In recent decades, the coupled nonlinear partial differential equations (CNPDE) [1] - [15] have received much more attention in plasma physics [16] , nonlinear optics [17] - [19] , Bose-Einstein condensation [20] - [22] , biophysics [23] [24] , finance [25] [26] , oceanography [27] and other fields. The equations provide mathematical models for studying complex systems and help us to better understand and control these systems. In addition, by solving the equations, we can better reveal the internal laws of many phenomena. Thus, the studying of the exact solutions of CNPDE is of great significance both in theory and practice. However, the studying of the exact solutions of CNPDE is difficult.

Now, nonlinear evolution equations have been a topic of deep theoretical research, hereby, a massive number of mathematicians, physicists and engineers have attempted to invent various approaches by which one can obtain the exact solutions of such equations. In the 1880s, after the French mathematician Darboux proposed the Darboux transformation method, many effective and reliable methods for nonlinear evolution equations were proposed. For example, the Hirota bilinear method [28] , the Painlevé expansion method [29] [30] , Bäcklund transformation [31] [32] , the Riccati equation method [33] [34] , unification and general unity [35] , the Exp-function method [36] , the homogeneous balance method [37] , the $\frac{G^{\prime }}{G}$-polynomial expansion method [38] - [40] , the tanh-function method [41] - [43] , the Jacobian elliptic function expansion method [43] - [45] , the F-expansion method [46] [47] , the auxiliary equation method [48] , the first integration method [49] [50] , etc. Due to the complexity and diversity of partial differential equations, the research methods for the exact solutions of partial differential equations are different until now, and the methods used for different equations are also different.

In 2010, Deng-Shan Wang et al. proposed a similarity transformation for solving the coupled Gross-Pitaevskii equations and derived two types of explicit exact solutions, where the coupled equations are denoted as [51]

$\begin{array}{l}i\frac{\partial {\psi }_1}{\partial t}=\left(-\frac{1}{2}\frac{{\partial }^2}{\partial x^2}+\frac{{\gamma }^2}{2}{x}^2+b_{11}{\left|{\psi }_1\right|}^2+b_{12}{\left|{\psi }_2\right|}^2\right){\psi }_1,\\ i\frac{\partial {\psi }_2}{\partial t}=\left(-\frac{1}{2}\frac{{\partial }^2}{\partial x^2}+\frac{{\gamma }^2}{2}{x}^2+b_{21}{\left|{\psi }_1\right|}^2+b_{22}{\left|{\psi }_2\right|}^2\right){\psi }_2.\end{array}$(1)

In 2012, Fei Jin-Xi et al. employed the improved homogeneous balance principle and mapping method to derive the periodic wave solutions and solitary wave solutions of the generalized (3 + 1)-dimensional Gross-Pitaevskii equation [52]

$i{\psi }_t+\frac{1}{2}\rho \left(t\right){\nabla }^2\psi -g\left(t\right){\left|\psi \right|}^2\psi -V\left(t\right)r^2\psi =0,$(2)

where ${\nabla }^2={\partial }_{xx}+{\partial }_{yy}+{\partial }_{zz}$ and $r^2=x^2+y^2+z^2$.

In 2017, Tao Xu et al. extended the single-component Gross-Pitaevskii equation to a two-component coupled case with damping terms, linear and parabolic density distributions. The coupled equations are given by

$\begin{array}{l}i{q}_{1t}+q_{1xx}+2{\mu }^2\left({\left|q_1\right|}^2+{\left|q_2\right|}^2\right)q_1+\left(i\beta -\alpha x+{\beta }^2{x}^2\right)q_1=0,\\ i{q}_{2t}+q_{2xx}+2{\mu }^2\left({\left|q_1\right|}^2+{\left|q_2\right|}^2\right)q_2+\left(i\beta -\alpha x+{\beta }^2{x}^2\right)q_2=0,\end{array}$(3)

where they derived the Lax pair for the two-component coupled Gross-Pitaevskii equations and obtained multi-nonautonomous solitons, a single breather soliton, and first-order rogue waves using the Darboux transformation [53] .

