Classical and nonclassical Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to a 3D-modified nonlinear wave equation in liquid involving gas bubbles

Abstract

The current paper undertakes an in-depth exploration of the dynamics of nonlinear waves governed by a 3D-modified nonlinear wave equation, a significant model in the study of complex wave phenomena. To this end, the study employs both classical and nonclassical Lie symmetries for rigorously deriving invariant solutions of the governing equation. These symmetries enable the formal construction of exact solutions, which are crucial for understanding the complex behavior of the model. Furthermore, the research extends into the realm of bifurcation analysis through the application of planar dynamical system theory. Such an analysis reveals the conditions under which the 3D-modified nonlinear wave equation admits Jacobi elliptic function solutions. The study also delves into the impact of the nonlinear parameter on the physical characteristics of bright and kink solitary waves as well as continuous periodic waves using Maple. Overall, the comprehensive analysis presented not only enhances the understanding of complex nonlinear wave dynamics but also sets the stage for future advancements in vast areas of fluid dynamics and plasma physics.

1 Introduction

Nonlinear partial differential equations (NLPDEs) are crucial in modeling numerous physical phenomena, from plasma physics to fluid mechanics. Nowadays, by improving systematic packages like Maple and Mathematica, seeking exact solutions for NLPDEs has become more convenient. In recent decades, many effective methods, such as $(G'/G)$-expansion method [1–4], Kudryashov method [5–8], sine-Gordon method [9–12], and others, have been proposed and applied to deal with this class of equations. Mohanty et al. [13] employed the $(G'/G)$-expansion method to construct solitary waves of the mKdV equation. Hosseini et al. [14] found dark solitons of a nonlinear Schrödinger equation with parabolic law using the Kudryashov method. In [15], Akbar et al. utilized the sine-Gordon method to obtain solitons of the Boussinesq equation.

The utilization of Lie group theory [16–18] provides a robust mathematical framework for investigating the classical and nonclassical symmetries and invariance properties of NLPDEs, enabling us to derive exact solutions and gain insights into their behaviors. For a long time, such a method has been a reliable tool for solving different types of NLPDEs. Numerous studies have been conducted on Lie groups and their applications in dealing with NLPDEs. For example, the authors in [19] discovered soliton waves of the Zakharov–Kuznetsov modified equal-width equation using the Lie symmetry method. Yang and Wang [20] employed the Lie symmetry method to construct invariant solutions of the Benjam–Bona–Mahony–Peregrine equation. Hussain et al. [21] extracted kink solitary waves of Riabouchinsky–Proudman–Johnson equation via the Lie symmetry method. In another work conducted in [22], Jarad et al. utilized the Lie symmetry method to acquire traveling waves of the Calogero–Bogoyavlenskii–Schiff equation. It should be pointed out that, in addition to classical symmetries, nonclassical symmetries [23–26] can also be determined, which lead to finding a new class of invariant solutions. Some well-known NLPDEs and their nonclassical symmetries are:

• the thin film model [27],

• nonlinear lattice equations [28],

• and the macroscopic production model [29].

In light of the fact that academics have to deal with NLPDEs and their dynamical systems, qualitative theory is an essential part of the studies. Today’s studies of dynamical systems rely on bifurcation analysis, a crucial subject in the qualitative theory of dynamical systems. The following are some famous NLPDEs and their bifurcation analysis:

• generalized Schrödinger equation [30],

• paraxial Schrödinger equation [31],

• perturbed Schrödinger equation [32],

• and Radhakrishnan–Kundu–Lakshmanan equation [33].

A wide variety of phenomena can be explained by 3D-nonlinear wave (3D-NLW) equations, ranging from plasma physics to fluid mechanics. The study of the 3D-modified nonlinear wave equation in liquids containing gas bubbles is significant due to its potential to enhance our understanding of complex wave phenomena in multiphase media. Liquids with gas bubbles [34] are prevalent in various natural and industrial settings, such as underwater acoustics, biomedical ultrasound, and cavitation processes. The presence of gas bubbles significantly changes the propagation of waves, leading to nonlinear effects that are not captured by traditional wave equations. Recently, in the study by Li et al. [35], the authors investigated a 2D-NLW equation using symbolic methods. By transforming the equation into the one-dimensional domain, the authors obtained multiple exact solutions, including triangular and hyperbolic functions, which revealed the dynamics of nonlinear waves. Wang et al. [36] focused on novel exact solutions for a generalized 2D-NLW equation, including soliton molecules, periodic waves, and other diverse wave solutions. In another research [37], authors explored the integrability of a new 2D-NLW equation by examining its Painlevé property and derived multiorder breather and lump solutions as well as hybrid solutions. While much of the previous research has focused on one- or two-dimensional models, some researchers have dealt with 3D-NLW equations. This is important because many practical applications, such as underwater sonar or medical ultrasound, involve inherently three-dimensional wave propagation. Wang et al. [38] dealt with a 3D-NLW equation in liquid involving gas bubbles, i.e.,

