The Extended Weierstrass Transformation Method for the Biswas–Arshed Equation with Beta Time Derivative

Abstract

In this article, exact solutions of the Biswas–Arshed equation are obtained using the extended Weierstrass transformation method (EWTM). This method is widely used in solid-state physics, electrodynamics, and mathematical physics, and it yields exact solution functions involving trigonometric, rational trigonometric, Weierstrass elliptic, wave, and rational functions. The process involves expanding the solution functions of an elliptic differential equation into finite series by transforming them into Weierstrass functions. Furthermore, it generates parametric solutions for nonlinear algebraic equation systems, which are particularly useful in mathematical physics. These solutions are derived using the Mathematica package program. To analyze the behavior of these determined solution functions, the article employs separate two- and three-dimensional graphs showing the real and imaginary components, along with contour and density graphs. These visuals aid in comprehending the physical characteristics exhibited by these solution functions.

1. Introduction

Many phenomena in nature and interdisciplinary science are described by nonlinear partial differential equations. Considering that these equations have applications in many fields such as physics, agriculture, medicine, fiber optics, plasma physics, engineering, aviation and biology, it is of great importance to model real-life problems and determine the numerical, analytical, or exact solutions of these models. On the other hand, the fact that these equations contain various types of integer or fractional derivatives and that the solution spaces are infinitely dimensional have made it necessary to investigate their solutions and determine the physical behavior of the solutions. The nonlinearity of the partial differential equations considered makes it almost impossible to calculate their analytical solutions. For this reason, the numerical and the exact solutions of such problems have been mainly investigated recently. In order to demonstrate the effectiveness of the results obtained by numerical solutions of these problems, it is necessary to calculate the exact solutions and thus determine how little error can be approached to the analytical solution. The exact solutions not only express wave solutions of nonlinear problems, but they also enable analysis of their physical behavior. For this reason, calculating the exact solutions of any nonlinear integer or fractional partial differential equation is very important for studies in mathematical physics [1,2,3]. On the other hand, these equations have different properties in terms of obtaining their solutions compared to linear equations. This is so much so that, while the superposition rule is valid for linear equations, integrability properties are examined for nonlinear equations. Some methods such as inverse scattering transform [4,5], infinite symmetries [6,7], Bäcklund and Darboux transforms [8,9,10], Lax pairs [11,12,13], and Painlevé analysis [14,15] are used in determining integrability, which is of great importance in demonstrating the existence of solutions. These methods play an active role in both investigating the existence of solutions to nonlinear partial differential equations and finding their exact solutions.

In recent years, the studies on the concepts of the fractional derivative and fractional integral, which have been observed to be considered as alternatives to the classical derivative and integral concepts, and the exact solutions of nonlinear partial differential equations defined by these concepts have gained great momentum. This has led to the introduction of new concepts of the fractional derivatives and integrals in order to present the alternative suggestions to the literature, as they enable the definition of new approaches and the achievement of new results in modeling real-life problems and researching the solutions of the obtained models. In this context, when the development processes of different fractional derivatives and fractional integrals are reviewed, it can be seen that there are a large number of alternative definitions. The concepts of the fractional derivative and fractional integral were studied in detail by famous mathematicians such as Riemann, Liouville, Abel, Laurent, Hardy, and Littlewood during the 18th and 19th centuries. There are various definitions of fractional derivatives in the literature, such as Riemann–Liouville [16], Caputo [17], Atangana–Baleanu [18], Riesz [19], Weyl [20], and Hadamard [21]. Since the fractional derivatives are not uniform, as in the definition of the classical derivative with integer order, different fractional derivative definitions have been used to present alternative approaches to solve the encountered physical problems. Apart from these different known fractional derivatives, a new fractional derivative called the conformable fractional derivative, based on the classical integer-order derivative definition, has been defined in order to eliminate the complexity of fractional derivatives and make them understandable [22]. Many nonlinear physical models have been defined on conformable derivatives, and many studies have been conducted on the numerical and the exact solutions of these problems [23,24,25,26]. Then, a new conformable derivative idea called the M-truncated derivative, which was applied for many exact solution methods, and it was introduced in [27,28,29]. On the other hand, the conformable beta derivative developed by Atangana and his colleagues has been widely used in modeling various real-life problems and for searching for the exact solutions of these models [30,31,32,33]. This new derivative, also called Atangana’s beta derivative, has attracted great attention from many authors because it also provides the properties of the classical integer-order derivative. There are methods that can be used to classify the exact solutions of many differential equations, especially beta derivative differential equations. Some of these methods can be listed as the trial equation method [34,35,36], the extended trial equation method [37,38,39,40], the generalized tanh method [41,42], the bilinear neural network method [43,44], the homogeneous balance method [45,46], the Jacobi elliptic function method [47,48], the Kudryashov method [49,50,51], the simplest equation method [52,53], the F-expansion method [54,55], bifurcation analysis, and soliton solutions [56,57,58]. In this study, the exact solutions of a differential equation with the beta conformable derivative were investigated using the Weierstrass functions obtained from the solution of the elliptic differential equation. For this purpose, by considering a finite series approach called the extended Weierstrass transformation method in the literature [59,60,61], it has been attempted to determine whether there are exact solutions of the relevant equation, including the trigonometric, the Weierstrass elliptic, and the rational functions.

The rest of the paper is organized as follows: Section 2 describes the Weierstrass transformation method and the general form of nonlinear partial differential equations with beta time derivative; it additionally gives the beta derivative and some of its basic properties. Section 3 includes an application for calculating wave solutions of the Biswas equation with a beta derivative. Section 4 presents visual descriptions of certain solution functions in the form of two-dimensional, three-dimensional, and contour plots to analyze the results. In Section 5, the results obtained in this study are included.

