Solitary Wave Solutions for the Higher Dimensional Jimo-Miwa Dynamical Equation via New Mathematical Techniques

Abstract

In this study, under the considerations of symbolic computation with the help of Mathematica software, various types of solitary wave solutions for the (3 + 1)-dimensional Jimo–Miwa (JM) equation are successfully constructed based on the extended modified rational expansion method. The constructed solutions are novel and more general for the JM equation named kink wave solutions, anti-kink wave solutions, bright and dark solutions, mixed solutions in the shape of bright-dark solutions, and periodic waves, which do not exist in the existing literature. The physical phenomena of the demonstrated results is represented graphically by two-dimensional, three-dimensional, and contour images with the help of Mathematica. The obtained results will be widely used to explain the various interesting physical structures in the areas of optics, plasma, gas, acoustics, classical mechanics, fluid dynamics, heat transfer, and many others.

1. Introduction

A lot of research have been conducted on the investigation of nonlinear evolution equations (NLEEs), which involves the study of nonlinear structures in applied and natural sciences. The researcher investigate the various kinds of nonlinear evolution equations to describe some interesting physical phenomena. For instance, the Kortewege–de Vries (KdV) equation describes the development of one dimensional long waves in many physical phenomena such as waves on the surface of shallow water with weakly nonlinear restoring forces, long internal waves in a density stratified ocean, ion acoustic waves in plasmas, and acoustic waves on a crystal lattice [1,2,3,4]. The Schrödinger equation is established to model the surface waves in deep waters as well as electromagnetic waves in optical fibers [5]. The Kadomtsev–Petviashvili (KP) equation explain the dynamics of long wavelength and small amplitude solitary waves in two dimensions [6]. The Whitham–Broer–Kaup (WBK) equation describes the small amplitude regime for dispersive long waves in shallow water. The WBK equation also describes the propagation of waves in shallow water with different dispersion relations [7].

Moreover, it is of more significant importance to determine techniques for solving the NLEEs, if the exact solutions of the NLEEs are available, then they can provide enhanced knowledge of certain application problems such as optical communications. With the solutions of these NLEEs people can know the physical phenomena of master and matter its routes, for example, in the areas of fiber optics, heat transfer, electromagnetism, classical mechanics, field physics, fluid dynamics, acoustics, aerospace, and so on. The partial differential equations have been importantly established and solving the PDEs has become more and more popular [8,9,10,11,12,13,14,15,16,17,18,19]. From the solutions of PDEs, it is very difficult to solve the nonlinear partial differential equations. With the passage of time increasing the power of computing tools, such as the computer softwares Mathematica, Maple, Matlab, etc., we can determine rogue waves, lump solutions, and solitary wave solutions successfully by symbolic computations.

The NLPDEs have marvelous applications in the area of nonlinear sciences to understand the behavior of physical phenomena. The difficulties and problems are that for the construction of exact and solitary wave solutions of NLPDEs, the efficient and powerful methods are required to analytically investigate the NLPDEs. According to this aspect, large groups of mathematicians and engineers have developed various types of methods, including the Sine-Gorden expansion method [20], the F-expansion method [21], the lamped Galerkin method [22], the extended simple equation method, the Exp-function method [23], the Backlund transformation [24], the modified Kudryashov method [25], the spectral collection method [26], extended direct algebraic technique [27,28], Modified auxiliary mapping technique [29,30,31,32], the binary bell polynomials [33], and the (G${}^{\prime }$/G)-expansion method [34].

The (3 + 1)-dimensional Jimo–Miwa equation is the second equation in the well-known KP hierarchy of integrable systems. The JM equation describes the interesting (3 + 1)-dimensional phenomena of waves in physics. Many researchers have investigated and found various solutions for the (3 + 1)-dimensional Jimo–Miwa equation in previous literature. Zhuosheng Lü et al. constructed the exact solutions in the form of bi-solutions for the (3 + 1)-dim JM equation with Bäcklund transformation [35]. Runfa Zhang et al. found the exact solutions for (3 + 1)-dim JM equation named cross-kink wave, periodic wave with generalized bilinear method in [36] and demonstrated the periodic lump wave and interaction solutions via the Hirota bilinear method in [37]. Yong Zhang et al. found the rational and semi-rational solutions named lump, breather, and rogue wave [38] for (3 + 1)-dim JM equation via the Hirota bilinear method. Bintao Cao calculated the two families of explicit exact solutions with the help of two methods named stable-range and logarithmic stable-range methods in [39,40,41,42,43,44].