$$ (u_{t} +h_{1}uu_{x} +h_{2}u_{xxx} +h_{3}u_{x})_{x} +h_{4}u_{yy} +h_{5}u_{zz} = 0, $$

and derived its lump and interaction solutions using the Hirota bilinear method. Adeyemo and Khalique [39] considered another 3D-NLW equation in a fluid medium containing gas bubbles

$$ (u_{t} +\alpha u_{x} +\beta uu_{x} +\gamma u_{xxx})_{x} +\nu u_{yy} + \epsilon u_{zz} = 0, $$

and extracted soliton waves of it using the Lie symmetry method. Newly, Wazwaz et al. [40] constructed solitary waves of the following 3D-NLW equation

$$ (u_{t} +6uu_{x} +u_{xxx})_{xx} +\alpha u_{xx}+\beta u_{xy}+\gamma u_{xz}+ \lambda u_{xt}+\mu u_{xxx} = 0. $$

In the present paper, the authors aim to investigate a 3D-modified nonlinear wave (3D-mNLW) equation in liquid involving gas bubbles as follows

$$ \Delta \equiv (u_{t} +a_{1}u_{x} +a_{2}u^{2}u_{x} +a_{3}u_{xxx})_{x} +a_{4}u_{yy} +a_{5}u_{zz}+a_{6}u_{yz} = 0, $$

(1)

where u is the wave function dependent on time t and spatial coordinates x, y, and z. The parameters $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$, and $a_{6}$ are constants that characterize the properties of the medium and the nature of the wave propagation. For the other terms, the reader can see Table 1.

Table 1 Description of the terms

Generally, the goal is to find classical and nonclassical Lie symmetries and apply them to construct invariant solutions of Eq. (1). Additionally, by utilizing the theory of planar dynamical systems, bifurcation analysis of the 3D-mNLW equation is accomplished, resulting in Jacobi elliptic function solutions. An in-depth discussion of the physical characteristics of bright and kink solitary waves, as well as continuous periodic waves, is presented systematically.

2 Lie symmetry analysis of 3D-mNLW equation

The authors begin this section by describing a method for establishing the Lie point symmetry for the 3D-mNLW equation (1). Following this, they derive both classical and nonclassical symmetries associated with the aforementioned equation and explore their respective exact solutions. This approach enables a comprehensive analysis of the inherent symmetries of the equation, revealing a range of solutions and facilitating a deeper understanding of its dynamics and implications.

2.1 Lie group analysis for Eq. (1)

In this subsection, the authors address the 3D-mNLW equation and conduct its Lie symmetry analysis. To achieve this, Lie group transformations are constructed with ϵ as a group parameter, as follows

$$\begin{aligned}& \begin{aligned} \hat{x}&\mapsto x+\epsilon \xi (x,y,z,t,u)+O(\epsilon ^{2}), \\ \hat{y}&\mapsto y+\epsilon \zeta (x,y,z,t,u)+O(\epsilon ^{2}), \\ \hat{z}&\mapsto z+\epsilon \varsigma (x,y,z,t,u)+O(\epsilon ^{2}), \\ \hat{t}&\mapsto t+\epsilon \tau (x,y,z,t,u)+O(\epsilon ^{2}), \\ \hat{u}&\mapsto u+\epsilon \eta (x,y,z,t,u)+O(\epsilon ^{2}), \end{aligned} \end{aligned}$$

which the vector field corresponding to the Lie group symmetry of the above transformations is considered as follows

$$ \begin{aligned} \mathbf{V}&=\xi (x,y,z,t,u)\frac{\partial}{\partial x}+\zeta (x,y,z,t,u) \frac{\partial}{\partial y}+\varsigma (x,y,z,t,u) \frac{\partial}{\partial z}+\tau (x,y,z,t,u) \frac{\partial}{\partial t}\\ &\quad +\eta (x,y,z,t,u) \frac{\partial}{\partial u}. \end{aligned} $$

(2)

