On the Wave Structures to the (3+1)-Dimensional Boiti–Leon–Manna–Pempinelli Equation in Incompressible Fluid

Abstract

In the present study, two effective methods, the Exp-function method and He’s frequency formulation, are employed to investigate the dynamic behaviors of the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation, which is used widely to describe the incompressible fluid. A variety of the wave structures, including the dark wave, bright-dark wave and periodic wave solutions, are successfully constructed. Compared with the results attained by the methods, the obtained solutions are all new and have not been presented in the other literature. The diverse wave structures of the solutions are presented through numerical results in the form of three-dimensional plots and two-dimensional curves. It reveals that the proposed methods are powerful and straightforward, which are expected to be helpful for the study of travelling-wave theory in fluid.

1. Introduction

Exploration of the exact solutions of the partial differential equations (PDEs) has obtained wide attention from scientists and mathematicians. Until now, some effective approaches thus far are: the auxiliary equation mapping method [1,2], Wang’s Bäcklund transformation based method [3,4], the modified extended direct algebraic method [5,6], Sardar subequation method [7,8], variational method [9,10], trial equation method [11,12], logarithmic transformation [13] and so on. As is common knowledge, the fluid, as a phase of matter including liquid, gas and plasma, is considered to be one of the most common substances in nature. Significant research on fluids has provided an important guide for the design of related industries. Additionally, there are many famous partial–differential equations available to model fluid characteristics. In this paper, we consider the famous (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation, which is provided by [14,15]:

$${\psi }_{yt}+{\psi }_{zt}+{\psi }_{xxxy}+{\psi }_{xxxz}-3{\psi }_x\left({\psi }_{xy}+{\psi }_{xz}\right)-3{\psi }_{xx}\left({\psi }_y+{\psi }_z\right)=0$$

(1)

where $\psi =\psi \left(x,y,z,t\right)$ is a well-known function that is used widely in the liquid engendering and incompressible liquid. Additionally, many scholars have made outstanding contributions to the solution of Equation (1). MelikeKaplan used the transformed rational function to construct different types of analytical solutions [16]. In [17], Mohamed R. Ali et al, obtained the lump solutions and mixed solution involving lump waves and solitons by the bilinear method and symbolic computation. Tang et al. found the periodic wave solutions for Equation (1) via the extended homoclinic test approach [18]. Additionally, the double-periodic soliton solutions are constructed by Liu in [19]. The sine Gordon expansion method and the extended tanh function method are used to seek the new soliton solutions in [20]. Mohammad Najafi et al. studied the problem via the semi-inverse method [21]. The Hirota’s bilinear algorithm is applied to seek the lump solutions by Ahmad M.Alenezi in [22]. Considering the importance of Equation (1) in the field of fluids, and being encouraged by recent research results, we aim to construct abundant exact solutions of Equation (1) by the Exp-function method (EFM) and He’s frequency formulation (HFF). The structure of this paper is as follows. In Section 2, we will provide a brief introduction to the EFM and HFF. In Section 3, the EFM and HFF are used to find the exact solutions and the behaviors of the different solutions are presented by the three-dimensional and two-dimensional contours. The physical explanation of the different solutions are provided in Section 4. The conclusion is presented in Section 5.

2. The Two Methods

Considering the general nonlinear FDEs as:

$$F\left(u,u_x,u_y,u_z,u_t,\dots \right)=0$$

(2)

Introducing the following transformation:

$$u\left(x,y,z,t\right)=U\left(\xi \right),\xi =\omega x+my+kz+vt$$

(3)

where $\omega$, $m$, $k$, $v$ are all non-zero constant. using Equation (3), the nonlinear PDE of Equation (2) can be reduced into the ODE as:

$$F\left(U,U_{\xi },U_{\xi \xi },U_{\xi \xi \xi },\dots \right)=0$$

(4)

2.1. The EFM

The EFM is a powerful tool to construct the abundant traveling wave solutions of the studied equation. In this section, we will provide a brief introduction to the EFM [23,24,25,26,27]. Its primary steps are presented as follows:

Assuming the solution of Equation (4) with the form as:

$$U\left(\xi \right)=\frac{{\displaystyle \sum _{i=-p}^u}{a}_i\exp \left(i\xi \right)}{{\displaystyle \sum _{j=-g}^s}{b}_j\exp \left(j\xi \right)}$$

(5)

where u, p, s, g are positive integers that can be determined later, $a_i$, $b_j$ are unknown constants.