In this study, we investigate the JM equation with the modified mathematical technique named the extended modified rational expansion method [45]. The constructed solutions are novel and more general for the JM equation named kink wave solutions, anti-kink wave solutions, bright and dark solutions, mixed solutions in the shape of bright-dark solutions, and periodic waves, which does not exist in the literature.

The rest of the article is organized as follows: In Section 2, we explain the algorithm of the applied technique. In Section 3, we applied the proposed technique on the (3 + 1)-dimensional Jimo–Miwa equation for the construction of solitary wave solutions. We discuss the obtained results in Section 4. In the last section, the conclusion is given (Section 5) (See Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14).

2. Algorithm of Proposed Technique

We consider the NLPDEs as:

$$\Delta (u_{yt},u_{xxxy},u_{xx}{u}_y,u_x{u}_{xy},u_{xz},......)=0.$$

(1)

where $\Delta$ denotes the polynomial function of $u(x,y,z,t)$ and their derivatives. The main structures of the EMRE method (extended modified rational expansion method) is described as:

Step 1. We take the transformations of traveling wave as:

$$u(x,y,z,t)=\phi \left(\xi \right),\xi =(kx+ly+mz-vt).$$

(2)

The ODE of Equation (1), is obtained as:

$$\Lambda \left(-vl{\phi }^{{}^{\prime \prime }},k^{3}l{\phi }^{\left(4\right)},k^{2}l{\phi }^{{}^{\prime \prime \prime }},km{\phi }^{{}^{\prime \prime }},...\right)=0.$$

(3)

In Equation (3), $\Lambda$ represents the polynomial function in $\phi \left(\xi \right)$ and its derivatives.

Step 2. The series solution of Equation (3), is considered as:

$$\phi \left(\xi \right)=a_0+\sum _{i=1}^n\frac{a_i\Psi {\left(\xi \right)}^i+b_i\Psi {\left(\xi \right)}^{i-1}{\Psi }^{\prime }\left(\xi \right)}{{(\mu \Psi \left(\xi \right)+1)}^i}+\sum _{i=2}^n{c}_i\Psi {\left(\xi \right)}^{i-2}{\Psi }^{\prime }\left(\xi \right)+\sum _{i=1}^n{d}_i{\left(\frac{{\Psi }^{\prime }\left(\xi \right)}{\Psi \left(\xi \right)}\right)}^i.$$

(4)

Here $\mu ,a_0,a_i,b_i,c_i,d_i$ are constants, which are determined to be latter, the $\Psi \left(\xi \right)$ and their derivatives satisfy the following auxiliary equation:

$${\left(\frac{d\Psi }{d\xi }\right)}^2={\eta }_1{\Psi }^2\left(\xi \right)+{\eta }_2{\Psi }^3\left(\xi \right)+{\eta }_3{\Psi }^4\left(\xi \right).$$

(5)

In Equation (5), ${\eta }_i^{,}s$ are real constants which are found later.

Step 3. For determining the value of n for Equation (4), we balance the nonlinear terms and derivative of higher order in Equation (3).

Step 4. Putting Equation (5) in Equation (4), and collecting the coefficients of ${\Psi }^i\left(\xi \right){\Psi }^{{}^{\prime }k}\left(\xi \right)(i=1,2,3,.....n,k=0,1),$ then making every coefficient equal to zero and obtaining a family of equations, we solve these families of equations by any computer software, and the values of these parameters $\mu ,a_0,a_i,b_i,c_i,d_i$ can be calculated.

Step 5. Putting the $\mu ,a_0,a_i,b_i,c_i,d_i$ values and $\Psi \left(\xi \right)$ in Equation (4), then we obtain the required solutions of Equation (1).