The vector field (2) induces a symmetry in Eq. (1) when the following infinitesimal invariance condition is satisfied

$$\begin{aligned}& \begin{aligned} &\mathcal{P}r^{(4)}\mathbf{V}(\Delta )\bigg\vert _{\Delta =0}=0, \end{aligned} \end{aligned}$$

(3)

where the extended prolongation operator $\mathcal{P}r^{(4)}$ is defined as follows

$$\begin{aligned}& \begin{aligned} \mathcal{P}r^{(4)}\mathbf{V}=&\mathbf{V}+\eta ^{x} \frac{\partial}{\partial u_{x}}+\eta ^{xx} \frac{\partial}{\partial u_{xx}}+\eta ^{xt} \frac{\partial}{\partial u_{xt}}+\cdots+\eta ^{i_{1}i_{2}...i_{k}} \frac{\partial}{\partial u_{i_{1}i_{2}...i_{k}}}, \end{aligned} \end{aligned}$$

and the symbol $\eta ^{j}$ represents the extended infinitesimals of integer order, expressed as follows

$$\begin{aligned}& \begin{aligned} &\eta ^{i}=D_{i}\eta -(D_{i}\xi _{j})u_{j}, \hspace{2cm} i,j=1,2,..,n \\ &\eta ^{i_{1}i_{2}...i_{k}}=D_{i_{k}}\eta ^{i_{1}i_{2}...i_{k-1}}-(D_{i_{k}} \xi _{j})u_{i_{1}...i_{k-1}j}, \hspace{1cm} \end{aligned} \end{aligned}$$

with $k=2,3,4,\ldots$ , $\ell =1,2,\ldots$ , $k ~~ \text{for} ~~ i_{\ell}=1,2,\ldots,n$.

In the given expression, $D_{i_{k}}$ denotes the total derivative with respect to x, y, z, and t. This derivative can be defined as

$$ D_{i_{k}}=\frac{\partial}{\partial i_{k}}+u_{i_{k}} \frac{\partial}{\partial u}+u_{i_{k}j_{k}} \frac{\partial}{\partial u_{i_{k}}}+\cdots . $$

2.2 Computing classical symmetries to the main equation

According to the above explanations, we are searching for symmetries here, and to achieve this we have to follow the process that has been explained before. The infinitesimal invariance criterion for Eq. (1) is as follows

$$ \mathcal{P}r^{(4)}\mathbf{V}\bigg((u_{t} +a_{1}u_{x} +a_{2}u^{2}u_{x} +a_{3}u_{xxx})_{x} +a_{4}u_{yy} +a_{5}u_{zz}+a_{6}u_{yz}\bigg)\bigg\vert _{\text{(1)}}=0. $$

(4)

Therefore, we need to verify the following equation

$$ \begin{aligned} &2a_{2}(u_{x}^{2}+uu_{xx})\eta +4a_{2}uu_{x}\eta ^{x}+(a_{1}+a_{2}u^{2}) \eta ^{xx}+\eta ^{xt}+a_{3}\eta ^{xxxx}+a_{4}\eta ^{yy}\\ &\quad +a_{5}\eta ^{zz}+a_{6} \eta ^{yz}=0. \end{aligned} $$

(5)