Taking Equation (5) into Equation (4) and balancing the linear term of highest and lowest orders, respectively, we can determine the u, p, s, g.

Then, substituting the obtained results into Equation (4), setting the coefficients of $\exp \left(i\xi \right)$ to be zero, we can determine the coefficients $a_i$ and $b_j$. With this, we can obtain the exact solutions of Equation (2) via Equation (3).

2.2. The HFF

The HFF [28,29,30], which was first proposed by Chinese mathematician Ji-Huan He, is a powerful tool to seek the periodic solution.

Considering Equation (4) with the form as:

$$U^{″}+f(U)=0$$

(6)

In Equation (6), $f\left(U\right)$ is a function of $U$. Here, the periodic solution of Equation (6) can be assumed as:

$$U\left(\xi \right)=M\cos \left(\Omega \xi \right),\Omega >0$$

(7)

where $\Omega$ represents angular frequency. In the view of He’s frequency formulation [31,32], the amplitude–frequency relationship of Equation (7) can be determined via one step, which is:

$${\Omega }^2={\frac{df\left(U\right)}{dU}|}_{U=\frac{M}{2}}$$

(8)

With this, we can obtain the periodic wave solution of Equation (1).

3. Applications

In order to obtain the abundant solutions, the following transformation is introduced:

$$\psi \left(x,y,z,t\right)=\mathsf{\Psi }\left(\xi \right),\xi =\omega x+my+kz+vt+{\xi }_0$$

(9)

In Equation (9), $\omega$, $m$, $k$ and $v$ are non-zero real numbers, and there is $m+k\ne 0$. With the help of Equation (9), Equation (1) becomes an ordinary differential equation, which is:

$$\left(m+k\right)v{\mathsf{\Psi }}^{″}+{\omega }^3\left(m+k\right){\mathsf{\Psi }}^{\left(4\right)}-6{\omega }^2\left(m+k\right){\mathsf{\Psi }}^{\prime }{\mathsf{\Psi }}^{″}=0$$

(10)

Additionally, there are ${\mathsf{\Psi }}^{\prime }=\frac{d\mathsf{\Psi }}{d\xi }$, $\mathsf{\Psi }=\frac{d^2\mathsf{\Psi }}{d{\xi }^2}$ and ${\mathsf{\Psi }}^{\left(4\right)}=\frac{d^4\mathsf{\Psi }}{d{\xi }^4}$.

Integrating Equation (10) once, with respect to $\xi$, and setting the integral constant to be zero, yields:

$$\left(m+k\right)v{\mathsf{\Psi }}^{\prime }+{\omega }^3\left(m+k\right){\mathsf{\Psi }}^{\left(3\right)}-3{\omega }^2\left(m+k\right){\left({\mathsf{\Psi }}^{\prime }\right)}^2=0$$

(11)

which is:

$$v{\mathsf{\Psi }}^{\prime }+{\omega }^3{\mathsf{\Psi }}^{\left(3\right)}-3{\omega }^2{\left({\mathsf{\Psi }}^{\prime }\right)}^2=0$$

(12)

3.1. Application of the EFM

Based on the EFM, we assume the solution of Equation (12) with the form as:

$$\mathsf{\Psi }\left(\xi \right)=\frac{{\displaystyle \sum _{i=-u}^c{a}_i\exp \left(i\xi \right)}}{{\displaystyle \sum _{j=-s}^g{b}_j\exp \left(j\xi \right)}}$$

(13)

which can be expressed as:

$$\mathsf{\Psi }\left(\xi \right)=\frac{a_c\exp \left(c\xi \right)+\dots +a_{-u}\exp \left(-u\xi \right)}{b_g\exp \left(g\xi \right)+\dots +b_{-s}\exp \left(-s\xi \right)}$$

(14)

Putting Equation (14) into Equation (12) and balancing the linear term of highest order ${\mathsf{\Psi }}^{\left(3\right)}$ and ${\left({\mathsf{\Psi }}^{\prime }\right)}^2$ in Equation (12) with the highest order nonlinear term, we obtain:

$${\mathsf{\Psi }}^{\left(3\right)}=\frac{p_1\exp \left[\left(3g+c\right)\xi \right]+\dots }{p_2\exp \left(4g\xi \right)+\dots }$$