2. Definition of EWTM for Nonlinear Models with Beta Time Derivative

In this section, we will introduce the EWTM for determining the traveling wave solutions, including the various Weierstrass elliptic functions of the nonlinear physical problems, especially the nonlinear partial differential equations with beta time derivatives. Although this method is used in the literature to calculate the exact solutions of nonlinear partial differential equations, it will be applied for the first time to a nonlinear partial differential equation containing the conformable beta derivative in this study. Therefore, the results obtained in this research are of great importance in the field of mathematical physics. Let us provide a brief explanation of this method in detail as follows. Consider the general form of nonlinear partial differential equations with beta time derivatives in variables $x_i$, $i=1,2,3,\dots ,n$m and t as follows:

$$\begin{array}{c}\hfill \mathrm{\Phi }\left(u,{}_0{}^{A}D_t^{\beta }\left\{u\right\},{}_0{}^{A}D_t^{\beta }\left\{u_{x_i}\right\},{}_0{}^{A}D_t^{\beta }\left\{u_{x_i{x}_i}\right\},u_{x_i},u_{x_i{x}_i},u_{x_{i}t},u_{x_i{x}_i{x}_i},\dots \right)=0,\end{array}$$

(1)

where $\mathrm{\Phi }$ is, in general, a polynomial in $u(x_i,t)$ and its various partial derivatives. The definition of the beta derivative used in Equation (1) and some of its basic properties are given below.

Definition 1.

The definition of the beta derivative is described as follows [62]:

$$\begin{array}{c}\hfill {}_0{}^{A}D_t^{\beta }\left\{\mathrm{ϝ}\left(t\right)\right\}=\underset{\epsilon \to 0}{lim}{\displaystyle \frac{\mathrm{ϝ}\left(t+\epsilon {\left(t+\left(1/\mathrm{\Gamma }\left(\beta \right)\right)\right)}^{1-\beta }\right)-\mathrm{ϝ}\left(t\right)}{\epsilon }}.\end{array}$$

(2)

This derivative is considered because it preserves the basic rules and properties required between the partial derivative and the ordinary derivative, which are used to transform the nonlinear partial differential equations containing beta derivatives as in Equation (1) into nonlinear ordinary differential equations with the traveling wave transform. According to these aspects, the various effective properties of the beta derivative can be listed as follows:

Let $\mathrm{\Xi }\ne 0$ and $\chi$ be functions that are differentiable with respect to $\beta$ in the range $\beta \in \left(0,1\right].$ Accordingly, the equality that can satisfy all the real numbers $\kappa$ and $\sigma$ is as follows:

$$\begin{array}{c}\hfill {}_0{}^{A}D_t^{\beta }\left\{\kappa \chi \left(t\right)+\sigma \mathrm{\Xi }\left(t\right)\right\}=\kappa {}_0{}^{A}D_t^{\beta }\left\{\chi \left(t\right)\right\}+\sigma {}_0{}^{A}D_t^{\beta }\left\{\mathrm{\Xi }\left(t\right)\right\}.\end{array}$$

(3)

For any constant $\zeta$, the following equality is satisfied:

$$\begin{array}{c}\hfill {}_0{}^{A}D_t^{\beta }\left\{\zeta \right\}=0.\end{array}$$

(4)

Also, the folllowing features can be easily obtained:

$$\begin{array}{c}\hfill {}_0{}^{A}D_t^{\beta }\left\{\chi \left(t\right)\mathrm{\Xi }\left(t\right)\right\}=\mathrm{\Xi }\left(t\right){}_0{}^{A}D_t^{\beta }\left\{\chi \left(t\right)\right\}+\chi \left(t\right){}_0{}^{A}D_t^{\beta }\left\{\mathrm{\Xi }\left(t\right)\right\},\end{array}$$

(5)

$$\begin{array}{c}\hfill {}_0{}^{A}D_t^{\beta }\left\{{\displaystyle \frac{\chi \left(t\right)}{\mathrm{\Xi }\left(t\right)}}\right\}={\displaystyle \frac{\mathrm{\Xi }\left(t\right){}_0{}^{A}D_t^{\beta }\left\{\chi \left(t\right)\right\}-\chi \left(t\right){}_0{}^{A}D_t^{\beta }\left\{\mathrm{\Xi }\left(t\right)\right\}}{{\mathrm{\Xi }}^2\left(t\right)}}.\end{array}$$

(6)

If $\phi =\aleph {\left(t+\left(1/\mathrm{\Gamma }\left(\beta \right)\right)\right)}^{\beta -1}$ is written instead of $\phi$ in Equation (2) and $\aleph \to 0$, when $\phi \to 0$, it is taken as follows:

$$\begin{array}{c}\hfill {}_0{}^{A}D_t^{\beta }\left\{\chi \left(t\right)\right\}={\left(t+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\beta \right)}}\right)}^{1-\beta }{\displaystyle \frac{d\chi \left(t\right)}{dt}},\end{array}$$

(7)

with

$$\begin{array}{c}\hfill \xi ={\displaystyle \frac{\varrho }{\beta }}{\left(t+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\beta \right)}}\right)}^{\beta },\end{array}$$

(8)

where $\varrho$ is a constant, and thus, the relation is found as follows:

$$\begin{array}{c}\hfill {}_0{}^{A}D_t^{\beta }\left\{\chi \left(\xi \right)\right\}=\varrho {\displaystyle \frac{d\chi \left(\xi \right)}{d\xi }}.\end{array}$$

(9)

Seeking the traveling wave solution of Equation (1), taking $u(x_i,t)=u\left(\eta \right)$ and $\eta ={\sum }_i\left(x_i-{\displaystyle \frac{c}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\beta \right)}}\right)}^{\beta }\right)$, and then using the properties of Definition 1 leads to an ordinary differential equation as

$$\begin{array}{c}\hfill \mathrm{\Phi }\left(u,-c{u}^{\prime },c{u}^{″},u^{\prime },u^{″},u^{\prime ″},\dots \right)=0.\end{array}$$

(10)

