcal{S}}_3(x,t)+\frac{1}{{\mathcal{S}}_3(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(206)

$${\mathit{v}}_{1.3,20}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+{\mathcal{S}}_3(x,t)+\frac{1}{{\mathcal{S}}_3(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(207)

or

$${\mathit{u}}_{1.3,21}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{2\sqrt{D_2^2+D_3^2}}{D_{2}sin[x+\gamma t]+D_{3}cos[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(208)

$${\mathit{v}}_{1.3,21}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{2\sqrt{D_2^2+D_3^2}}{D_{2}sin[x+\gamma t]+D_{3}cos[x+\gamma t]}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(209)

where ${\mathcal{S}}_3(x,t)=\frac{D_{2}cosh[x+\gamma t]+\sqrt{D_2^2+D_3^2}}{D_{2}sinh[x+\gamma t]+D_3}$.

(1.3,10) 

If $s_0=s_4=\frac{1}{4},s_2=\frac{1-2r^2}{2}$ and $0\le r\le 1$, we obtain Jacobi elliptic solutions as follows:

$${\mathit{u}}_{1.3,22}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+{\mathcal{S}}_4(x,t)+\frac{1}{{\mathcal{S}}_4(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(210)

$${\mathit{v}}_{1.3,22}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+{\mathcal{S}}_4(x,t)+\frac{1}{{\mathcal{S}}_4(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(211)

or

$${\mathit{u}}_{1.3,23}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+{\mathcal{S}}_5(x,t)+\frac{1}{{\mathcal{S}}_5(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(212)

$${\mathit{v}}_{1.3,23}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+{\mathcal{S}}_5(x,t)+\frac{1}{{\mathcal{S}}_5(x,t)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(213)

where ${\mathcal{S}}_4(x,t)=\frac{\mathrm{sn}[x+\gamma t]}{1+\mathrm{cn}[x+\gamma t]}$, and ${\mathcal{S}}_5(x,t)=\frac{\mathrm{cn}[x+\gamma t]}{\sqrt{1-r^2}\mathrm{sn}[x+\gamma t]+\mathrm{dn}[x+\gamma t]}$.

(1.3,11) 

If $s_0=\frac{1}{4},s_2=\frac{1+r^2}{2},s_4=\frac{{\left(1-r^2\right)}^2}{4}$ and $0\le r\le 1$, we obtain Jacobi elliptic solutions as

$${\mathit{u}}_{1.3,24}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{{(\mathrm{dn}[x+\gamma t]+\mathrm{cn}[x+\gamma t])}^2+\left(1-r^2\right){\mathrm{sn}}^2[x+\gamma t]}{\mathrm{sn}[x+\gamma t]\left(\mathrm{dn}\right[x+\gamma t]+\mathrm{cn}[x+\gamma t\left]\right)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(214)

$${\mathit{v}}_{1.3,24}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{{(\mathrm{dn}[x+\gamma t]+\mathrm{cn}[x+\gamma t])}^2+\left(1-r^2\right){\mathrm{sn}}^2[x+\gamma t]}{\mathrm{sn}[x+\gamma t]\left(\mathrm{dn}\right[x+\gamma t]+\mathrm{cn}[x+\gamma t\left]\right)}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(215)

Special case. In the case of $r=0$, we derive combo periodic solutions as

$${\mathit{u}}_{1.3,25}={\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{2cot\left[\frac{x+\gamma t}{2}\right]}{cos[x+\gamma t]+1}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(216)

$${\mathit{v}}_{1.3,25}=\mathcal{A}{\left(\frac{\sqrt{-\left({\mathcal{L}}_1+2\right)}}{2\sqrt{{\mathcal{L}}_6}}\left(\frac{{\mathcal{L}}_5\sqrt{-\left({\mathcal{L}}_1+2\right)}}{\left(2{\mathcal{L}}_1+3\right)\sqrt{{\mathcal{L}}_6}}+\frac{2cot\left[\frac{x+\gamma t}{2}\right]}{cos[x+\gamma t]+1}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(217)

Case: 2 $s_0=s_1=0$.