(15)

$${\left({\mathsf{\Psi }}^{\prime }\right)}^2=\frac{p_3\exp \left[\left(2g+2c\right)\xi \right]+\dots }{p_4\exp \left(4g\xi \right)+\dots }$$

(16)

By balancing the highest order of Exp-function in Equations (15) and (16), it yields:

$$3g+c=2g+2c$$

(17)

which leads to:

$$g=c$$

(18)

In the same way, we balance the linear term of lowest order in Equation (12), there is:

$${\mathsf{\Psi }}^{\left(3\right)}=\frac{\dots +q_1\exp \left[-\left(3s+u\right)\xi \right]}{\dots +q_2\exp \left(-4s\xi \right)}$$

(19)

$${\left({\mathsf{\Psi }}^{\prime }\right)}^2=\frac{\dots +q_3\exp \left[-\left(2s+2u\right)\xi \right]}{\dots +q_4\exp \left(-4s\xi \right)}$$

(20)

Balancing the lowest order of Exp-function in Equations (19) and (20), there is:

$$-3s-u=-2s-2u$$

(21)

which results in:

$$u=s$$

(22)

For simplicity, we set $g=c=1$, $u=s=1$ and $b_1=1$, Equation (14) can be written as:

$$\mathsf{\Psi }\left(\xi \right)=\frac{a_1\exp \left(\xi \right)+a_0+a_{-1}\exp \left(-\xi \right)}{\exp \left(\xi \right)+b_0+b_{-1}\exp \left(-\xi \right)}$$

(23)

Taking Equation (23) into Equation (12), we can obtain the following form:

$$\frac{\left[\begin{array}{l}{M}_3\exp \left(3\xi \right)+M_2\exp \left(2\xi \right)+M_1\exp \left(\xi \right)+M_0+M_{-1}\exp \left(-\xi \right)+M_{-2}\exp \left(-2\xi \right)\\ +M_{-3}\exp \left(-3\xi \right)\end{array}\right]}{{\left[\exp \left(\xi \right)+b_0+b_{-1}\exp \left(-\xi \right)\right]}^4}=0$$

(24)