In the next step, we assume that the solution to Equation (1) can be represented in a general form:

$$\begin{array}{c}\hfill u\left(\eta \right)=\sum _{k=-N}^N{c}_k{w}^k\left(\eta \right),\end{array}$$

(11)

where $c_k$ are real constants to be determined later, N is fixed by balancing the linear term of the highest-order derivative with the highest nonlinear term in Equation (10), and $w\left(\eta \right)$ satisfies the general elliptic equation [59,60,61]

$$\begin{array}{c}\hfill w^{\prime }\left(\eta \right)={\displaystyle \frac{d}{d\eta }}w\left(\eta \right)=\sqrt{d_0+d_{1}w\left(\eta \right)+d_2{w}^2\left(\eta \right)+d_3{w}^3\left(\eta \right)+d_4{w}^4\left(\eta \right)}.\end{array}$$

(12)

By substituting Equation (11) into Equation (10), along with Equation (12), and equating the coefficients of all powers of $w^j\left(\eta \right)(j=0,\pm 1,\dots )$ to zero, we obtain a system of algebraic equations. Solving this system of nonlinear algebraic equations using Mathematica provides us with explicit expressions for the real constants $c_k$, $d_l(l=0,1,2,3,4)$ and c. The success of algebraic methods relies on the solubility of the nonlinear algebraic system, as trivial solutions would only lead to unhelpful results. Furthermore, solving Equation (1) in a general sense is a challenging task. However, the solutions obtained from Equation (12) correspond to the solution classes of Equation (1). Some specific cases of Equation (1), depending on $d_l$ values, have been provided in reference [61] and are presented here as follows:

For $d_1=d_3=0$, we have the following:

The solutions of Equation (12) in this case are

$$\begin{array}{c}\hfill w_1\left(\eta \right)=\sqrt{{\displaystyle \frac{3\wp (\eta ;g_2,g_3)-d_2}{3d_4}}},\end{array}$$

(13)

and

$$\begin{array}{c}\hfill w_2\left(\eta \right)=\sqrt{{\displaystyle \frac{3d_0}{\wp (\eta ;g_2,g_3)-d_2}}},\end{array}$$

(14)

where the invariants of the Weierstrass function $\wp (\eta ;g_2,g_3)$ are expressed by

$$\begin{array}{c}\hfill g_2={\displaystyle \frac{4}{3}}\left(d_2^2-3d_0{d}_4\right),g_3={\displaystyle \frac{4d_2^2}{27}}(9d_4-2d_2).\end{array}$$

(15)

Another type of solution allows

$$\begin{array}{c}\hfill w_3\left(\eta \right)=\sqrt{{\displaystyle \frac{2d_0[6\wp (\eta ;g_2,g_3)+2d_2+D_{\pm }]}{12\wp (\eta ;g_2,g_3)+D_{\pm }}}},\end{array}$$

(16)

where the quantity D is given by

$$\begin{array}{c}\hfill D\pm ={\displaystyle \frac{-5d_2\pm \sqrt{9d_2^2-36d_0{d}_4}}{2}},\end{array}$$

(17)

and the invariants of the Weierstrass function are as follows:

$$\begin{array}{c}\hfill g_2=-{\displaystyle \frac{d_2}{12}}(5D_{\pm }+4d_2+33d_0{d}_4),\end{array}$$

(18)

and

$$\begin{array}{c}\hfill g_3={\displaystyle \frac{1}{216}}[d_2^2(21D_{\pm }+20d_2)-3d_0{d}_4(21D_{\pm }+9d_2)].\end{array}$$

(19)

Also, there are two separate solutions, which have been identified as

$$\begin{array}{c}\hfill w_4\left(\eta \right)={\displaystyle \frac{\sqrt{d_0}}{3}}{\displaystyle \frac{6\wp (\eta ;g_2,g_3)+d_2}{\wp \prime (\eta ;g_2,g_3)}}\end{array}$$

(20)

and

$$\begin{array}{c}\hfill w_5\left(\eta \right)={\displaystyle \frac{3\wp \prime (\eta ;g_2,g_3)}{\sqrt{d_4}(6\wp (\eta ;g_2,g_3)+d_2)}},\end{array}$$

(21)

where $\wp \prime (\eta ;g_2,g_3)={\displaystyle \frac{d\wp (\eta ;g_2,g_3)}{d\eta }}$ and the invariants of the Weierstrass function are

$$\begin{array}{c}\hfill g_2={\displaystyle \frac{d_2^2-d_0{d}_4}{12}},g_3={\displaystyle \frac{36d_0{d}_4-d_2^2}{216}}.\end{array}$$

(22)

For $d_1=d_3=0$ and $d_0={\displaystyle \frac{5d_2^2}{36d_4}}$:

The solution of Equation (12) in this case is given as

$$\begin{array}{c}\hfill w_6\left(\eta \right)=\sqrt{-{\displaystyle \frac{15d_2}{2d_4}}}{\displaystyle \frac{\wp (\eta ;g_2,g_3)}{d_2+3\wp (\eta ;g_2,g_3)}},\end{array}$$

(23)

with the invariants are written in the forms

$$\begin{array}{c}\hfill g_2={\displaystyle \frac{2d_2^2}{9}},g_3={\displaystyle \frac{d_2^3}{54}}.\end{array}$$

(24)

For $d_0=d_1=0$, we have the following:

$$\begin{array}{c}\hfill w_7\left(\eta \right)={\displaystyle \frac{-8d_2{d}_3{k}_0{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(\sqrt{d_2}\eta \right)}{\left(B+4d_3^2\right)tanh\left(\sqrt{d_2}\eta \right)+4d_3^2{\sec \mathrm{h}}^2\left(2\sqrt{d_2}\eta \right)+4d_3^2-B}},\end{array}$$

(25)

where $k_0$ is a constant, and B is represented as

$$\begin{array}{c}\hfill B=\left(4d_2{d}_4-d_3^2\right)k_0^2.\end{array}$$

(26)