$${\alpha }_0=-\frac{\left({\mathcal{L}}_1+2\right){\mathcal{L}}_5}{2\left(2{\mathcal{L}}_1+3\right){\mathcal{L}}_6}+\frac{\sqrt{-s_3^2{s}_4\left({\mathcal{L}}_1+2\right){\mathcal{L}}_6}}{4s_4{\mathcal{L}}_6},{\alpha }_1=\frac{\sqrt{-s_3^2{s}_4\left({\mathcal{L}}_1+2\right){\mathcal{L}}_6}}{s_3{\mathcal{L}}_6},{\beta }_1=0,$$

$$s_2=\frac{3s_3^2\left({\mathcal{L}}_1+1\right){\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right){}^2+4s_4\left(2{\mathcal{L}}_4{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right){}^2+3\left({\mathcal{L}}_1^2+3{\mathcal{L}}_1+2\right){\mathcal{L}}_5^2\right)}{8s_4\left({\mathcal{L}}_1+1\right)\left(2{\mathcal{L}}_1+3\right){}^2{\mathcal{L}}_6},$$

${\mathcal{L}}_2=\frac{{\mathcal{L}}_1\left({\mathcal{L}}_1+2\right)}{256s_4^2({\mathcal{L}}_1+1){(2{\mathcal{L}}_1+3)}^5{\mathcal{L}}_6^4}(-16s_4^2{\mathcal{L}}_5^2{\mathcal{L}}_6(2{\mathcal{L}}_1^2+7{\mathcal{L}}_1+6)(4{\mathcal{L}}_4{\mathcal{L}}_6{(2{\mathcal{L}}_1+3)}^2$

$+5({\mathcal{L}}_1^2+3{\mathcal{L}}_1+2){\mathcal{L}}_5^2\left)\right)+s_3^2{\mathcal{L}}_6({\mathcal{L}}_1+1){(2{\mathcal{L}}_1+3)}^2(3s_3^2{\mathcal{L}}_6^2{(2{\mathcal{L}}_1+3)}^3$

$+8{\mathcal{L}}_5\sqrt{-s_3^2{s}_4({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^4{\mathcal{L}}_6^3})+8s_4(2{\mathcal{L}}_4{\mathcal{L}}_6{(2{\mathcal{L}}_1+3)}^2$

$+3({\mathcal{L}}_1^2+3{\mathcal{L}}_1+2){\mathcal{L}}_5^2\left)\right(s_3^2{\mathcal{L}}_6^2{(2{\mathcal{L}}_1+3)}^3+4{\mathcal{L}}_5\sqrt{-s_3^2{s}_4({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^4{\mathcal{L}}_6^3})$,

${\mathcal{L}}_3=\frac{(2{\mathcal{L}}_1+1)}{32s_4^2\left({\mathcal{L}}_1+1\right)\left(2{\mathcal{L}}_1+3\right){}^4{\mathcal{L}}_6^3}(-16s_4^2\left(2{\mathcal{L}}_1^2+7{\mathcal{L}}_1+6\right){\mathcal{L}}_5{\mathcal{L}}_6\left({\mathcal{L}}_4{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right){}^2+\left({\mathcal{L}}_1^2+3{\mathcal{L}}_1+2\right){\mathcal{L}}_5^2\right)$

$+4s_4\left(2{\mathcal{L}}_4{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right){}^2+3\left({\mathcal{L}}_1^2+3{\mathcal{L}}_1+2\right){\mathcal{L}}_5^2\right)\sqrt{-s_3^2{s}_4\left({\mathcal{L}}_1+2\right)\left(2{\mathcal{L}}_1+3\right){}^4{\mathcal{L}}_6^3}+s_3^2({\mathcal{L}}_1+1){(2{\mathcal{L}}_1+3)}^2{\mathcal{L}}_6\sqrt{-s_3^2{s}_4({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^4{\mathcal{L}}_6^3})$.

Through this case, if $({\mathcal{L}}_1+2)<0$ and $\left(2{\mathcal{L}}_1+3\right),{\mathcal{L}}_5,s_4,s_3\mathrm{and}{\mathcal{L}}_6>0$, the exact solutions of (1) and (2) are in the following forms:

(2.1) 