In 2019, T. Kanna et al. obtained solutions for the coupled two-component and three-component non-autonomous Gross-Pitaevskii systems by introducing appropriate rotation and similarity transformations. The two-component system is given by

$i\frac{\partial {\psi }_j}{\partial t}+\gamma \left(x\right)\frac{{\partial }^2{\psi }_j}{\partial x^2}+2\underset{l=1}{\overset{2}{{\displaystyle \sum }}}{g}_{jl}\left(x\right){\left|{\psi }_l\right|}^2{\psi }_j+V\left(x,t\right){\psi }_j-\underset{l=1,l\ne j}{\overset{2}{{\displaystyle \sum }}}\sigma {\psi }_l=0\left(j=1,2\right),$(4)

where $\gamma \left(x\right)$, $g_{jl}\left(x\right)$, and $V\left(x,t\right)$ are functions characterizing the non-autonomous features of the system, and $\sigma$ is a coupling constant [54] .

In 2020, H. Chaachoua Sameut et al. solved two one-dimensional coupled Gross-Pitaevskii equations with time-varying harmonic traps and found soliton solutions that can be effectively controlled by modulating the frequency of the external potential. The equations are given by

$\begin{array}{l}i{\psi }_{1t}+\frac{1}{2}{\psi }_{1xx}+\left(V\left(x,t\right)+R_{11}{\left|{\psi }_1\right|}^2+R_{12}{\left|{\psi }_2\right|}^2\right){\psi }_1=0,\\ i{\psi }_{2t}+\frac{1}{2}{\psi }_{2xx}+\left(V\left(x,t\right)+R_{21}{\left|{\psi }_1\right|}^2+R_{22}{\left|{\psi }_2\right|}^2\right){\psi }_2=0,\end{array}$(5)

where $V\left(x,t\right)$ represents the time-varying harmonic trap potential, and $R_{ij}$ are interaction coefficients [55] .

In recent years, the properties of Bose-Einstein condensates (BEC) and nonlinear optics under the modulation of scattering length have attracted much attention. For example, the research of Fischbach resonance is well known and has led to the proposal of many new nonlinear phenomena. The Gross-Pitaevskii equation (GPE) is the basic model describing the phenomenon of BEC. The mathematical theoretical study of GPE is not only helpful to understand and analyze the BEC phenomena in physical experiments, but also to predict new BEC phenomena theoretically. This paper studies the following system of the coupled Gross-Pitaevskii equations (CGPE), without an external potential [56] .

$\begin{array}{l}i\frac{\partial u}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}u}{\partial x^2}-\left(\beta {\left|u\right|}^2+\gamma {\left|v\right|}^2\right)u=0,\\ i\frac{\partial v}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}v}{\partial x^2}-\left(\delta {\left|v\right|}^2+\gamma {\left|u\right|}^2\right)v=0,\end{array}$(6)

where, $i=\sqrt{-1}$, $u,v$ are the function of $t$ and $x$. In optics, $t$ represents the propagation time variable, $x$ represents the propagation space variable, $u$ and $v$ represent two unrelated beams of light, respectively. However, in order to better understand the nature and laws of the phenomena of CGPE (6), it is still necessary to study the exact solutions for the equations. In this paper, we transform (6) into a nonlinear coupled ordinary differential system by using coupled traveling wave solution transformation, and then obtain new exact solutions by using the F-expansion method.

The organization of this paper is as follows. Section 1 gives an introduction. Section 2 gives a brief description of the algorithm by using the F-expansion method. Section 3 gives the exact solutions of CGPE (6). Section 4 gives some numerical results and their figures to illustrate the solutions. Finally, the paper ends with a conclusion in Section 5.

In this section, we describe the algorithm of polynomial expansion method by modified polynomial expansion method for finding the exact solutions of the coupled nonlinear partial differential equations. Suppose that the coupled nonlinear partial differential equations, which have independent space variable $x$ and time variable $t$, are given by

$\{\begin{array}{l}P\left(u,v,u_x,v_x,u_t,v_t,u_{x}v,u{v}_x,u_{t}v,u{v}_t,u_{xx},v_{xx},u_{xt},v_{xt},u_{tt},\cdots \right)=0,\hfill \\ Q\left(u,v,u_x,v_x,u_t,v_t,u_{x}v,u{v}_x,u_{t}v,u{v}_t,u_{xx},v_{xx},u_{xt},v_{xt},u_{tt},\cdots \right)=0,\hfill \end{array}$(7)

where $u=u\left(x,t\right),v=v\left(x,t\right)$ are unknown functions of the coupled system (7), $P,Q$ are polynomials of $u\left(x,t\right),v\left(x,t\right)$ and their partial derivatives in which the highest order partial derivatives and the nonlinear terms are involved and the subscripts stand for the partial derivatives.

Next, we will discuss the specific steps of the modified polynomial expansion method.