3. Application to the EWTM for Biswas–Arshed Equation

In this section, we consider the Biswas–Arshed equation with beta derivative

$$\begin{array}{c}\hfill i{}_0{}^{A}D_t^{\beta }\left\{u\right\}+a_1{u}_{xx}+a_2{}_0{}^{A}D_t^{\beta }\left\{u_x\right\}+i\left(b_1{u}_{xxx}+b_2{}_0{}^{A}D_t^{\beta }\left\{u_{xx}\right\}\right)-i\left(\lambda {\left({\left|u\right|}^{2}u\right)}_x+\mu u{\left({\left|u\right|}^2\right)}_x+\delta {\left|u\right|}^2{u}_x\right)=0,\end{array}$$

(27)

where $a_1,a_2,b_1,b_2,\lambda ,\mu$, and $\delta$ are arbitrary real constants [26,55]. The traveling wave transformations are defined as follows

$$\begin{array}{c}\hfill u\left(x,t\right)=\mathrm{\Phi }\left(\eta \right)e^{i\mathrm{\Psi }\left(x,t\right)},\end{array}$$

$$\begin{array}{c}\hfill \eta =x-{\displaystyle \frac{\alpha }{\beta }}{\left(t+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\beta \right)}}\right)}^{\beta },\end{array}$$

(28)

$$\begin{array}{c}\hfill \mathrm{\Psi }\left(x,t\right)=-\gamma x+{\displaystyle \frac{c}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\beta \right)}}\right)}^{\beta }+\varpi ,\end{array}$$

where $\alpha ,\gamma ,c$, and $\varpi$ are real constants. When the terms containing derivatives required in Equation (27) are obtained from the wave transform Equation (28) and written in their place, we obtain the following system of nonlinear ordinary differential equations. The real part

$$\begin{array}{c}\hfill \left(a_1-\alpha a_2+\gamma \left(3b_1-2\alpha b_2\right)-c{b}_2\right){\mathrm{\Phi }}^{″}+\left(-{\gamma }^3{b}_1+\left(c{b}_2-a_1\right){\gamma }^2+c\gamma a_2-c\right)\mathrm{\Phi }+\left(-\gamma \lambda +\gamma \delta \right){\mathrm{\Phi }}^3=0,\end{array}$$

(29)

and the imaginary part

$$\begin{array}{c}\hfill \left(b_1-\alpha b_2\right){\mathrm{\Phi }}^{\prime ″}+\left(-\alpha -2\gamma a_1+\alpha \gamma a_2+c{a}_2-3{\gamma }^2{b}_1+\alpha {\gamma }^2{b}_2+2c\gamma b_2\right)\mathrm{\Phi }+\left(-3\lambda -2\mu -\delta \right){\mathrm{\Phi }}^2{\mathrm{\Phi }}^{\prime }=0.\end{array}$$

(30)

By equating the coefficients of Equation (30) to zero, the following results are obtained:

$$\begin{array}{c}\hfill \alpha ={\displaystyle \frac{b_1}{b_2}},\end{array}$$

$$\begin{array}{c}\hfill \lambda ={\displaystyle \frac{1}{3}}(-\delta -2\mu ),\end{array}$$

(31)

$$\begin{array}{c}\hfill c={\displaystyle \frac{2b_1{b}_2{\gamma }^2+\gamma \left(2a_1{b}_2-a_2{b}_1\right)+b_1}{b_2\left(a_2+2b_2\gamma \right)}}.\end{array}$$

A nonlinear ordinary differential equation is obtained by substituting the values in Equation (31) into Equation (29) as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{a_2^2{b}_1+b_2\left(b_1-a_1{a}_2\right)}{b_2\left(a_2+2b_2\gamma \right)}}{\mathrm{\Phi }}^{″}+{\displaystyle \frac{b_1{\left(a_2\gamma -1\right)}^2+b_2\gamma \left(a_1\left(2-a_2\gamma \right)+b_1\gamma \right)}{b_2\left(a_2+2b_2\gamma \right)}}\mathrm{\Phi }+{\displaystyle \frac{2}{3}}\gamma (\delta -\mu ){\mathrm{\Phi }}^3=0.\end{array}$$

(32)

According to the balance procedure between the term ${\mathrm{\Phi }}^{″}$ with the highest-order derivative and the term ${\mathrm{\Phi }}^3$ with the highest-order nonlinearity in Equation (32), the value of N is obtained:

$$\begin{array}{c}\hfill N+2=3N\Rightarrow N=1.\end{array}$$

(33)

Therefore, we can seek the traveling wave solutions of Equation (32) in the form below:

$$\begin{array}{c}\hfill \mathrm{\Phi }\left(\eta \right)=c_{-1}{w}^{-1}\left(\eta \right)+c_0+c_{1}w\left(\eta \right).\end{array}$$

(34)

Case A.

In this case, we can find two different statements for $d_1=d_3=0$:

State 1.

$$\begin{array}{c}\hfill c_{-1}=0,c_0=-\sqrt{{\displaystyle \frac{-3\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{2\gamma (\delta -\mu )b_2\left(a_2+2\gamma b_2\right)}}},c_1=0.\end{array}$$

(35)

Substituting Equation (35) into Equation (34) gives the following function:

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_{1,1}\left(\eta \right)=-\sqrt{{\displaystyle \frac{-3\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{2\gamma (\delta -\mu )b_2\left(a_2+2b_2\gamma \right)}}}.\end{array}$$

(36)

If Equation (36) is substituted into Equation (28), then the following solution can be obtained as

$$\begin{array}{c}\hfill u_{1,1}(x,t)=-\sqrt{{\displaystyle \frac{-3\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{2\gamma (\delta -\mu )b_2\left(a_2+2\gamma b_2\right)}}}{e}^{i\mathrm{\Psi }(x,t)}.\end{array}$$