3. Application of Proposed Technique for the (3 + 1)-Dimensional Jimo–Miwa Equation

Here, we apply the proposed method (extended modified rational expansion method) on the (3 + 1)-dimensional Jimo–Miwa Equation for the constructions of solitary wave solutions. The (3 + 1)-dimensional Jimbo–Miwa equation is given as:

$$2u_{yt}+u_{xxxy}+3u_{xx}{u}_y+3u_x{u}_{xy}-3u_{xz}=0.$$

(6)

Applying the traveling wave transformation on Equation (1):

$$u(x,y,z,t)=\phi \left(\xi \right),\xi =(kx+ly+mz-vt).$$

(7)

Substituting Equation (11) into Equation (1) and integratating once with respect to $\xi$, then Equation (1) changes into the ODE as:

$$k^{3}l{\phi }^{{}^{\prime \prime \prime }}+6k^{2}l{\left({\phi }^2\right)}^{{}^{\prime }}-(2lv+3km){\phi }^{{}^{\prime }}=0.$$

(8)

Using the homogeneous principle on Equation (16), we obtain n = 1. The trial solution of Equation (16), is taken as:

$$\phi \left(\xi \right)=a_0+\frac{a_1\Psi \left(\xi \right)+b_1{\Psi }^{\prime }\left(\xi \right)}{\mu \Psi \left(\xi \right)+1}+d_1\frac{{\Psi }^{\prime }\left(\xi \right)}{\Psi \left(\xi \right)}.$$

(9)

Substituting Equation (9) in Equation (8) and collecting the coefficients of ${\Psi }^i\left(\xi \right){\Psi }^{{}^{\prime }k}$ $\left(\xi \right)(i=1,2,3,.....N,k=0,1),$ then making every coefficient equal to zero, obtaining a family of equations, and solving these families of equations by Mathematica software, the parameter values are obtained as:

Case-I

$$\begin{array}{cc}\hfill a_0=& a_0,a_1=\frac{\sqrt{{\eta }_2{k}^3(-l)\mu +{\eta }_3{k}^{3}l+3k{\mu }^{2}m+2l{\mu }^{2}v}}{2\sqrt{k}\sqrt{l}},\hfill \\ \hfill & b_1=\frac{k\mu }{2},d_1=-\frac{k}{2},\mu =\mu ,{\eta }_1=\frac{3km+2\mathrm{lv}}{k^{3}l}.\hfill \end{array}$$

(10)

Substituting Equation (10) into Equation (9), the solution for Equation (6), is given as:

$$\begin{array}{cc}\hfill u_1(x,y,z,t)=& a_0+\left(\sqrt{{\eta }_1}{\eta }_2\left(-k^{3/2}\right)\sqrt{l}ϵ\mathrm{csch}{\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]}^2+\right.\hfill \\ \hfill & 2{\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)}^2\left.{\eta }_1\sqrt{{\eta }_2{k}^3(-l)\mu +{\eta }_3{k}^{3}l+{\mu }^2(3km+2lv)}\right)/\hfill \\ \hfill & \left(4\sqrt{k}\sqrt{l}\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\right.\hfill \\ \hfill & \left.{\eta }_1\left(\mu ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\mu \right)-{\eta }_2\right).\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill u_2(x,y,z,t)=& a_0-\sqrt{{\eta }_1}kϵ\left(\Gamma cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)/\hfill \\ \hfill & \left(2\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right.\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\hfill \\ \hfill & \left.\left.ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right)+\hfill \\ \hfill & \left(\sqrt{\frac{{\eta }_1}{{\eta }_3}}\left(\sqrt{{\eta }_1}{k}^{3/2}\sqrt{l}\mu ϵ\left(\Gamma cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)+\right.\right.\hfill \\ \hfill & \Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\hfill \\ \hfill & \left.ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\left.\left.\sqrt{{\eta }_2{k}^3(-l)\mu +{\eta }_3{k}^{3}l+{\mu }^2(3km+2lv)}\right)\right)/\hfill \\ \hfill & \left(2\sqrt{k}\sqrt{l}\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right.\hfill \\ \hfill & \left(\Gamma \left(\sqrt{\frac{{\eta }_1}{{\eta }_3}}\mu -2\right)+\left(\sqrt{\frac{{\eta }_1}{{\eta }_3}}\mu -2\right)cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\hfill \\ \hfill & \left.\left.\sqrt{\frac{{\eta }_1}{{\eta }_3}}\mu ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right).\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill u_3(x,y,z,t)=& \left.\left.\sqrt{\frac{{\eta }_1}{{\eta }_3}}\mu ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right)\left.qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\left.\sqrt{{\eta }_1}\right)/\hfill \\ \hfill & \left(2\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right)\right.\hfill \\ \hfill & \left(\Gamma \sqrt{q^2+1}+qϵ+cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\left.\left.ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right)+\hfill \\ \hfill & 2{\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right)}^2+\left.qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\left.\sqrt{{\eta }_1}\right)/\hfill \\ \hfill & 2{\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right)}^2+\frac{-\frac{ϵ\left(sinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+q\right)}{cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}}-1}{2\sqrt{k}\sqrt{l}}\hfill \\ \hfill & \left.\sqrt{{\eta }_2{k}^3(-l)\mu +{\eta }_3{k}^{3}l+{\mu }^2(3km+2lv)}\right)/\hfill \\ \hfill & \left(1+\mu \left(-\frac{ϵ\left(sinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+q\right)}{cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}}-1\right)\right).\hfill \end{array}$$