If $s_2<0$; thus, the combo periodic solutions are obtained as

$${\mathit{u}}_{2.1}={\left(\frac{\sqrt{-s_3^2{s}_4{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{{\mathcal{L}}_6}\left(\frac{\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}+2{\mathcal{L}}_5\sqrt{-s_4\left({\mathcal{L}}_1+2\right)}}{4s_4\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}}-\frac{s_2{sec}^2\left[\frac{(x+\gamma t)\sqrt{-s_2}}{2}\right]}{s_3\left(s_3+2\sqrt{-s_2{s}_4}tan\left[(\frac{(x+\gamma t)\sqrt{-s_2}}{2}\right]\right)}\right)\right)}^{\frac{1}{n}}$$

$$\times e^{i(\rho x+\Im t+\vartheta )},$$

(218)

$${\mathit{v}}_{2.1}=\mathcal{A}{\left(\frac{\sqrt{-s_3^2{s}_4{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{{\mathcal{L}}_6}\left(\frac{\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}+2{\mathcal{L}}_5\sqrt{-s_4\left({\mathcal{L}}_1+2\right)}}{4s_4\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}}-\frac{s_2{sec}^2\left[\frac{(x+\gamma t)\sqrt{-s_2}}{2}\right]}{s_3\left(s_3+2\sqrt{-s_2{s}_4}tan\left[(\frac{(x+\gamma t)\sqrt{-s_2}}{2}\right]\right)}\right)\right)}^{\frac{1}{n}}$$

$$\times e^{i(\rho x+\Im t+\vartheta )},$$

(219)

and

$${\mathit{u}}_{2.2}={\left(\frac{\sqrt{-s_3^2{s}_4{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{{\mathcal{L}}_6}\left(\frac{\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}+2{\mathcal{L}}_5\sqrt{-s_4\left({\mathcal{L}}_1+2\right)}}{4s_4\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}}-\frac{s_2{csc}^2\left[\frac{(x+\gamma t)\sqrt{-s_2}}{2}\right]}{s_3\left(s_3+2\sqrt{-s_2{s}_4}cot\left[(\frac{(x+\gamma t)\sqrt{-s_2}}{2}\right]\right)}\right)\right)}^{\frac{1}{n}}$$

$$\times e^{i(\rho x+\Im t+\vartheta )},$$

(220)

$${\mathit{v}}_{2.2}=\mathcal{A}{\left(\frac{\sqrt{-s_3^2{s}_4{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{{\mathcal{L}}_6}\left(\frac{\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}+2{\mathcal{L}}_5\sqrt{-s_4\left({\mathcal{L}}_1+2\right)}}{4s_4\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}}-\frac{s_2{csc}^2\left[\frac{(x+\gamma t)\sqrt{-s_2}}{2}\right]}{s_3\left(s_3+2\sqrt{-s_2{s}_4}cot\left[(\frac{(x+\gamma t)\sqrt{-s_2}}{2}\right]\right)}\right)\right)}^{\frac{1}{n}}$$

$$\times e^{i(\rho x+\Im t+\vartheta )}.$$

(221)

(2.2) 

If $s_2>0$, we obtain singular combo solitons as

$${\mathit{u}}_{2.3}={\left(\frac{\sqrt{-s_3^2{s}_4{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{{\mathcal{L}}_6}\left(\frac{\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}+2{\mathcal{L}}_5\sqrt{-s_4\left({\mathcal{L}}_1+2\right)}}{4s_4\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}}+\frac{s_{2}csc{h}^2\left[\frac{(x+\gamma t)\sqrt{s_2}}{2}\right]}{s_3\left(s_3+2\sqrt{s_2{s}_4}coth\left[(\frac{(x+\gamma t)\sqrt{s_2}}{2}\right]\right)}\right)\right)}^{\frac{1}{n}}$$

$$\times e^{i(\rho x+\Im t+\vartheta )},$$

(222)

$${\mathit{v}}_{2.3}=\mathcal{A}{\left(\frac{\sqrt{-s_3^2{s}_4{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{{\mathcal{L}}_6}\left(\frac{\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}+2{\mathcal{L}}_5\sqrt{-s_4\left({\mathcal{L}}_1+2\right)}}{4s_4\left(2{\mathcal{L}}_1+3\right)\sqrt{s_3^2{\mathcal{L}}_6}}+\frac{s_{2}csc{h}^2\left[\frac{(x+\gamma t)\sqrt{s_2}}{2}\right]}{s_3\left(s_3+2\sqrt{s_2{s}_4}coth\left[(\frac{(x+\gamma t)\sqrt{s_2}}{2}\right]\right)}\right)\right)}^{\frac{1}{n}}$$

$$\times e^{i(\rho x+\Im t+\vartheta )}.$$

(223)