(37)

State 2.

$$\begin{array}{c}\hfill c_0=0,c_1=-{\displaystyle \frac{\left({\gamma }^2+d_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1}{2\gamma (\delta -\mu )b_2{c}_{-1}\left(a_2+2b_2\gamma \right)}},\end{array}$$

$$\begin{array}{c}\hfill d_0=-{\displaystyle \frac{\gamma (\delta -\mu )b_2{c}_{-1}^2\left(a_2+2\gamma b_2\right)}{3\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)}},\end{array}$$

(38)

$$\begin{array}{c}\hfill d_4={\displaystyle \frac{{\left(\left({\gamma }^2+d_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^2}{12\gamma (\delta -\mu )b_2{c}_{-1}^2\left(a_1{a}_2{b}_2-a_2^2{b}_1-b_1{b}_2\right)\left(a_2+2\gamma b_2\right)}}.\end{array}$$

The Weierstrass functions (13) and (14) are calculated in the forms

$$\begin{array}{c}\hfill w_{1,2}\left(\eta \right)=\sqrt{{\displaystyle \frac{3\wp \left(\eta ;g_2,g_3\right)-d_2}{3d_4}}},\end{array}$$

(39)

$$\begin{array}{c}\hfill w_{2,2}\left(\eta \right)=\sqrt{{\displaystyle \frac{3d_0}{\wp \left(\eta ;g_2,g_3\right)-d_2}}},\end{array}$$

(40)

where the invariants of the first Weierstrass function are given by

$$\begin{array}{c}\hfill g_2={\displaystyle \frac{4}{3}}\left(d_2^2-{\displaystyle \frac{{\left(\left({\gamma }^2+d_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^2}{12{\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)}^2}}\right),\end{array}$$

(41)

$$\begin{array}{c}\hfill g_3={\displaystyle \frac{4d_2^2}{27}}\left(-2d_2-{\displaystyle \frac{3{\left(\left({\gamma }^2+d_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^2}{4\gamma (\delta -\mu )b_2{c}_{-1}^2\left(a_2+2\gamma b_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)}}\right).\end{array}$$

(42)

If Equations (38)–(42) are first written into Equation (34), and then the functions obtained by this way are substituted into Equation (28), the solutions can be constructed as follows:

$$\begin{array}{c}\hfill u_{1,2}\left(x,t\right)=\left(c_{-1}\sqrt{{\displaystyle \frac{3d_4}{3\wp \left(\eta ;g_2,g_3\right)-d_2}}}-{\displaystyle \frac{\left({\gamma }^2+d_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1}{2\gamma (\delta -\mu )b_2{c}_{-1}\left(a_2+2\gamma b_2\right)}}\sqrt{{\displaystyle \frac{3\wp \left(\eta ;g_2,g_3\right)-d_2}{3d_4}}}\right)e^{i\mathrm{\Psi }\left(x,t\right)},\end{array}$$

(43)

$$u_{2,2}\left(x,t\right)=\left(c_{-1}\sqrt{{\displaystyle \frac{\wp \left(\eta ;g_2,g_3\right)-d_2}{3d_0}}}-{\displaystyle \frac{\left({\gamma }^2+d_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1}{2\gamma (\delta -\mu )b_2{c}_{-1}\left(a_2+2\gamma b_2\right)}}\sqrt{{\displaystyle \frac{3d_0}{\wp \left(\eta ;g_2,g_3\right)-d_2}}}\right)e^{i\mathrm{\Psi }\left(x,t\right)}.$$

(44)

Another type of solutions admits

$$\begin{array}{c}\hfill w_{3,2}\left(\eta \right)=\sqrt{{\displaystyle \frac{2d_0\left(2d_2+D_{\pm }+6\wp \left(\eta ;g_2,g_3\right)\right)}{D_{\pm }+12\wp \left(\eta ;g_2,g_3\right)}}},\end{array}$$

(45)

where the quantity D is given by

$$\begin{array}{c}\hfill D_{\pm }={\displaystyle \frac{1}{2}}\left(-5d_2\pm \sqrt{9d_2^2-L}\right),\end{array}$$

(46)

and the Weierstrass function invariants are

$$\begin{array}{c}\hfill g_2=-{\displaystyle \frac{d_2}{12}}\left(4d_2+{\displaystyle \frac{11}{12}}L+{\displaystyle \frac{5}{2}}\left(-5d_2\pm \sqrt{9d_2^2-L}\right)\right),\end{array}$$

(47)

$$\begin{array}{c}\hfill g_3={\displaystyle \frac{1}{216}}\left(d_2^2\left({\displaystyle \frac{21}{2}}\left(-5d_2\pm \sqrt{9d_2^2-L}\right)+20d_2\right)-{\displaystyle \frac{L}{12}}\left({\displaystyle \frac{21}{2}}\left(-5d_2\pm \sqrt{9d_2^2-L}\right)+9d_2\right)\right),\end{array}$$

(48)

where $L={\displaystyle \frac{{\left(\left({\gamma }^2+d_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^2}{{\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)}^2}}.$

When Equations (38) and (45)–(48) are written into Equation (34), and the functions obtained in this way are substituted into Equation (28), the solutions are obtained as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_{3,2}(x,t)& =\left(c_{-1}\sqrt{{\displaystyle \frac{D_{\pm }+12\wp \left(\eta ;g_2,g_3\right)}{2d_0\left(2d_2+D_{\pm }+6\wp \left(\eta ;g_2,g_3\right)\right)}}}\right.\hfill \\ & \left.-{\displaystyle \frac{\left({\gamma }^2+d_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1}{2\gamma (\delta -\mu )b_2{c}_{-1}\left(a_2+2b_2\gamma \right)}}\sqrt{{\displaystyle \frac{2d_0\left(2d_2+D_{\pm }+6\wp \left(\eta ;g_2,g_3\right)\right)}{D_{\pm }+12\wp \left(\eta ;g_2,g_3\right)}}}\right)e^{i\mathrm{\Psi }\left(x,t\right)}.\hfill \end{array}\end{array}$$