where

$$\begin{array}{ll}{M}_3=& -{\omega }^3{a}_0-v{a}_0+{\omega }^3{a}_1{b}_0+v{a}_1{b}_0\\ M_2=& -8{\omega }^3{a}_{-1}-2v{a}_{-1}-3{\omega }^2{a}_0^2+8{\omega }^3{a}_1{b}_{-1}+2v{a}_1{b}_{-1}+4{\omega }^3{a}_0{b}_0-2v{a}_0{b}_0+6{\omega }^2{a}_0{a}_1{b}_0\\ & -4{\omega }^3{a}_1{b}_0^2+2v{a}_1{b}_0^2-3{\omega }^2{a}_1^2{b}_0^2\\ M_1=& -12{\omega }^2{a}_{-1}^{}{a}_0+23{\omega }^3{a}_0{b}_{-1}-v{a}_0{b}_{-1}+12{\omega }^2{a}_0{a}_1{b}_{-1}-5{\omega }^3{a}_{-1}{b}_0-5v{a}_{-1}{b}_0+12{\omega }^2{a}_{-1}{a}_1{b}_0\\ & -18{\omega }^3{a}_1{b}_{-1}{b}_0+6v{a}_1{b}_0{b}_{-1}-12{\omega }^2{a}_1^2{b}_{-1}^{}{b}_0^{}-{\omega }^3{a}_0^{}{b}_0^2-v{a}_0^{}{b}_0^2+{\omega }^3{a}_1^{}{b}_0^3+v{a}_1^{}{b}_0^3\\ M_0=& -12{\omega }^2{a}_{-1}^2+32{\omega }^3{a}_{-1}{b}_{-1}-4v{a}_{-1}{b}_{-1}+6{\omega }^2{a}_0^2{b}_{-1}+24{\omega }^2{a}_{-1}{a}_1{b}_{-1}-32{\omega }^3{a}_1{b}_{-1}^2+4v{a}_{-1}{b}_{-1}^2\\ & -12{\omega }^2{a}_1^2{b}_{-1}^2-6{\omega }^2{a}_{-1}{a}_0{b}_0-6{\omega }^2{a}_0{a}_1{b}_0{b}_{-1}+4{\omega }^3{a}_{-1}^{}{b}_0^2-4v{a}_{-1}{b}_0^2\\ & +6{\omega }^2{a}_{-1}{a}_1{b}_0^2+4{\omega }^3{a}_1{b}_{-1}{b}_0^2+4v{a}_1{b}_{-1}{b}_0^2\\ M_{-1}=& 12{\omega }^2{a}_{-1}^{}{a}_0{b}_{-1}^{}-23{\omega }^3{a}_0{b}_{-1}^2+v{a}_0{b}_{-1}^2-12{\omega }^2{a}_0{a}_1{b}_{-1}^2-12{\omega }^2{a}_{-1}^2{b}_0+18{\omega }^3{a}_{-1}{b}_0{b}_{-1}-6v{a}_{-1}{b}_{-1}{b}_0\\ & -12{\omega }^2{a}_{-1}{a}_1{b}_{-1}{b}_0+5{\omega }^3{a}_1{b}_0{b}_{-1}^2+5v{a}_1^{}{b}_{-1}^2{b}_0^{}+{\omega }^3{a}_0^{}{b}_{-1}^{}{b}_0^2+v{a}_0^{}{b}_{-1}^{}{b}_0^2-{\omega }^3{a}_1^{}{b}_0^3-v{a}_{-1}^{}{b}_0^3\\ M_{-2}=& -8{\omega }^3{a}_{-1}^{}{b}_{-1}^2-2v{a}_{-1}{b}_{-1}^2-3{\omega }^2{a}_0^2{b}_{-1}^2+8{\omega }^3{a}_1{b}_{-1}^3+2v{a}_1{b}_{-1}^3+6{\omega }^2{a}_{-1}{a}_0{b}_{-1}{b}_0-4{\omega }^3{a}_0{b}_{-1}^2{b}_0\\ & +2v{a}_0{b}_{-1}^2{b}_0-3{\omega }^2{a}_{-1}^2{b}_0^2+4{\omega }^3{a}_{-1}{b}_{-1}{b}_0^2-2v{a}_{-1}^{}{b}_{-1}{b}_0^2\\ M_{-3}=& {\omega }^3{a}_0{b}_{-1}^3+v{a}_0{b}_{-1}^3-{\omega }^3{a}_{-1}{b}_{-1}^2{b}_0-v{a}_{-1}^{}{b}_{-1}^2{b}_0^{}\end{array}$$

Letting the coefficients of $\exp \left(i\xi \right)$ to be zero, we obtain:

$$\{\begin{array}{l}{M}_3=0,M_2=0,M_1=0\\ M_0=0\\ M_{-1}=0,M_{-2}=0,M_{-3}=0\end{array}$$

Solving the above systems, we obtain the following three families:

Family 1

Set 1:$a_1=a_1$, $a_0=\omega b_0+a_1{b}_0+\sqrt{{\omega }^2{b}_0^2-4{\omega }^2{b}_{-1}}$, $a_{-1}=\left(2\omega +a_1\right)b_{-1}$, $b_0=b_0$, $b_{-1}=b_{-1}$, $v=-{\omega }^3$, $\omega =\omega$.

Therefore, we can get the first exact traveling wave solution (dark-wave solution Form 1) as:

$${\psi }_1\left(x,y,t,z\right)=\frac{\left[\begin{array}{l}{a}_1\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+\omega b_0+a_1{b}_0+\sqrt{{\omega }^2{b}_0^2-4{\omega }^2{b}_{-1}}\\ +\left(2\omega +a_1\right)b_{-1}\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}{\left[\begin{array}{l}\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+b_0\\ +b_{-1}\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}$$

(25)

Here, $a_1$, $b_0$, $b_{-1}$ and $\omega$ are free parameters. If we select $m=1$, $k=1$, ${\xi }_0$ = 0, the behavior of solution ${\psi }_1\left(x,y,z,t\right)$ is plotted in Figure 1.

Set 2:$a_1=a_1$, $a_0=\omega b_0+a_1{b}_0-\sqrt{{\omega }^2{b}_0^2-4{\omega }^2{b}_{-1}}$, $a_{-1}=\left(2\omega +a_1\right)b_{-1}$, $b_0=b_0$, $b_{-1}=b_{-1}$, $v=-{\omega }^3$, $\omega =\omega$.