(13)

Case-II

$$\begin{array}{cc}\hfill a_0=& a_0,a_1=0,b_1=\frac{2{\eta }_2{k}^{4}l}{3km+2lv},d_1=-k,\hfill \\ \hfill & \mu =\frac{2{\eta }_2{k}^{3}l}{3km+2lv},{\eta }_1=\frac{3km+2lv}{4k^{3}l}.\hfill \end{array}$$

(14)

Substituting Equation (14) into Equation (9), then solutions for Equation (6), are obtained as:

$$\begin{array}{cc}\hfill u_4(x,y,z,t)=& a_0-\sqrt{{\eta }_1}kϵ(3km+2lv)\csc \mathrm{h}{\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]}^2/\hfill \\ \hfill & \left(2\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\right.\hfill \\ \hfill & \left.2{\eta }_1{k}^{3}l\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)-3km-2lv\right).\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill u_5(x,y,z,t)=& a_0+\sqrt{{\eta }_1}kϵ(3km+2lv)\left(\Gamma cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)/\hfill \\ \hfill & \left(\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\right.\left.ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\hfill \\ \hfill & \left(-(3km+2lv)\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)+\right.k^{3}l\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right.\hfill \\ \hfill & +\left.ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\left.\left.{\eta }_2\sqrt{\frac{{\eta }_1}{{\eta }_3}}\right)\right).\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill u_6(x,y,z,t)=& a_0+\left(kϵ\right(-1-\Gamma \sqrt{q^2+1}cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\left.qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\hfill \\ \hfill & \left.\sqrt{{\eta }_1}\right)/\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right.\hfill \\ \hfill & \left(\Gamma \sqrt{q^2+1}+qϵ+cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\left.\left.ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right)+\hfill \\ \hfill & \left(2k^{4}lϵ\left(-1-\Gamma \sqrt{q^2+1}cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\right.\left.qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\hfill \\ \hfill & \left.\sqrt{{\eta }_1}{\eta }_2\right)/cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\hfill \\ \hfill & \left((3km+2lv)\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right)-\right.2k^{3}l\hfill \\ \hfill & \left(\Gamma \sqrt{q^2+1}+qϵ+cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\left.ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\left.\left.{\eta }_2\right)\right).\hfill \end{array}$$

(17)

Case-III

$$\begin{array}{cc}\hfill a_0=& a_0,a_1=0,b_1=\frac{{\eta }_2{k}^{4}l-k^{5/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2\mathrm{lv})}}{12km+8lv},d_1=-\frac{k}{2},\hfill \\ \hfill & \mu =\frac{{\eta }_2{k}^{3}l-k^{3/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}}{6km+4lv},{\eta }_1=\frac{3km+2lv}{k^{3}l}.\hfill \end{array}$$

(18)

Substituting Equation (18) into Equation (9), then we obtain solutions for Equation (6), as:

$$\begin{array}{cc}\hfill u_7(x,y,z,t)=& a_0+\sqrt{{\eta }_1}{\eta }_{2}kϵ(3km+2lv)\csc \mathrm{h}{\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]}^2/\hfill \\ \hfill & \left(2\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\right.\hfill \\ \hfill & \left({\eta }_1{k}^3(-l)\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)+6km+4lv\right.\hfill \\ \hfill & {\eta }_2+k^{3/2}\sqrt{l}\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\left.\left.{\eta }_1\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}\right)\right).\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill u_8(x,y,z,t)=& a_0-\sqrt{{\eta }_1}kϵ\left(\Gamma cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)/\hfill \\ \hfill & \left(2\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right.\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\hfill \\ \hfill & \left.\left.ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right)-\left(\sqrt{{\eta }_1}\sqrt{\frac{{\eta }_1}{{\eta }_3}}ϵ\left(\Gamma cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\right.\hfill \\ \hfill & \left.{\eta }_2{k}^{4}l-k^{5/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}\right)/\left(2(12km+8lv){\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)}^2\right.\hfill \\ \hfill & \left(1-\frac{\frac{ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]}{\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]}+1}{2(6km+4lv)}\right.\left.\left.\sqrt{\frac{{\eta }_1}{{\eta }_3}}\left({\eta }_2{k}^{3}l-k^{3/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}\right)\right)\right).\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill u_9(x,y,z,t)=& a_0-\left(kϵ\right(1+\Gamma \sqrt{q^2+1}cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]-\left.qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\hfill \\ \hfill & \left.\sqrt{{\eta }_1}\right)/\left(2\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right)\right.\hfill \\ \hfill & \left(\Gamma \sqrt{q^2+1}+qϵ+cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\left.\left.sinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right)+\hfill \\ \hfill & \left(ϵ\left(-1-\Gamma \sqrt{q^2+1}cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\right.\left.qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\hfill \\ \hfill & \left.\sqrt{{\eta }_1}\left({\eta }_2{k}^{4}l-k^{5/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}\right)\right)/\hfill \\ \hfill & \left(12km+8lv{\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right)}^2\left(\frac{1}{6km+4lv}+1\right.\right.\hfill \\ \hfill & -\frac{ϵ\left(sinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+q\right)}{cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}}-1\left.\left.{\eta }_2{k}^{3}l-k^{3/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}\right)\right).\hfill \end{array}$$

(21)

Case-IV

$$\begin{array}{cc}\hfill a_0=& a_0,a_1=0,b_1\to \frac{{\eta }_2{k}^{4}l+k^{5/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}}{12km+8lv},d_1=-\frac{k}{2},\hfill \\ \hfill & \mu =\frac{{\eta }_2{k}^{3}l+k^{3/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}}{6km+4lv},{\eta }_1=\frac{3km+2lv}{k^{3}l}.\hfill \end{array}$$

(22)

Substituting Equation (22) in Equation (9), then solutions for Equation (6), we obtain:

$$\begin{array}{cc}\hfill u_{10}(x,y,z,t)=& a_0-\sqrt{{\eta }_1}{\eta }_{2}kϵ(3km+2lv)\csc \mathrm{h}{\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]}^2/\hfill \\ \hfill & \left(2\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\right.\hfill \\ \hfill & \left({\eta }_1{k}^{3}l\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\right.-6km-4lv{\eta }_2+k^{3/2}\sqrt{l}\hfill \\ \hfill & \left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\left.\left.{\eta }_1\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}\right)\right).\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill u_{11}(x,y,z,t)=& a_0-\sqrt{{\eta }_1}kϵ\left(\Gamma cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)/\left(2\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right.\hfill \\ \hfill & \left.\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)-\hfill \\ \hfill & \sqrt{{\eta }_1}\sqrt{\frac{{\eta }_1}{{\eta }_3}}ϵ\left(\Gamma cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\left.{\eta }_2{k}^{4}l+k^{5/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}\right)/\hfill \\ \hfill & \left(2(12km+8lv){\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)}^2\right.\hfill \\ \hfill & \left(1-\frac{\frac{ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]}{\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]}+1}{2(6km+4lv)}\right.\left.\left.\sqrt{\frac{{\eta }_1}{{\eta }_3}}\left({\eta }_2{k}^{3}l+k^{3/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}\right)\right)\right).\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill u_{12}(x,y,z,t)=& a_0-\left(kϵ\right(1+\Gamma \sqrt{q^2+1}cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]-\hfill \\ \hfill & \left.qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\left.\sqrt{{\eta }_1}\right)/\left(2\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right)\right.\hfill \\ \hfill & \left.cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}+qϵ\right)+\hfill \\ \hfill & \left(ϵ\left(\Gamma \left(-\sqrt{q^2+1}\right)cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]-1\right)\right.\hfill \\ \hfill & \left.\sqrt{{\eta }_1}\left({\eta }_2{k}^{4}l+k^{5/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}\right)\right)/\hfill \\ \hfill & \left((12km+8lv){\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right)}^2\right.\hfill \\ \hfill & \left(\frac{-\frac{ϵ\left(sinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+q\right)}{cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}}-1}{6km+4lv}+1\right.\left.\left.{\eta }_2{k}^{3}l+k^{3/2}\sqrt{l}\sqrt{{\eta }_2^2{k}^{3}l-4{\eta }_3(3km+2lv)}\right)\right).\hfill \end{array}$$