(49)

There are two different solutions, which are found to be

$$\begin{array}{c}\hfill w_{4,2}\left(\eta \right)={\displaystyle \frac{\sqrt{d_0}\left(d_2+6\wp \left(\eta ;g_2,g_3\right)\right)}{3{\wp }^{\prime }\left(\eta ;g_2,g_3\right)}},\end{array}$$

(50)

$$\begin{array}{c}\hfill w_{5,2}\left(\eta \right)={\displaystyle \frac{3{\wp }^{\prime }\left(\eta ;g_2,g_3\right)}{\sqrt{d_4}\left(d_2+6\wp \left(\eta ;g_2,g_3\right)\right)}},\end{array}$$

(51)

where

$$\begin{array}{c}\hfill {\wp }^{\prime }\left(\eta ;g_2,g_3\right)={\displaystyle \frac{d\wp \left(\eta ;g_2,g_3\right)}{d\eta }},\end{array}$$

(52)

and the invariants of the Weierstrass functions are

$$\begin{array}{c}\hfill g_2={\displaystyle \frac{1}{12}}\left(d_2^2-{\displaystyle \frac{1}{36}}L\right),g_3={\displaystyle \frac{1}{216}}\left(L-d_2^2\right).\end{array}$$

(53)

If Equations (38) and (50)–(53) are first written into Equation (34), and the resulting functions are substituted in Equation (28), we have the following solutions as follows:

$$\begin{array}{c}\hfill u_{4,2}\left(x,t\right)=\left({\displaystyle \frac{3c_{-1}{\wp }^{\prime }\left(\eta ;g_2,g_3\right)}{\sqrt{d_0}\left(d_2+6\wp \left(\eta ;g_2,g_3\right)\right)}}-{\displaystyle \frac{\left({\gamma }^2+d_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1}{2\gamma (\delta -\mu )b_2{c}_{-1}\left(a_2+2b_2\gamma \right)}}{\displaystyle \frac{\sqrt{d_0}\left(d_2+6\wp \left(\eta ;g_2,g_3\right)\right)}{3\wp {\left(\eta ;g_2,g_3\right)}^{\prime }}}\right)e^{i\mathrm{\Psi }\left(x,t\right)},\end{array}$$

(54)

$$\begin{array}{c}\hfill u_{5,2}\left(x,t\right)=\left({\displaystyle \frac{\sqrt{d_4}{c}_{-1}\left(d_2+6\wp \left(\eta ;g_2,g_3\right)\right)}{3{\wp }^{\prime }\left(\eta ;g_2,g_3\right)}}-{\displaystyle \frac{\left({\gamma }^2+d_2\right)\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1}{2\gamma (\delta -\mu )b_2{c}_{-1}\left(a_2+2b_2\gamma \right)}}{\displaystyle \frac{3{\wp }^{\prime }\left(\eta ;g_2,g_3\right)}{\sqrt{d_4}\left(d_2+6\wp \left(\eta ;g_2,g_3\right)\right)}}\right)e^{i\mathrm{\Psi }\left(x,t\right)}.\end{array}$$

(55)

Case B.

In this case, we can find four different statements for $d_1=d_3=0,d_0={\displaystyle \frac{5d_2^2}{36d_4}}$:

State 1.

$$\begin{array}{c}\hfill c_{-1}=0,c_0=\pm \sqrt{{\displaystyle \frac{-3\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{2\gamma (\delta -\mu )b_2\left(a_2+2\gamma b_2\right)}}},c_1=0.\end{array}$$

(56)

If Equation (56) is first written into Equation (34), and the functions obtained by this way are substituted into Equation (28), the wave solutions can be constructed as follows:

$$\begin{array}{c}\hfill u_{6,1}(x,t)=\pm \sqrt{{\displaystyle \frac{-3\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{2b_2\gamma (\delta -\mu )\left(a_2+2b_2\gamma \right)}}}{e}^{i\mathrm{\Psi }(x,t)}.\end{array}$$

(57)

State 2.

$$\begin{array}{c}\hfill c_{-1}=c_0=0,d_4=-{\displaystyle \frac{\gamma (\delta -\mu )b_2{c}_1^2\left(a_2+2b_2\gamma \right)}{3\left(a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2\right)}},\end{array}$$

$$\begin{array}{c}\hfill d_2={\displaystyle \frac{-\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2}}.\end{array}$$

(58)

According to the results in Equation (58), Equation (23) can be rewritten as

$$\begin{array}{c}\hfill w_{6,2}\left(\eta \right)=\sqrt{-{\displaystyle \frac{15d_2}{2d_4}}}{\displaystyle \frac{\wp \left(\eta ;g_2,g_3\right)}{d_2+3\wp \left(\eta ;g_2,g_3\right)}},\end{array}$$

(59)

where the invariants of the Weierstrass function are given by

$$\begin{array}{c}\hfill g_2={\displaystyle \frac{2{\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^2}{9{\left(a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2\right)}^2}},\end{array}$$

(60)

$$\begin{array}{c}\hfill g_3=-{\displaystyle \frac{{\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^3}{54{\left(a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2\right)}^3}}.\end{array}$$

(61)

If Equations (58)–(61) are substituted into Equation (34), and when the resulting functions are written into Equation (28), the following wave solutions can be found:

$$\begin{array}{c}\hfill u_{6,2}\left(x,t\right)=\left(c_1\sqrt{-{\displaystyle \frac{15d_2}{2d_4}}}{\displaystyle \frac{\wp \left(\eta ;g_2,g_3\right)}{d_2+3\wp \left(\eta ;g_2,g_3\right)}}\right)e^{i\mathrm{\Psi }(x,t)}.\end{array}$$

(62)

State 3.