Furthermore, we can obtain the second exact traveling solution (bright-dark wave solution) as:

$${\psi }_2\left(x,y,t,z\right)=\frac{\left[\begin{array}{l}{a}_1\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+\omega b_0+a_1{b}_0-\sqrt{{\omega }^2{b}_0^2-4{\omega }^2{b}_{-1}}\\ +\left(2\omega +a_1\right)b_{-1}\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}{\left[\begin{array}{l}\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+b_0\\ +b_{-1}\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}$$

(26)

where $a_1$, $b_0$, $b_{-1}$ and $\omega$ are free parameters. By selecting $m=1$, $k=1$, ${\xi }_0=0$, the propagation of solution ${\psi }_1\left(x,y,z,t\right)$ is described in Figure 2.

Family 2

Set 3:$a_1=\frac{a_0{b}_0-\omega b_0^2-\sqrt{{\omega }^2{b}_0^4-4{\omega }^2{b}_{-1}{b}_0^2}}{b_0^2}$, $a_0=a_0$, $a_{-1}=\omega b_{-1}+\frac{a_0{b}_{-1}}{b_0}-\frac{\omega b_{-1}\sqrt{-{\omega }^2{b}_0^2\left(4b_{-1}-b_0^2\right)}}{b_0^2}$, $b_0=b_0$, $b_{-1}=b_{-1}$, $v=-{\omega }^3$, $\omega =\omega$.

Next, we can obtain the third exact traveling wave solution (dark-solitary wave solution Form 2):

$${\psi }_3\left(x,y,t,z\right)=\frac{\left[\begin{array}{l}\frac{a_0{b}_0-\omega b_0^2-\sqrt{{\omega }^2{b}_0^4-4{\omega }^2{b}_{-1}{b}_0^2}}{b_0^2}\exp \left(\omega x^{\alpha }+m{y}^{\beta }+k{z}^{\gamma }-{\omega }^3{t}^{\gamma }+{\xi }_0\right)+a_0\\ +\left(\omega b_{-1}+\frac{a_0{b}_{-1}}{b_0}-\frac{b_{-1}\sqrt{-{\omega }^2{b}_0^2\left(4b_{-1}-b_0^2\right)}}{b_0^2}\right)\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}{\left[\begin{array}{l}\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+b_0\\ +b_{-1}\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}$$

(27)

where $a_0$, $b_0$, $b_{-1}$ and $\omega$ are free parameters. The performance of the solution ${\psi }_3\left(x,y,z,t\right)$ is presented in Figure 3 for $m=1$, $k=1$, ${\xi }_0=0$.

Set 4:$a_1=\frac{a_0{b}_0-\omega b_0^2+\sqrt{{\omega }^2{b}_0^4-4{\omega }^2{b}_{-1}{b}_0^2}}{b_0^2}$, $a_0=a_0$, $a_{-1}=\omega b_{-1}+\frac{a_0{b}_{-1}}{b_0}+\frac{b_{-1}\sqrt{-{\omega }^2{b}_0^2\left(4b_{-1}-b_0^2\right)}}{b_0^2}$, $b_0=b_0$, $b_{-1}=b_{-1}$, $v=-{\omega }^3$, $\omega =\omega$.

Then, we can obtain the fourth exact traveling wave solution (bright-dark wave solution) as:

$${\psi }_4\left(x,y,t,z\right)=\frac{\left[\begin{array}{l}\frac{a_0{b}_0-\omega b_0^2+\sqrt{{\omega }^2{b}_0^4-4{\omega }^2{b}_{-1}{b}_0^2}}{b_0^2}\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+a_0\\ +\left(\omega b_{-1}+\frac{a_0{b}_{-1}}{b_0}+\frac{b_{-1}\sqrt{-{\omega }^2{b}_0^2\left(4b_{-1}-b_0^2\right)}}{b_0^2}\right)\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}{\left[\begin{array}{l}\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+b_0\\ +b_{-1}\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}$$

(28)

where $a_0$, $b_0$, $b_{-1}$ and $\omega$ are free parameters. We plot the behavior of the solution ${\psi }_4\left(x,y,z,t\right)$ in Figure 4 with $m=1$, $k=1$, ${\xi }_0=0$.