(25)

Case-V

$$\begin{array}{c}\hfill a_0=a_0,a_1=0,b_1=-\frac{2d_1{\eta }_3}{{\eta }_2},d_1=d_1,\mu =\frac{2{\eta }_3}{{\eta }_2},{\eta }_1=\frac{{\eta }_2^2}{4{\eta }_3}.\end{array}$$

(26)

Substituting Equation (26) into Equation (9), then we obtain the solutions for Equation (6), as:

$$\begin{array}{cc}\hfill u_{13}(x,y,z,t)=& a_0+d_1\sqrt{{\eta }_1}{\eta }_2^2ϵ\csc \mathrm{h}{\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz+{\xi }_0-tv\right)\right]}^2/\hfill \\ \hfill & \left(2\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz+{\xi }_0-tv\right)\right]+1\right)\right.\hfill \\ \hfill & \left.2{\eta }_1{\eta }_3\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz+{\xi }_0-tv\right)\right]+1\right)-{\eta }_2^2\right).\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill u_{14}(x,y,z,t)=& -\left(d_1\sqrt{{\eta }_1}{\eta }_2ϵ\left(\Gamma cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)-\right.\hfill \\ \hfill & \Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\hfill \\ \hfill & a_0\left({\eta }_2\left(-\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right)+\right.\hfill \\ \hfill & \left.\left.\sqrt{\frac{{\eta }_1}{{\eta }_3}}{\eta }_3\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right)\right)/\hfill \\ \hfill & \left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right.\hfill \\ \hfill & \left({\eta }_2\left(-\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right)+\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\right.\right.\hfill \\ \hfill & \left.ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\left.\left.\sqrt{\frac{{\eta }_1}{{\eta }_3}}{\eta }_3\right)\right).\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill u_{15}(x,y,z,t)=& a_0+\left(ϵ\left(\Gamma \sqrt{q^2+1}cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]-qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\right.\hfill \\ \hfill & \left.d_1\sqrt{{\eta }_1}\right)/\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right.\hfill \\ \hfill & \left.cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}+qϵ\right)+\hfill \\ \hfill & \left(2ϵ\left(1+\Gamma \sqrt{q^2+1}cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]-\right.\right.\left.qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\hfill \\ \hfill & \left.d_1\sqrt{{\eta }_1}{\eta }_3\right)/\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right.\hfill \\ \hfill & \left({\eta }_2\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right)-\right.\hfill \\ \hfill & \left.\left.2{\eta }_3\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}+qϵ\right)\right)\right).\hfill \end{array}$$

(29)

Case-VI

$$\begin{array}{c}\hfill a_0=a_0,a_1=0,b_1=\frac{d_1\sqrt{{\eta }_3}}{\sqrt{{\eta }_1}},d_1=d_1,\mu =-\frac{\sqrt{{\eta }_3}}{\sqrt{{\eta }_1}},{\eta }_2=-2\sqrt{{\eta }_1}\sqrt{{\eta }_3}.\end{array}$$

(30)

Substituting Equation (30) into Equation (9), then solutions for Equation (6), we obtain:

$$\begin{array}{cc}\hfill u_{16}(x,y,z,t)=& a_0+\frac{1}{2}ϵ\mathrm{csch}{\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]}^2\hfill \\ \hfill & d_1\sqrt{{\eta }_1}\left(-\frac{1}{ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1}+\right.\hfill \\ \hfill & \left.\frac{\sqrt{{\eta }_1}\sqrt{{\eta }_3}}{{\eta }_2-\sqrt{{\eta }_1}\sqrt{{\eta }_3}\left(ϵcoth\left[\frac{1}{2}\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)}\right).\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill u_{17}(x,y,z,t)=& a_0+d_1\sqrt{{\eta }_1}ϵ\left(\Gamma cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)/\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right.\hfill \\ \hfill & \left.\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)-\hfill \\ \hfill & d_1\sqrt{{\eta }_1}\sqrt{\frac{{\eta }_1}{{\eta }_3}}\sqrt{{\eta }_3}ϵ\left(\Gamma cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)/\hfill \\ \hfill & \left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\left(2\sqrt{{\eta }_1}\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)-\right.\right.\hfill \\ \hfill & \left.\left.\sqrt{\frac{{\eta }_1}{{\eta }_3}}\sqrt{{\eta }_3}\left(\Gamma +cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]\right)\right)\right).\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill u_{18}(x,y,z,t)=& a_0+\left(ϵ\left(\Gamma \sqrt{q^2+1}cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]-qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+1\right)\right.\hfill \\ \hfill & \left.d_1\sqrt{{\eta }_1}\right)/\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right.\hfill \\ \hfill & \left.cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}+qϵ\right)+\hfill \\ \hfill & \left(ϵ\left(\Gamma \left(-\sqrt{q^2+1}\right)cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+qsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]-1\right)\right.\hfill \\ \hfill & \left.d_1\sqrt{{\eta }_1}\sqrt{{\eta }_3}\right)/\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right.\hfill \\ \hfill & \left(\sqrt{{\eta }_1}\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}\right)-\right.\sqrt{{\eta }_3}\hfill \\ \hfill & \left.\left.\left(cosh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+ϵsinh\left[\sqrt{{\eta }_1}\left(kx+ly+mz-vt+{\xi }_0\right)\right]+\Gamma \sqrt{q^2+1}+qϵ\right)\right)\right).\hfill \end{array}$$

(33)

4. Results and Discussion

In this section we describe the similarities and differences between constructed solutions for the (3 + 1)-dimensional Jimo–Miwa equation in this study which have already been demonstrated in the previous literature. In this work, we have constructed novel and interesting results named solitary solutions. The series solution $\left(4\right),$ is the significant result in this study with the variety of five parameters, which have diverse constructions. The constant parameter $\mu ,a_0,a_i,b_i,c_i,d_i$ values were collected with symbolic computation, then Equation (5) had different kinds of results named rational, trigonometric, and elliptic functions. The constructed results are novel, more general, and interesting solutions via this powerful and efficient method. Now, we compare our calculated results with other techniques (See Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14).

1. Bäcklund transformation:

Zhuosheng Lü et al. constructed the exact solutions in the form of bi-solutions for the (3 + 1)-dimensional Jimo–Miwa equation with Bäcklund transformation in [35]. However, our constructed results are novel, interesting, and have different physical structures.

2. Generalized bilinear method:

Runfa Zhang et al. found the exact solutions for the JM equation named cross-kink wave and periodic wave with generalized bilinear methods in [36]. However, our constructed results are novel, interesting, and have different physical structure.

3. Hirota bilinear method:

Runfa Zhang et al. demonstrated the periodic lump wave and interaction solutions in [37] and Yong Zhang et al. found the rational and semi-rational solutions named lump, breather, and rogue wave [38] for (3 + 1)-dim JM equation via the Hirota bilinear method. However, our constructed results are novel, interesting, and have different physical structures.

4. Stable-range and logarithmic stable-range methods:

Bintao Cao calculated the two families of explicit exact solutions with the help of two methods named stable-range and logarithmic stable-range methods in [39]. However, our constructed results are novel, interesting, and have different physical structures.

In this work, the demonstrated new results named elliptic, rational, and trigonometric kink wave solutions, anti-kink wave solutions, bright and dark solutions, and mixed solutions in the shape of bright-dark solutions and periodic solitary waves for (3 + 1)-dim JM equations.

From the above detailed discussion and compression, we accomplish that the constructed results are novel, interesting, and more general, which have not been calculated in the previous literature by other methods. The current study shows that applied method is more powerful, efficient, and reliable for the investigation of other nonlinear evolution equations.

5. Conclusions

Under the investigation of the EMRE method (extended modified rational expansion method), we have successfully constructed the novel, more general, and interesting results for the the (3 + 1)-dimensional Jimo–Miwa (JM) equation. In this current work, the demonstrated new results named elliptic, rational, and trigonometric, in the shape of kink wave solutions, anti-kink wave solutions, bright and dark solutions, mixed solutions in the shape of bright-dark solutons and periodic solitary waves. The physical behavior of the demonstrated results is represented by images via two-dimensional, three-dimensional, and contour plots with Mathematica. The constructed solutions are novel, more general, interesting, and do not exist in the existing literatures, which will be widely used to describe the many interesting physical phenomena in the area of optics, plasma, gas, acoustics, classical mechanics, fluid dynamics, heat transfer, and many others. The current research shows that the proposed method is more powerful, reliable, and efficient for the study of other nonlinear evolution equations.