$$\begin{array}{c}\hfill c_0=0,c_1=0,d_2=-{\displaystyle \frac{{\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1}{a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2}},\end{array}$$

$$\begin{array}{c}\hfill d_4={\displaystyle \frac{5{\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^2}{12\gamma (\delta -\mu )b_2{c}_{-1}^2\left(a_1{a}_2{b}_2-a_2^2{b}_1-b_1{b}_2\right)\left(a_2+2b_2\gamma \right)}}.\end{array}$$

(63)

From Equation (23), the Weierstrass function can be calculated in the form

$$\begin{array}{c}\hfill w_{6,3}\left(\eta \right)=\sqrt{-{\displaystyle \frac{15d_2}{2d_4}}}{\displaystyle \frac{\wp \left(\eta ;g_2,g_3\right)}{d_2+3\wp \left(\eta ;g_2,g_3\right)}},\end{array}$$

(64)

where the invariants of the Weierstrass function are given by

$$\begin{array}{c}\hfill g_2={\displaystyle \frac{2{\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^2}{9{\left(a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2\right)}^2}},\end{array}$$

(65)

and

$$\begin{array}{c}\hfill g_3=-{\displaystyle \frac{{\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^3}{54{\left(a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2\right)}^3}}.\end{array}$$

(66)

If Equations (63)–(66) are initially introduced into Equation (34), and then the functions obtained through this process are substituted into Equation (28), the wave solutions can be derived as follows:

$$\begin{array}{c}\hfill u_{6,3}\left(x,t\right)=\left(c_{-1}\sqrt{-{\displaystyle \frac{2d_4}{15d_2}}}{\displaystyle \frac{d_2+3\wp \left(\eta ;g_2,g_3\right)}{\wp \left(\eta ;g_2,g_3\right)}}\right)e^{i\mathrm{\Psi }(x,t)}.\end{array}$$

(67)

State 4.

$$\begin{array}{c}\hfill c_0=0,c_1=-{\displaystyle \frac{\left(5+\sqrt{5}\right)\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{8\gamma (\delta -\mu )b_2{c}_{-1}\left(a_2+2\gamma b_2\right)}},\end{array}$$

$$\begin{array}{c}\hfill d_2=\pm {\displaystyle \frac{(\sqrt{5}+1)\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{a_1{a}_2{b}_2-a_2^2{b}_1-b_1{b}_2}},\end{array}$$

$$\begin{array}{c}\hfill d_4={\displaystyle \frac{{(\sqrt{5}+5)}^2}{192\gamma (\delta -\mu )b_2{c}_{-1}^2\left(a_1{a}_2{b}_2-a_2^2{b}_1-b_1{b}_2\right)\left(a_2+2b_2\gamma \right)}}.\end{array}$$

(68)

Substituting Equation (68) into Equation (23), the Weierstrass function is reconstructed as follows:

$$\begin{array}{c}\hfill w_{6,4}\left(\eta \right)=\sqrt{-{\displaystyle \frac{15d_2}{2d_4}}}{\displaystyle \frac{\wp \left(\eta ;g_2,g_3\right)}{d_2+3\wp \left(\eta ;g_2,g_3\right)}},\end{array}$$

(69)

where the invariants of the Weierstrass function are given by

$$\begin{array}{c}\hfill g_2={\displaystyle \frac{{(\sqrt{5}+1)}^2{\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^2}{72{\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)}^2}},\end{array}$$

(70)

$$\begin{array}{c}\hfill g_3=\pm {\displaystyle \frac{{(\sqrt{5}+1)}^3{\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}^3}{3456{\left(a_1{a}_2{b}_2-a_2^2{b}_1-b_1{b}_2\right)}^3}}.\end{array}$$

(71)

When Equations (68)–(71) are written into Equation (34), and the resulting functions written into Equation (28), the wave solutions can be obtained as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_{6,4}\left(x,t\right)& =\left(c_{-1}\sqrt{-{\displaystyle \frac{2d_4}{15d_2}}}{\displaystyle \frac{d_2+3\wp \left(\eta ;g_2,g_3\right)}{\wp \left(\eta ;g_2,g_3\right)}}\right.\hfill \\ & \left.-(5+\sqrt{5})\sqrt{-{\displaystyle \frac{15d_2}{2d_4}}}{\displaystyle \frac{{\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1}{8\gamma (\delta -\mu )b_2{c}_{-1}\left(a_2+2b_2\gamma \right)}}{\displaystyle \frac{\wp (\eta ;g_2,g_3)}{d_2+3\wp (\eta ;g_2,g_3)}}\right)e^{i\mathrm{\Psi }(x,t)}.\hfill \end{array}\end{array}$$

(72)

Case C.

In this case, we can find one different statement for $d_0=d_1=0$:

State 1.

$$\begin{array}{c}\hfill c_{-1}=0,c_0=\pm \sqrt{{\displaystyle \frac{-3\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{2\gamma (\delta -\mu )b_2\left(a_2+2\gamma b_2\right)}}},\end{array}$$

$$\begin{array}{c}\hfill c_1=\mp \sqrt{{\displaystyle \frac{3d_3^2{\left(a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2\right)}^2}{8\gamma (\delta -\mu )b_2\left(a_2+2b_2\gamma \right)\left({\gamma }^2\left(-a_2^2{b}_1+a_1{a}_2{b}_2-b_1{b}_2\right)+2\gamma \left(a_2{b}_1-a_1{b}_2\right)-b_1\right)}}},\end{array}$$

$$\begin{array}{c}\hfill d_2={\displaystyle \frac{2\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2}},\end{array}$$

$$\begin{array}{c}\hfill d_4={\displaystyle \frac{d_3^2\left(a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2\right)}{8\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}}\end{array}$$

(73)