(i) Transforming the coupled nonlinear partial differential system (7) into the coupled ordinary differential equations (ODE). Suppose that

$\{\begin{array}{l}u=u\left(x-ct\right)=\varphi \left(\xi \right),\hfill \\ v=v\left(x-ct\right)=\phi \left(\xi \right),\hfill \end{array}$(8)

where $\xi =x-ct$ and $c\in \left(R-\left\{0\right\}\right)$ is any real number. Substituting (8) into the coupled system (7), we can obtain

$\{\begin{array}{l}P\left(\varphi ,\phi ,{\varphi }^{\prime },{\phi }^{\prime },{\varphi }^{″},{\phi }^{″},{\varphi }^{\prime }{\phi }^{\prime },{\varphi }^{‴},\cdots \right)=0,\hfill \\ Q\left(\varphi ,\phi ,{\varphi }^{\prime },{\phi }^{\prime },{\varphi }^{″},{\phi }^{″},{\varphi }^{\prime }{\phi }^{\prime },{\varphi }^{‴},\cdots \right)=0,\hfill \end{array}$(9)

where the prime is the derivative with respect to $\xi$.

(ii) Assume that the expressions of the exact solutions of the system (9) are

$\{\begin{array}{l}\varphi \left(\xi \right)={\displaystyle {\sum }_{i=0}^M{a}_i}{\left(F\left(\xi \right)\right)}^i,\hfill \\ \phi \left(\xi \right)={\displaystyle {\sum }_{i=0}^N{b}_i}{\left(F\left(\xi \right)\right)}^i,\hfill \end{array}$(10)

where $F\left(\xi \right)$ satisfies the auxiliary differential equation

$F^{\prime }=\sqrt{c_0+c_2{F}^2+c_4{F}^4},$(11)

where $F=F\left(\xi \right)$, $c_0,c_2$, and $c_4$ are real constants related to the elliptic modulus of elliptic Jacobian function $F\left(\xi \right)$.

(iii) Confirming the value of $M$ and $N$ in (10). To confirm the parameters $M$ and $N$, we use the balance coefficient method by balancing the highest-order derivative term and the highest-degree nonlinear term in the coupled system (9).

(iv) Substituting the value of $M,N$ and the solutions $F$ of Equation (11) into the coupled solutions (9), with the help of the software Maple, we can obtain all the values of $a_i,\left(i=1,2,\cdots ,M\right)$ and the values $b_i,\left(i=1,2,\cdots ,N\right)$, by setting the coefficients of all terms with the same power exponent about $\varphi \left(\xi \right)$ and $\phi \left(\xi \right)$ in (9) to zeros.

(v) Substituting the values of $a_i,\left(i=1,2,\cdots ,M\right)$ and the values $b_i,\left(i=1,2,\cdots ,N\right)$ into (8), we get the exact solutions of the coupled system (7).

Next, we give the exact solutions for CGPE (6). Firstly, we need to transform CGPE (6) into a coupled ordinary differential system.

Suppose the modified coupled traveling wave transformation is as follows

$\{\begin{array}{l}u=u\left(x-ct\right)\text{exp}\left(i\left(n_{1}x-{\omega }_{1}t\right)\right)=\varphi \left(\xi \right)\text{exp}\left(i\left(n_{1}x-{\omega }_{1}t\right)\right),\hfill \\ v=v\left(x-ct\right)\text{exp}\left(ic\left(n_{2}x-{\omega }_{2}t\right)\right)=\phi \left(\xi \right)\text{exp}\left(i\left(n_{2}x-{\omega }_{2}t\right)\right),\hfill \end{array}$(12)

where $\varphi \left(\xi \right)=u\left(x-ct\right),\phi \left(\xi \right)=v\left(x-ct\right)$ are real-valued functions about $\xi =x-ct$, $c\in \left(R-\left\{0\right\}\right)$, $n_1,n_2$ are the wave numbers, and ${\omega }_1,{\omega }_2$ are the frequencies, respectively.

Substituting the modified coupled traveling wave transformation (12) into CGPE (6), we get $n_1=n_2=c$, for simplicity, letting ${\omega }_1=c\mu$ and ${\omega }_2=c\lambda$, then CGPE can be changed into the following coupled ordinary differential system,

$\{\begin{array}{l}\frac{1}{2}{\varphi }^{″}+c\left(\mu -\frac{c}{2}\right)\varphi -\beta {\varphi }^3-\gamma \varphi {\phi }^2=0,\hfill \\ \frac{1}{2}{\phi }^{″}+c\left(\lambda -\frac{c}{2}\right)\phi -\delta {\phi }^3-\gamma \phi {\varphi }^2=0,\hfill \end{array}$(13)

where ${\varphi }^{″}=\frac{{\text{d}}^2\varphi }{\text{d}{\xi }^2},{\phi }^{″}=\frac{{\text{d}}^2\phi }{\text{d}{\xi }^2}$, by balancing the order of the highest derivative term and the highest order nonlinear term of (13), we obtain that $M=1,N=1$.