Case: 3 $s_0=s_1=s_2=0$.

$${\alpha }_0=-\frac{{\mathcal{L}}_5\left({\mathcal{L}}_1+2\right)}{2{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)}+\frac{\sqrt{-s_3^2{s}_4{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_4{\mathcal{L}}_6},{\alpha }_1=\frac{\sqrt{-s_3^2{s}_4{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{s_3{\mathcal{L}}_6},{\beta }_1=0,$$

${\mathcal{L}}_2=\frac{{\mathcal{L}}_1({\mathcal{L}}_1+2)}{256s_4^2{\mathcal{L}}_6^3{(2{\mathcal{L}}_1+3)}^4}(16s_4^2{\mathcal{L}}_5^4{({\mathcal{L}}_1+2)}^2+24s_3^2{s}_4{\mathcal{L}}_6{\mathcal{L}}_5^2({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^2$

$-s_3^2(2{\mathcal{L}}_1+3)(3s_3^2{\mathcal{L}}_6^2{(2{\mathcal{L}}_1+3)}^3+16{\mathcal{L}}_5\sqrt{-s_3^2{s}_4{\mathcal{L}}_6^3({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^4})),$

${\mathcal{L}}_3=\frac{\left(2{\mathcal{L}}_1+1\right)}{16s_4^2\left(2{\mathcal{L}}_1+3\right){}^3{\mathcal{L}}_6^2}(4s_4^2\left({\mathcal{L}}_1+2\right){}^2{\mathcal{L}}_5^3+3s_3^2{s}_4\left({\mathcal{L}}_1+2\right)\left(2{\mathcal{L}}_1+3\right){}^2{\mathcal{L}}_6{\mathcal{L}}_5$

$-s_3^2\left(2{\mathcal{L}}_1+3\right)\sqrt{-s_3^2{s}_4\left({\mathcal{L}}_1+2\right)\left(2{\mathcal{L}}_1+3\right){}^4{\mathcal{L}}_6^3}),$

$${\mathcal{L}}_4=-\frac{3\left({\mathcal{L}}_1+1\right)\left(s_3^2{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right){}^2+4s_4\left({\mathcal{L}}_1+2\right){\mathcal{L}}_5^2\right)}{8s_4\left(2{\mathcal{L}}_1+3\right){}^2{\mathcal{L}}_6}.$$

Through this case, if $s_4\ne 0$, we obtain rational solutions to (1) and (2), under the conditions $s_4\mathrm{and}({\mathcal{L}}_1+2)>0,s_3,{\mathcal{L}}_6,{\mathcal{L}}_5\mathrm{and}\left(2{\mathcal{L}}_1+3\right)<0$, and $s_3^2{(x+\gamma t)}^2>4s_4$, in the following forms:

$${\mathit{u}}_3={\left(\frac{\sqrt{-s_4{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_4{\mathcal{L}}_6}\left(\frac{s_3\left(2{\mathcal{L}}_1+3\right){\mathcal{L}}_6+2{\mathcal{L}}_5\sqrt{-s_4\left({\mathcal{L}}_1+2\right){\mathcal{L}}_6}}{\left(2{\mathcal{L}}_1+3\right){\mathcal{L}}_6}+\frac{16s_3{s}_4}{s_3^2{(x+\gamma t)}^2-4s_4}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(224)

$${\mathit{v}}_3=\mathcal{A}{\left(\frac{\sqrt{-s_4{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_4{\mathcal{L}}_6}\left(\frac{s_3\left(2{\mathcal{L}}_1+3\right){\mathcal{L}}_6+2{\mathcal{L}}_5\sqrt{-s_4\left({\mathcal{L}}_1+2\right){\mathcal{L}}_6}}{\left(2{\mathcal{L}}_1+3\right){\mathcal{L}}_6}+\frac{16s_3{s}_4}{s_3^2{(x+\gamma t)}^2-4s_4}\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(225)

Case: 4 $s_3=s_4=0$.