Using the above results, Equation (34) can be given as

$$\begin{array}{c}\hfill w_{7,1}\left(\eta \right)={\displaystyle \frac{-8d_2{d}_3{k}_0{\sec \mathrm{h}}^2\left(\sqrt{d_2}\eta \right)}{4d_3^2{\sec \mathrm{h}}^2\left(2\sqrt{d_2}\eta \right)+\left(B+4d_3^2\right)tanh\left(\sqrt{d_2}\eta \right)+4d_3^2-B}},\end{array}$$

(74)

where $k_0$ is a constant, and B is represented as $B=k_0^2\left(4d_2{d}_4-d_3^2\right)$. If Equation (28) is substituted into Equation (74), then the following wave solution can be found as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_{7,1}(x,t)& =\left(\pm \sqrt{{\displaystyle \frac{-3\left({\gamma }^2\left(a_2^2{b}_1-a_1{a}_2{b}_2+b_1{b}_2\right)+2\gamma \left(a_1{b}_2-a_2{b}_1\right)+b_1\right)}{2\gamma (\delta -\mu )b_2\left(a_2+2\gamma b_2\right)}}}\right.\hfill \\ & \mp \sqrt{{\displaystyle \frac{3d_3^2{\left(a_2^2{b}_1+b_1{b}_2-a_1{a}_2{b}_2\right)}^2}{8\gamma (\delta -\mu )b_2\left(a_2+2b_2\gamma \right)\left({\gamma }^2\left(-a_2^2{b}_1+a_1{a}_2{b}_2-b_1{b}_2\right)+2\gamma \left(a_2{b}_1-a_1{b}_2\right)-b_1\right)}}}\times \hfill \\ & \times \left.{\displaystyle \frac{-8d_2{d}_3{k}_0{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(\sqrt{d_2}\eta \right)}{4d_3^2{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(2\sqrt{d_2}\eta \right)+\left(B+4d_3^2\right)tanh\left(\sqrt{d_2}\eta \right)+4d_3^2-B}}\right)e^{i\mathrm{\Psi }(x,t)}.\hfill \end{array}\end{array}$$

(75)

4. Graphical Description of Solutions

In this section, we present visual descriptions of certain solution functions in the form of two-dimensional, three-dimensional, and contour plots. Some of the obtained solutions such as wave solutions (37), (43), (44), (62), and (72) are expressed graphically to depict the physical behavior. The real and imaginary parts of the solutions are displayed with the help of two-dimensional, three-dimensional, and contour plots by selecting appropriate values of arbitrary parameters. When looking at the graphs, it has been determined that doubly periodic and hyperbolic behaviors are obtained.

• Figure 1 shows the graph of the real and imaginary parts of Equation (37) for the values of $\beta =0.5,\gamma =0.1,\delta =0.7,\mu =0.6,a_1=-20.5,a_2=10,\alpha =0.5,b_1=0.9,b_2=0.5,c=0.7,d_2=0.4$, and $t=1$.
• Figure 2 shows the graph of the real and imaginary parts of Equation (43) for the values of $\beta =0.5,\alpha =0.5,\gamma =0.1,\mu =0.6,\delta =0.7,a_1=-20.5,a_2=10,b_1=0.9,b_2=0.5,c_{-1}=0.2,c=0.7,d_2=0.4$, and $t=1.$
• Figure 3 shows the graph of the real and imaginary parts of Equation (44) for the values of $\gamma =0.1,c=0.7,\delta =0.7,\alpha =0.5,\beta =0.5,\mu =0.6,a_1=-20.5,a_2=10,b_1=0.9,b_2=0.5,c_{-1}=0.2,d_2=0.4$, and $t=1$.
• Figure 4 shows the graph of the real and imaginary parts of Equation (62) for the values of $\alpha =0.5,\beta =0.5,\gamma =0.1,\mu =0.6,a_1=-20.5,a_2=10,b_1=0.9,b_2=0.5,c_1=0.2,c=0.7,\delta =0.7,d_2=0.4$, and $t=1$.
• Figure 5 shows the graph of the real and imaginary parts of Equation (72) for the values of $\delta =0.7,\beta =0.5,\alpha =0.5,\gamma =0.1,\mu =0.6,a_1=-20.5,a_2=10,b_1=0.9,b_2=0.5,c_{-1}=0.2,c=0.7,d_2=0.4$, and $t=1$.

5. Conclusions

In this study, the generalized Weierstrass transformation method has been used to determine the wave solutions, including the Weierstrass elliptic functions, of the Biswas–Arshed equation with a beta derivative. In the application of this method, we have the following: Firstly, the reduction of the nonlinear partial differential equation to a nonlinear ordinary differential equation with the help of wave transform is done by using the definition of the beta derivative and some basic properties of this derivative. Thus, the solution function of this equation was expanded to the finite series of Weierstrass solution functions of the elliptic differential equation, and a system of nonlinear algebraic equations was obtained. This system has many parameters and equations, but sometimes trivial solutions can only be reached by writing various codes in package programs such as Mathematica, and sometimes there are an infinite number of solutions depending on parameters. By examining all the processes that guarantee the compatibility of the solution functions obtained by using all these approaches with the mathematical model, the doubly periodic and hyperbolic traveling wave solutions, including Weierstrass elliptic functions, have been derived for this mathematical model. In the literature, it has been observed that there is no study on finding Weierstrass elliptic function solutions of nonlinear partial differential equations with the beta derivative, and therefore, such solution functions have been obtained for the first time in this study. Due to the complex nature of these resulting solution functions, the graphs depicting their behavior, focusing separately on the real and imaginary components, have been presented in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5. It can be said that analyzing these results and working on deriving the doubly periodic solution functions from these results is of great importance. This research aims to contribute to a better understanding of the doubly periodic behavior in the mathematical model with the calculated solution functions and has shown that the intended physical behaviors are determined by obtaining Weierstrass solution functions. When all these issues are evaluated together, it has been shown that the Weiertrass transform method gives very effective results in determining the traveling wave solutions of nonlinear partial differential equations with beta derivatives.