Consequently, we yield the expression of the exact solutions for CGPE (6) as

$\{\begin{array}{l}\varphi \left(\xi \right)=a_1+a_{2}F\left(\xi \right),\hfill \\ \phi \left(\xi \right)=b_1+b_{2}F\left(\xi \right).\hfill \end{array}$(14)

Substituting (14) into (13), combining terms with the same power of $F\left(\xi \right)$ and setting the coefficient of the same power of $F\left(\xi \right)$ is to zero, we yield the system of algebraic equations about $a_1,a_2,b_1,b_2$ as shown below,

$\begin{array}{l}{a}_1=b_1=0,a_2=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}},\\ b_2=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}},c=c,\lambda =\mu =\frac{c^2-c_2}{2c}.\end{array}$(15)

Then, substituting the solution (15) about $a_1,a_2,b_1,b_2$ and Equation (14) into equation system (13), according to the auxiliary Equation (11), we can obtain the exact solutions for CGPE (6).

In order to obtain the solutions for CGPE (6), we need to give the main solutions $F\left(\xi \right)$ of the auxiliary differential Equation (11) according to the parameters $c_0,c_2,c_4$ as follows [57] [58] : (Note, $F_i\left(\xi \right)\left(i=1,2,\cdots ,37\right)$ are the solutions for Equation (11).)

Case 1:

$\text{If}{c}_0=1,c_2=-k^2-1,c_4=k^2,\text{then}{F}_1\left(\xi \right)=\text{sn}\left(\xi \right).$(16)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_1\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\text{sn}\left(x-ct\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_1\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\text{sn}\left(x-ct\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(17)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 2:

$\text{If}{c}_0=k^2,c_2=-1-k^2,c_4=1,\text{then}{F}_2\left(\xi \right)=\text{ns}\left(\xi \right).$(18)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_2\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\text{ns}\left(x-ct\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_2\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\text{ns}\left(x-ct\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(19)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 3:

$\text{If}{c}_0=k^2-1,c_2=2-k^2,c_4=-1,\text{then}{F}_3\left(\xi \right)=\text{dn}\left(\xi \right).$(20)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_3\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\text{dn}\left(x-ct\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_3\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\text{dn}\left(x-ct\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(21)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 4:

$\text{If}{c}_0=1-k^2,c_2=2k^2-1,c_4=-k^2,\text{then}{F}_4\left(\xi \right)=\text{cn}\left(\xi \right).$(22)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_4\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\text{cn}\left(x-ct\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_4\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\text{cn}\left(x-ct\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(23)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 5:

$\text{If}{c}_0=-k^2,c_2=-1+2k^2,c_4=1-k^2,\text{then}{F}_5\left(\xi \right)=\frac{1}{\text{cn}\left(\xi \right)}.$(24)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_5\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{1}{\text{cn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_5\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{1}{\text{cn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(25)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 6:

$\text{If}{c}_0=-1,c_2=2-k^2,c_4=k^2-1,\text{then}{F}_6\left(\xi \right)=\frac{1}{\text{dn}\left(\xi \right)}.$(26)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_6\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{1}{\text{dn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_6\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{1}{\text{dn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(27)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 7:

$\text{If}{c}_0=1-k^2,c_2=2-k^2,c_4=1,\text{then}{F}_7\left(\xi \right)=\frac{\text{cn}\left(\xi \right)}{\text{sn}\left(\xi \right)}.$(28)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_7\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{cn}\left(x-ct\right)}{\text{sn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_7\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{cn}\left(x-ct\right)}{\text{sn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(29)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 8:

$\text{If}{c}_0=1,c_2=2-k^2,c_4=1-k^2,\text{then}{F}_8\left(\xi \right)=\frac{\text{sn}\left(\xi \right)}{\text{cn}\left(\xi \right)}.$(30)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_8\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{sn}\left(x-ct\right)}{\text{cn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_8\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{sn}\left(x-ct\right)}{\text{cn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(31)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 9:

$\text{If}{c}_0=1,c_2=2k^2-1,c_4=k^2\left(k^2-1\right),\text{then}{F}_9\left(\xi \right)=\frac{\text{sn}\left(\xi \right)}{\text{dn}\left(\xi \right)}.$(32)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_9\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{sn}\left(x-ct\right)}{\text{dn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_9\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{sn}\left(x-ct\right)}{\text{dn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(33)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 10:

$\text{If}{c}_0=k^2\left(k^2-1\right),c_2=2k^2-1,c_4=1,\text{then}{F}_{10}\left(\xi \right)=\frac{\text{dn}\left(\xi \right)}{\text{sn}\left(\xi \right)}.$(34)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{10}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{dn}\left(x-ct\right)}{\text{sn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{10}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{dn}\left(x-ct\right)}{\text{sn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(35)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 11:

$\text{If}{c}_0=\frac{1}{4},c_2=\frac{1-2k^2}{2},c_4=\frac{1}{4},\text{then}{F}_{11}\left(\xi \right)=\text{ns}\left(\xi \right)\pm \text{cs}\left(\xi \right).$(36)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{11}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\left(\text{ns}\left(x-ct\right)\pm \text{cs}\left(x-ct\right)\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{11}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\left(\text{ns}\left(x-ct\right)\pm \text{cs}\left(x-ct\right)\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(37)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 12:

$\text{If}{c}_0=\frac{1-k^2}{4},c_2=\frac{1+k^2}{2},c_4=\frac{1-k^2}{4},\text{then}{F}_{12}\left(\xi \right)=\text{nc}\left(\xi \right)\pm \left(\xi \right).$(38)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{12}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\left(\text{nc}\left(x-ct\right)\pm \text{sc}\left(x-ct\right)\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{12}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\left(\text{nc}\left(x-ct\right)\pm \text{sc}\left(x-ct\right)\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(39)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 13:

$\text{If}{c}_0=\frac{k^2}{4},c_2=\frac{k^2-2}{2},c_4=\frac{1}{4},\text{then}{F}_{13}\left(\xi \right)=\text{ns}\left(\xi \right)+\text{ds}\left(\xi \right).$(40)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{13}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\left(\text{ns}\left(x-ct\right)+\text{ds}\left(x-ct\right)\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{13}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\left(\text{ns}\left(x-ct\right)+\text{ds}\left(x-ct\right)\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(41)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 14:

$\begin{array}{l}\text{If}{c}_0=\frac{k^2}{4},c_2=\frac{k^2-2}{2},c_4=\frac{k^2}{4},\\ \text{then}{F}_{14.1}\left(\xi \right)=\text{sn}\left(\xi \right)\pm i\text{cn}\left(\xi \right);\text{or}{F}_{14.2}\left(\xi \right)=\frac{\text{dn}\left(\xi \right)}{i\sqrt{1-k^2}\text{sn}\left(\xi \right)\pm \text{cn}\left(\xi \right)}.\end{array}$(42)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{14.1}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\left(\text{sn}\left(x-ct\right)\pm i\text{cn}\left(x-ct\right)\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{14.1}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\left(\text{sn}\left(x-ct\right)\pm i\text{cn}\left(x-ct\right)\right)\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right);\hfill \end{array}$(43)

or

$\{\begin{array}{l}{u}_{14.2}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{dn}\left(x-ct\right)}{i\sqrt{1-k^2}\text{sn}\left(x-ct\right)\pm \text{cn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{14.2}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{dn}\left(x-ct\right)}{i\sqrt{1-k^2}\text{sn}\left(x-ct\right)\pm \text{cn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(44)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 15:

$\text{If}{c}_0=1,c_2=2-4k^2,c_4=1,\text{then}{F}_{15}\left(\xi \right)=\frac{\text{sn}\left(\xi \right)\text{dn}\left(\xi \right)}{\text{cn}\left(\xi \right)}.$(45)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{15}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{sn}\left(x-ct\right)\text{dn}\left(x-ct\right)}{\text{cn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{15}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{sn}\left(x-ct\right)\text{dn}\left(x-ct\right)}{\text{cn}\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(46)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 16:

$\begin{array}{l}\text{If}{c}_0=\frac{{\left(k-1\right)}^2}{4D_1^2},c_2=\frac{1+k^2+6k}{2},c_4=\frac{D_1^2{\left(k-1\right)}^2}{4},\\ \text{then}{F}_{16}\left(\xi \right)=\frac{\text{dn}\left(\xi \right)\text{cn}\left(\xi \right)}{D_1\left(1+\text{sn}\left(\xi \right)\right)\left(1+k\text{sn}\left(\xi \right)\right)}.\end{array}$(47)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{16}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{dn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{D_1\left(1+\text{sn}\left(x-ct\right)\right)\left(1+k\text{sn}\left(x-ct\right)\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right)\hfill \\ v_{16}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{dn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{D_1\left(1+\text{sn}\left(x-ct\right)\right)\left(1+k\text{sn}\left(x-ct\right)\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(48)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 17:

$\begin{array}{l}\text{If}{c}_0=\frac{{\left(k+1\right)}^2}{4D_1^2},c_2=\frac{1+k^2-6k}{2},c_4=\frac{D_1^2{\left(k+1\right)}^2}{4},\\ \text{then}{F}_{17}\left(\xi \right)=\frac{\text{dn}\left(\xi \right)\text{cn}\left(\xi \right)}{D_1\left(1+\text{sn}\left(\xi \right)\right)\left(1-k\text{sn}\left(\xi \right)\right)}.\end{array}$(49)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{17}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{dn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{D_1\left(1+\text{sn}\left(x-ct\right)\right)\left(1-k\text{sn}\left(x-ct\right)\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{17}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{\text{dn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{D_1\left(1+\text{sn}\left(x-ct\right)\right)\left(1-k\text{sn}\left(x-ct\right)\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(50)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma ,D_1$ and $c$ are all real constants.

Case 18:

$\begin{array}{l}\text{If}{c}_0=-2k^3+k^4+k^2,c_2=6k-k^2-1,c_4=-\frac{4}{k},\\ \text{then}{F}_{18}\left(\xi \right)=\frac{k\text{dn}\left(\xi \right)\text{cn}\left(\xi \right)}{1+k{\text{sn}}^2\left(\xi \right)}.\end{array}$(51)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{18}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{k\text{dn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{1+k{\text{sn}}^2\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{18}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{k\text{dn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{1+k{\text{sn}}^2\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(52)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 19:

$\begin{array}{l}\text{If}{c}_0=2k^3+k^4+k^2,c_2=-6k-k^2-1,c_4=\frac{4}{k},\\ \text{then}{F}_{19}\left(\xi \right)=\frac{k\text{dn}\left(\xi \right)\text{cn}\left(\xi \right)}{k{\text{sn}}^2\left(\xi \right)-1}.\end{array}$(53)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{19}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{k\text{dn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{k{\text{sn}}^2\left(x-ct\right)-1}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{19}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{k\text{dn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{k{\text{sn}}^2\left(x-ct\right)-1}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(54)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 20:

$\begin{array}{l}\text{If}{c}_0=2+2k_1-k^2,c_2=6k_1-k^2-2,c_4=4k_1,\\ \text{then}{F}_{20}\left(\xi \right)=\frac{k^2\text{sn}\left(\xi \right)\text{cn}\left(\xi \right)}{k_1-{\text{dn}}^2\left(\xi \right)};\end{array}$(55)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{20}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{k^2\text{sn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{k_1-{\text{dn}}^2\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{20}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{k^2\text{sn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{k_1-{\text{dn}}^2\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right);\hfill \end{array}$(56)

where $0<k<1$ denotes the modulus of the Jacobian elliptic function, $k_1=\sqrt{1-k^2}$, $\beta ,\delta ,\gamma$ and $c$ are all real constants.

Case 21:

$\begin{array}{l}\text{If}{c}_0=2-2k_1-k^2,c_2=-6k_1-k^2+2,c_4=-4k_1,\\ \text{then}{F}_{21}\left(\xi \right)=-\frac{k^2\text{sn}\left(\xi \right)\text{cn}\left(\xi \right)}{k_1+{\text{dn}}^2\left(\xi \right)}.\end{array}$(57)

Thus, the solutions for CGPE (6) admit,

$\{\begin{array}{l}{u}_{21}\left(x,t\right)=\pm \sqrt{\frac{\delta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}\frac{k^2\text{sn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{k_1+{\text{dn}}^2\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \\ v_{21}\left(x,t\right)=\pm \sqrt{\frac{\beta c_4-c_4\gamma }{\beta \delta -{\gamma }^2}}-\frac{k^2\text{sn}\left(x-ct\right)\text{cn}\left(x-ct\right)}{k_1+{\text{dn}}^2\left(x-ct\right)}\text{exp}\left(ic\left(x-\frac{c^2-c_2}{2c}t\right)\right),\hfill \end{array}$(58)