$${\alpha }_0=-\frac{{\mathcal{L}}_5\left({\mathcal{L}}_1+2\right)}{2{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)}+\frac{\sqrt{-s_0{s}_1^2{\mathcal{L}}_6^3\left({\mathcal{L}}_1+2\right){\left(2{\mathcal{L}}_1+3\right)}^4}}{4s_0{\mathcal{L}}_6^2{\left(2{\mathcal{L}}_1+3\right)}^2},{\alpha }_1=0,$$

$${\beta }_1=\frac{\sqrt{-s_0{s}_1^2{\mathcal{L}}_6^3\left({\mathcal{L}}_1+2\right){\left(2{\mathcal{L}}_1+3\right)}^4}}{s_1{\mathcal{L}}_6^2{\left(2{\mathcal{L}}_1+3\right)}^2},$$

${\mathcal{L}}_2=\frac{{\mathcal{L}}_1({\mathcal{L}}_1+2)}{256s_0^2{\mathcal{L}}_6^3{(2{\mathcal{L}}_1+3)}^4}(16s_0^2{\mathcal{L}}_5^2({\mathcal{L}}_1+2)\left[{\mathcal{L}}_5^2({\mathcal{L}}_1+2)-4s_2{\mathcal{L}}_6{(2{\mathcal{L}}_1+3)}^2\right]$

$-s_1^2(2{\mathcal{L}}_1+3)(3s_1^2{\mathcal{L}}_6^2{(2{\mathcal{L}}_1+3)}^3+16{\mathcal{L}}_5\sqrt{-s_0{s}_1^2{\mathcal{L}}_6^3({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^4})$

$+8s_0(2{\mathcal{L}}_1+3)(s_1^2{\mathcal{L}}_6(2{\mathcal{L}}_1+3)(2s_2{\mathcal{L}}_6{(2{\mathcal{L}}_1+3)}^2+3({\mathcal{L}}_1+2){\mathcal{L}}_5^2)$

$+8s_2{\mathcal{L}}_5\sqrt{-s_0{s}_1^2{\mathcal{L}}_6^3({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^4}\left)\right),$

${\mathcal{L}}_3=\frac{2{\mathcal{L}}_1+1}{16s_0^2{\mathcal{L}}_6^2{(2{\mathcal{L}}_1+3)}^3}(4s_0^2({\mathcal{L}}_1+2){\mathcal{L}}_5({\mathcal{L}}_5^2({\mathcal{L}}_1+2)-2s_2{\mathcal{L}}_6{(2{\mathcal{L}}_1+3)}^2)$

$-s_1^2(2{\mathcal{L}}_1+3)\sqrt{-s_0{s}_1^2{\mathcal{L}}_6^3({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^4}+s_0(2{\mathcal{L}}_1+3)(3s_1^2{\mathcal{L}}_5{\mathcal{L}}_6(2{\mathcal{L}}_1^2+7{\mathcal{L}}_1+6)+4s_2\sqrt{-s_0{s}_1^2{\mathcal{L}}_6^3({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^4})),$

$${\mathcal{L}}_4=\frac{\left({\mathcal{L}}_1+1\right)\left[-3s_1^2{\mathcal{L}}_6{\left(2{\mathcal{L}}_1+3\right)}^2-4s_0\left(3\left({\mathcal{L}}_1+2\right){\mathcal{L}}_5^2-2s_2{\mathcal{L}}_6{\left(2{\mathcal{L}}_1+3\right)}^2\right)\right]}{8s_0{\mathcal{L}}_6{\left(2{\mathcal{L}}_1+3\right)}^2}.$$

Through this case, if $({\mathcal{L}}_1+2)<0,\left(2{\mathcal{L}}_1+3\right),{\mathcal{L}}_5\mathrm{and}{\mathcal{L}}_6>0,$ then the exact solutions of (1) and (2) are in the following forms:

(4.1) 

If $s_1<0,s_2>0$ and $s_0=\frac{s_1^2}{4s_2}$, we obtain exponential solutions as

$${\mathit{u}}_{4.1}={\left(\frac{\sqrt{-4s_1^2{s}_2{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_1^2{\mathcal{L}}_6}\left(\frac{2{\mathcal{L}}_5\sqrt{-s_1^2{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}+{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)\sqrt{4s_2{s}_1^2}}{{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)\sqrt{4s_2}}+\frac{2s_1^2}{-s_1+2s_{2}exp\left[(x+\gamma t)\sqrt{s_2}\right]}\right)\right)}^{\frac{1}{n}}$$

$$\times e^{i(\rho x+\Im t+\vartheta )},$$

(226)

$${\mathit{v}}_{4.1}=\mathcal{A}{\left(\frac{\sqrt{-4s_1^2{s}_2{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_1^2{\mathcal{L}}_6}\left(\frac{2{\mathcal{L}}_5\sqrt{-s_1^2{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}+{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)\sqrt{4s_2{s}_1^2}}{{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)\sqrt{4s_2}}+\frac{2s_1^2}{-s_1+2s_{2}exp\left[(x+\gamma t)\sqrt{s_2}\right]}\right)\right)}^{\frac{1}{n}}$$

$$\times e^{i(\rho x+\Im t+\vartheta )}.$$

(227)