Family 3

Set 5:$a_1=\frac{a_{-1}-2\omega b_{-1}}{b_{-1}}$, $a_0=\frac{a_{-1}{b}_{-1}{b}_0-\omega b_{-1}^2{b}_0-\sqrt{{\omega }^2{b}_{-1}^4{b}_{-1}^2-4{\omega }^2{b}_{-1}^5}}{b_{-1}^2}$, $a_{-1}=a_{-1}$, $b_0=b_0$, $b_{-1}=b_{-1}$, $v=-{\omega }^3$, $\omega =\omega$.

Next, we can get the fifth exact traveling wave solution (kinky bright-dark solitary wave solution Form 1) as:

$${\psi }_5\left(x,y,t,z\right)=\frac{\left[\begin{array}{l}\left(\frac{a_{-1}-2\omega b_{-1}}{b_{-1}}\right)\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+\frac{a_{-1}{b}_{-1}{b}_0-\omega b_{-1}^2{b}_0-\sqrt{{\omega }^2{b}_{-1}^4{b}_{-1}^2-4{\omega }^2{b}_{-1}^5}}{b_{-1}^2}\\ +a_{-1}\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}{\left[\begin{array}{l}\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+b_0\\ +b_{-1}\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}$$

(29)

Here, the $a_{-1}$, $b_0$, $b_{-1}$ and $\omega$ are free parameters. The solution ${\psi }_5\left(x,y,z,t\right)$ is presented in Figure 5 by using $m=1$, $k=1$, ${\xi }_0=0$.

Set 6:$$, $a_0=\frac{a_{-1}{b}_{-1}{b}_0-\omega b_{-1}^2{b}_0+\sqrt{{\omega }^2{b}_{-1}^4{b}_{-1}^2-4{\omega }^2{b}_{-1}^5}}{b_{-1}^2}$, $a_{-1}=a_{-1}$, $b_0=b_0$, $b_{-1}=b_{-1}$, $v=-{\omega }^3$, $\omega =\omega$.

Finally, we can get the sixth exact traveling wave solution (bright-dark wave solution) as

$${\psi }_6\left(x,y,t,z\right)=\frac{\left[\begin{array}{l}\left(\frac{a_{-1}-2\omega b_{-1}}{b_{-1}}\right)\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+\frac{a_{-1}{b}_{-1}{b}_0-\omega b_{-1}^2{b}_0+\sqrt{{\omega }^2{b}_{-1}^4{b}_{-1}^2-4{\omega }^2{b}_{-1}^5}}{b_{-1}^2}\\ +a_{-1}\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}{\left[\begin{array}{l}\exp \left(\omega x+my+kz-{\omega }^{3}t+{\xi }_0\right)+b_0\\ +b_{-1}\exp \left(-\omega x-my-kz+{\omega }^{3}t-{\xi }_0\right)\end{array}\right]}$$

(30)

where $a_{-1}$, $b_0$, $b_{-1}$ and $\omega$ are free parameters. In this case, the contour of solution ${\psi }_6\left(x,y,z,t\right)$ is illustrated in Figure 6 by selecting $m=1$, $k=1$, ${\xi }_0=0$.

3.2. Application of the HFF

In the view of Equation (12), we cannot solve it directly by the HFF. Here, we can make a transformation as:

$$\Im ={\psi }^{\prime }$$

(31)

So Equation (12) becomes:

$${\omega }^3{\Im }^{″}-3{\omega }^2{\Im }^2+v\Im =0$$

(32)

Which is

$${\Im }^{″}-\frac{3}{\omega }{\Im }^2+\frac{v}{{\omega }^3}\Im =0$$

(33)

From which, we obtain

$$f(\Im )=-\frac{3}{\omega }{\Im }^2+\frac{v}{{\omega }^3}\Im$$

(34)

The periodic solution of Equation (32) is supposed with the form as:

$$\Im \left(\xi \right)=\mathsf{\Lambda }\cos \left(\Omega \xi \right),\Omega >0$$

(35)

Based on the HFF, the amplitude frequency relationship can be easily determined as:

$${\Omega }^2={\frac{df(\Im )}{d\Im }|}_{\Im =\frac{\mathsf{\Lambda }}{2}}=\frac{1}{\omega }\left(\frac{v}{{\omega }^2}-3\mathsf{\Lambda }\right)$$