(4.2) 

If $s_2<0$, $s_1=0$ and $s_0>0$, we derive periodic solutions as

$${\mathit{u}}_{4.2}={\left(\frac{\sqrt{-s_0{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_0{\mathcal{L}}_6}\left(\frac{2{\mathcal{L}}_5\sqrt{-s_0{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)}+4\sqrt{-s_2{s}_0}csc\left[(x+\gamma t)\sqrt{-s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(228)

$${\mathit{v}}_{4.2}=\mathcal{A}{\left(\frac{\sqrt{-s_0{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_0{\mathcal{L}}_6}\left(\frac{2{\mathcal{L}}_5\sqrt{-s_0{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)}+4\sqrt{-s_2{s}_0}csc\left[(x+\gamma t)\sqrt{-s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(229)

(4.3) 

If $s_2>0$, $s_1=0$ and $s_0>0$, we extract singular solitons as

$${\mathit{u}}_{4.3}={\left(\frac{\sqrt{-s_0{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_0{\mathcal{L}}_6}\left(\frac{2{\mathcal{L}}_5\sqrt{-s_0{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)}+4\sqrt{s_2{s}_0}csch\left[(x+\gamma t)\sqrt{s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )},$$

(230)

$${\mathit{v}}_{4.3}=\mathcal{A}{\left(\frac{\sqrt{-s_0{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_0{\mathcal{L}}_6}\left(\frac{2{\mathcal{L}}_5\sqrt{-s_0{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)}+4\sqrt{s_2{s}_0}csch\left[(x+\gamma t)\sqrt{s_2}\right]\right)\right)}^{\frac{1}{n}}{e}^{i(\rho x+\Im t+\vartheta )}.$$

(231)

Case: 5 $s_2=s_4=0,s_3>0,s_0\mathrm{and}{s}_1\ne 0$.

$${\alpha }_0=-\frac{{\mathcal{L}}_5\left({\mathcal{L}}_1+2\right)}{2{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)}+\frac{s_1\sqrt{-s_0{s}_1^2{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_0{s}_1{\mathcal{L}}_6},{\alpha }_1=0,{\beta }_1=\frac{\sqrt{-s_0{s}_1^2{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{s_1{\mathcal{L}}_6},$$

${\mathcal{L}}_2=-\frac{{\mathcal{L}}_1({\mathcal{L}}_1+2)}{256s_0^2{s}_1{\mathcal{L}}_6^3{(2{\mathcal{L}}_1+3)}^4}(s_1^3(2{\mathcal{L}}_1+3)(3s_1^2{\mathcal{L}}_6^2{(2{\mathcal{L}}_1+3)}^3$

$+16{\mathcal{L}}_5\sqrt{-s_0{s}_1^2{\mathcal{L}}_6^3({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^4})$

$-24s_0{s}_1^3{\mathcal{L}}_5^2{\mathcal{L}}_6({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^2+16s_0^2(4s_1^2{s}_3{\mathcal{L}}_6^2{(2{\mathcal{L}}_1+3)}^4$

$+8s_3{\mathcal{L}}_5(2{\mathcal{L}}_1+3)\sqrt{-s_0{s}_1^2{\mathcal{L}}_6^3({\mathcal{L}}_1+2){(2{\mathcal{L}}_1+3)}^4}-s_1{\mathcal{L}}_5^4{({\mathcal{L}}_1+2)}^2\left)\right),$

${\mathcal{L}}_3=\frac{2{\mathcal{L}}_1+1}{16s_0^2{s}_1{\mathcal{L}}_6^2{(2{\mathcal{L}}_1+3)}^3}(3s_0{s}_1^3{\mathcal{L}}_5{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right){\left(2{\mathcal{L}}_1+3\right)}^2$

$-s_1^3\left(2{\mathcal{L}}_1+3\right)\sqrt{-s_0{s}_1^2{\mathcal{L}}_6^3\left({\mathcal{L}}_1+2\right){\left(2{\mathcal{L}}_1+3\right)}^4}+4s_0^2(s_1{\mathcal{L}}_5^3{\left({\mathcal{L}}_1+2\right)}^2-2s_3\left(2{\mathcal{L}}_1+3\right)\sqrt{-s_0{s}_1^2{\mathcal{L}}_6^3\left({\mathcal{L}}_1+2\right){\left(2{\mathcal{L}}_1+3\right)}^4})).$

$${\mathcal{L}}_4=-\frac{3\left({\mathcal{L}}_1+1\right)\left(s_1^2{\mathcal{L}}_6{\left(2{\mathcal{L}}_1+3\right)}^2+4s_0{\mathcal{L}}_5^2\left({\mathcal{L}}_1+2\right)\right)}{8s_0{\mathcal{L}}_6{\left(2{\mathcal{L}}_1+3\right)}^2}.$$

Through this case, if $({\mathcal{L}}_1+2)<0,{\mathcal{L}}_5\mathrm{and}\left(2{\mathcal{L}}_1+3\right)>0,{\mathcal{L}}_6\mathrm{and}{s}_i>0,i=0,1$, then we obtain Weierstrass solutions of (1) and (2) in the following forms:

$${\mathit{u}}_5={\left(\frac{\sqrt{-s_0{s}_1^2{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_0{s}_1{\mathcal{L}}_6}\left(\frac{s_1^2{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)+2{\mathcal{L}}_5\sqrt{-s_0{s}_1^2{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{s_1{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)}+\frac{4s_0}{\wp \left[\frac{(x+\gamma t)\sqrt{s_3}}{2},{\mathcal{E}}_2,{\mathcal{E}}_3\right]}\right)\right)}^{\frac{1}{n}}$$

$$\times e^{i(\rho x+\Im t+\vartheta )},$$

(232)

$${\mathit{v}}_5=\mathcal{A}{\left(\frac{\sqrt{-s_0{s}_1^2{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{4s_0{s}_1{\mathcal{L}}_6}\left(\frac{s_1^2{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)+2{\mathcal{L}}_5\sqrt{-s_0{s}_1^2{\mathcal{L}}_6\left({\mathcal{L}}_1+2\right)}}{s_1{\mathcal{L}}_6\left(2{\mathcal{L}}_1+3\right)}+\frac{4s_0}{\wp \left[\frac{(x+\gamma t)\sqrt{s_3}}{2},{\mathcal{E}}_2,{\mathcal{E}}_3\right]}\right)\right)}^{\frac{1}{n}}$$

$$\times e^{i(\rho x+\Im t+\vartheta )},$$

(233)

where ${\mathcal{E}}_2=-\frac{4s_1}{s_3}\mathrm{and}{\mathcal{E}}_3=-\frac{4s_0}{s_3}.$

4. Results and Discussion

The obtained wave solutions in this paper using the extended F-expansion method differ from the gained results of various researchers by some other existing methods. Different families of solutions were obtained from Equation (7) by giving specific values to the parameters. Dark soliton solutions, singular soliton solutions, the dark–singular combo soliton, singular combo soliton solutions, Jacobi elliptic solutions, periodic solutions, combo periodic solutions, hyperbolic solutions, rational solutions, exponential solutions and Weierstrass solutions were extracted. Zayed et al. [31] obtained solitons in magneto-optics waveguides for the nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index using the unified auxiliary equation method. Kengne [32] commented on “Solitons in magneto-optics waveguides for the nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index using the unified auxiliary equation method”. So, several innovative and corrected results have been achieved in this work, which have not been specified before.