(36)

With this, the periodic solution of Equation (32) can be obtained as:

$$\Im \left(\xi \right)=\mathsf{\Lambda }\cos \left(\sqrt{\frac{1}{\omega }\left(\frac{v}{{\omega }^2}-3\mathsf{\Lambda }\right)}\xi \right)$$

(37)

In the view of Equation (31), we obtain:

$$\psi \left(\xi \right)=\frac{\mathsf{\Lambda }}{\sqrt{\frac{1}{\omega }\left(\frac{v}{{\omega }^2}-3\mathsf{\Lambda }\right)}}\sin \left(\sqrt{\frac{1}{\omega }\left(\frac{v}{{\omega }^2}-3\mathsf{\Lambda }\right)}\xi \right)$$

(38)

Thus, the periodic wave solution of Equation (1) can be obtained as:

Set 7:

$${\psi }_7\left(x,y,z,t\right)=\frac{\mathsf{\Lambda }}{\sqrt{\frac{1}{\omega }\left(\frac{v}{{\omega }^2}-3\mathsf{\Lambda }\right)}}\sin \left[\sqrt{\frac{1}{\omega }\left(\frac{v}{{\omega }^2}-3\mathsf{\Lambda }\right)}\left(\omega x+my+k\mathrm{z}+vt+{\xi }_0\right)\right]$$

(39)

By using $\mathsf{\Lambda }=1$, $\omega =\frac{1}{2}$, $m=\frac{1}{2}$, $k=1$, $v=1$, ${\xi }_0$ = 0, the performances of Equation (39) are described in Figure 7.

4. Physical Explanation

This section provides the physical explanation of the results in Section 3. Figure 1 describes the contours of the solution ${\psi }_1\left(x,y,z,t\right)$ in the region $-8\le x\le 8$, $-8\le y\le 8$ for $z=1$, $t=1$. It can be seen that the wave is the dark-solitary wave. The behaviors of ${\psi }_2\left(x,y,z,t\right)$ are presented in Figure 2 through the three-dimensional contour and two-dimensional curve, where we can discover that the wave contour is the bright-dark wave. Figure 3 depicts the solution of ${\psi }_3\left(x,y,z,t\right)$ in the domain $-8\le x\le 8$, $-8\le y\le 8$ at $z=1$, $t=0$. By comparing Figure 3 with Figure 1, Figure 3 is similar with Figure 1, which is also the dark-solitary wave. We describe the performances of ${\psi }_4\left(x,y,z,t\right)$ in Figure 4 for $z=1$, $t=0$ in the interval $-8\le x\le 8$, $-8\le y\le 8$, which is the bright-dark wave in Figure 2. Figure 5 and Figure 6 illustrate the solutions ${\psi }_5\left(x,y,z,t\right)$ and ${\psi }_6\left(x,y,z,t\right)$, respectively. By comparing them with Figure 2 and Figure 4, we find that although Figure 5 and Figure 6 are somewhat similar to Figure 2 and Figure 4, they are still the bright-dark solitary wave. The behaviors of the solution ${\psi }_7\left(x,y,z,t\right)$ are plotted in Figure 7. It can be observed that the wave contour is a perfect periodic wave.

5. Conclusions and Future Recommendation

The (3+1)-dimensional BLMP equation that describes the incompressible fluid is investigated in this study. Two effective techniques, namely the EFM and HFF, are successfully applied to construct the abundant traveling wave solutions. Seven sets of the traveling wave solutions like the bright-dark wave, dark wave and periodic wave solutions are obtained. As we've seen from [14,15,16,17,18,19,20,21,22], the results disclosed in this work are all new, which can extend the study of the exact solutions of the (3+1)-dimensional BLMP equation. Finally, the solutions are all illustrated through the three-dimensional contours and two-dimensional curves. The results show that the proposed methods are concise and effective, which, in turn, are expected to be helpful for the study of nonlinear equations arising in physics.

Fractal and Fractional calculus has seen many advancements over the past years, and we have found that many classical models in current physics are being analyzed via this calculus [33,34,35,36,37,38,39]. Applying the Fractal and Fractional calculus to Equation (1) and extracting the exact solutions are the focus of future research.