Figure 1 presents the dark soliton solutions of Equation (36), when $m=3$, $n=8$, ${\mathit{a}}_2=0.9$, $\mathcal{A}=-0.5,{\xi }_1=0.8,{\xi }_2=0.6,{\varrho }_2=0.5,{\varrho }_1=0.6,{\mathit{b}}_1=0.9,{\mathit{f}}_1=0.8,{\mathit{c}}_1=0.5$, ${\mathit{g}}_1=1.1,{\mathit{d}}_1=-0.8,{\mathit{h}}_1=1.3,{\mathit{e}}_1=1.4,{\alpha }_1=0.9,{\mu }_1=1,{\mathit{k}}_1=0.5,\vartheta =0.8$ and $-15<x<15$. Figure 2 presents the singular soliton solutions of Equation (42), when $m=3,n=3.5,{\mathit{a}}_2=-1.9,\mathcal{A}=1.5,{\xi }_1=2.8,{\xi }_2=1.6,{\varrho }_2=1.5,{\varrho }_1=1.6,{\mathit{b}}_1=1.9$, ${\mathit{f}}_1=1.8,{\mathit{c}}_1=1.5,{\mathit{g}}_1=2.1,{\mathit{d}}_1=-2.8,{\mathit{h}}_1=0.6,{\mathit{e}}_1=1.4,{\alpha }_1=1.9,{\mu }_1=1.4,{\mathit{k}}_1=1.5$, $\vartheta =1.8$ and $-15<x<15$. Figure 3 presents the periodic solutions of Equation (46), when $m=3,n=3.5,{\mathit{a}}_2=-1.8,\mathcal{A}=1.4,{\xi }_1=2.7,{\xi }_2=1.6,{\varrho }_2=1.5,{\varrho }_1=1.6,{\mathit{b}}_1=1.8$, ${\mathit{f}}_1=1.7,{\mathit{c}}_1=1.5,{\mathit{g}}_1=2.2,{\mathit{d}}_1=-2.8,{\mathit{h}}_1=0.6,{\mathit{e}}_1=1.4,{\alpha }_1=1.8,{\mu }_1=1.5,{\mathit{k}}_1=1.4$, $\vartheta =1.7$ and $-15<x<15$. Figure 4 presents the singular combo soliton solutions of Equation (100), when $m=3,n=3.5,{\mathit{a}}_2=-1.7,\mathcal{A}=1.4,{\xi }_1=2.6,{\xi }_2=1.5$, ${\varrho }_2=1.4,{\varrho }_1=1.5,{\mathit{b}}_1=1.6,{\mathit{f}}_1=1.4,{\mathit{c}}_1=1.4,{\mathit{g}}_1=2.2,{\mathit{d}}_1=-2.8,{\mathit{h}}_1=0.6,{\mathit{e}}_1=1.4$, ${\alpha }_1=1.8,{\mu }_1=1.5,{\mathit{k}}_1=1.4,\vartheta =1.7,D_2=\sqrt{2},D_3=-\sqrt{2}$ and $-25<x<25$. Figure 5 presents the hyperbolic solution of Equation (160), when $m=3,n=3.5,{\mathit{a}}_2=-1.6,\mathcal{A}=1.5$, ${\xi }_1=2.7,{\xi }_2=1.5,{\varrho }_2=1.4,{\varrho }_1=1.5,{\mathit{b}}_1=1.6,{\mathit{f}}_1=1.5,{\mathit{c}}_1=1.5,{\mathit{g}}_1=2.1,{\mathit{d}}_1=-2.8$, ${\mathit{h}}_1=0.6,{\mathit{e}}_1=1.5,{\alpha }_1=1.8,{\mu }_1=1.5,{\mathit{k}}_1=1.4,\vartheta =1.7,D_2=\sqrt{2},D_3=-\sqrt{2}$ and $-15<x<15$.

Figure 1. Graphical simulation of dark soliton solutions to Equation (36).

Figure 2. Graphical simulation of singular soliton solutions to Equation (42).

Figure 3. Graphical simulation of periodic solutions to Equation (46).

Figure 4. Graphical simulation of singular combo soliton solutions to Equation (100).

Figure 5. Graphical simulation of hyperbolic solutions to Equation (160).

5. Conclusions

In this work, the coupled system of the nonlinear Biswas–Milovic equation in a magneto-optical waveguide with Kudryashov’s law was studied successfully by applying the extended F-expansion method. Various types of traveling wave solutions were extracted, such as the dark soliton solutions, singular soliton solutions, dark–singular combo soliton, singular combo soliton solutions, Jacobi elliptic solutions, periodic solutions, combo periodic solutions, hyperbolic solutions, rational solutions, exponential solutions and Weierstrass solutions. This paper shows that some of the obtained solutions by this effective method are new compared with the obtained solutions in [31,32]. For certain solutions, contour, three- and two-dimensional visualizations were provided for further illustration. Finally, our solutions were checked using Mathematica software by putting them back into the original equations.