Collecting the coefficients of various derivatives of u, we obtain a system as

$$\begin{aligned} \begin{aligned} &\xi _{u}=\zeta _{u}=\zeta _{x}=\varsigma _{u}=\varsigma _{x}=\tau _{u}= \tau _{x}=0, \\ &\eta _{uu}=0, \\ &2\eta _{xu}-3\xi _{xx}=0, \\ &2a_{5}\tau _{z}+a_{6}\tau _{y}=0, \\ &2a_{4}\tau _{y}+a_{6}\tau _{z}=0, \\ &4a_{5}\xi _{x}-2a_{5}\varsigma _{z}-a_{6}\varsigma _{y}=0, \\ &2a_{4}\xi _{y}+a_{6}\xi _{z}+\zeta _{t}=0, \\ &a_{6}\xi _{y}+2a_{5}\xi _{z}+\varsigma _{t}=0, \\ &a_{4}a_{6}\varsigma _{y}+2a_{4}a_{5}\varsigma _{z}-2a_{4}a_{5}\zeta _{y}-a_{5}a_{6} \zeta _{z}=0, \\ &2a_{5}\varsigma _{z}-a_{5}\xi _{x}+a_{6}\varsigma _{y}-a_{5}\tau _{t}=0, \\ &a_{6}\tau _{yz}+a_{4}\tau _{yy}+a_{5}\tau _{zz}-\eta _{xu}=0, \\ &a_{5}a_{6}\varsigma _{z}-a_{5}a_{6}\zeta _{y}-2a_{5}^{2}\zeta _{z}-2a_{4}a_{5} \varsigma _{y}+a_{6}^{2}\varsigma _{y}=0, \\ &a_{6}\zeta _{yz}-a_{6}\eta _{zu}+a_{4}\zeta _{yy}-2a_{4}\eta _{yu}+a_{5} \zeta _{zz}=0, \\ &a_{6}+2a_{5}\varsigma _{yz}-2a_{5}\eta{zu}+a_{4}+2a_{5}\varsigma _{yy}-a_{6} \eta _{yu}+a_{5}+2a_{5}\varsigma _{zz}=0, \\ &a_{5}(\eta _{u}-2\xi _{x}+2\varsigma _{z})u+a_{6}\varsigma _{y}u+a_{5} \eta =0, \\ &a_{2}\eta _{xx}u^{2}+a_{1}\eta _{xx}+a_{5}\eta _{zz}+a_{3}\eta _{xxxx}+a_{4} \eta _{yy}+a_{6}\eta _{yz}+\eta _{xt}=0, \\ &2a_{2}a_{5}(\varsigma _{z}-\xi _{x})u^{2}+a_{2} a_{6}\varsigma _{y} u^{2}+2a_{2}a_{5} \eta u-4a_{3}a_{5}\xi _{xxx}+6a_{3}a_{5}\eta _{xxu} \\ &+2a_{1}a_{5}(\varsigma _{z}-\xi _{x})+a_{1}a_{6}\varsigma _{y}-a_{5} \xi _{t}=0, \\ &a_{2}(2\eta _{xu}-\xi _{x}x)u^{2}+4a_{2}\eta _{x} u+4a_{3}\eta _{xxxu}-a_{4} \xi _{yy}-a_{6}\xi _{yz}-a_{3}\xi _{xxxx} \\ &-a_{5}\xi _{zz}+a_{1}(2\eta _{xu}-\xi _{xx})-\xi _{xt}+\eta _{tu}=0. \end{aligned} \end{aligned}$$

Therefore, according to the above system, the following vector fields are obtained

$$\begin{aligned} &\mathbf{V}_{1}=\frac{\partial}{\partial x}, \hspace{1cm} \mathbf{V}_{2}=\frac{\partial}{\partial y}, \hspace{8cm} \\ & \mathbf{V}_{3}=\frac{\partial}{\partial z}, \hspace{1cm} \mathbf{V}_{4}=\frac{\partial}{\partial t}, \\ &\mathbf{V}_{5}=(ya_{6}-2za_{4})\frac{\partial}{\partial y}+(2ya_{5}-za_{6}) \frac{\partial}{\partial z}, \\ &\mathbf{V}_{6}=3t\frac{\partial}{\partial t}a_{6}-u \frac{\partial}{\partial u}a_{6}+(2ta_{1}b_{6}+xb_{6}) \frac{\partial}{\partial x}+(4ya_{6}-4za_{4}) \frac{\partial}{\partial y}+4y\frac{\partial}{\partial z}a_{5}, \\ &\mathbf{V}_{7}=-2t\frac{\partial}{\partial y}a_{4}-t \frac{\partial}{\partial z}a_{6}+y\frac{\partial}{\partial x}, \\ &\mathbf{V}_{8}=-t\frac{\partial}{\partial y}a_{6}-2t \frac{\partial}{\partial z}a_{5}+z\frac{\partial}{\partial x}. \end{aligned}$$

Case 1:

The vector field $\mathbf{V}_{5}$ reduces the 3D-mNLW equation (1) to a PDE with three independent variables. Next, by solving the characteristic equations

$$ \dfrac{dx}{0 }=\dfrac{dy}{ya_{6}-2za_{4} }= \dfrac{dz}{2ya_{5}-za_{6} }=\dfrac{dt}{0 }=\dfrac{du}{0 }, $$

we have invariants, which are listed as

$$ \xi _{1}=x,~~~~ \xi _{2}={ \dfrac {1}{\sqrt {{y}^{2}a_{{5}}-yza_{{6}}+{z}^{2}a_{{4}}}}},~~~~\xi _{3}=t,~~~~ u(x,y,z,t)=F(\xi _{1},\xi _{2},\xi _{3}). $$

(6)

Using Eq. (6), Eq. (1) reduces to

$$ \begin{aligned} &\left (4a_{4}a_{5}-a_{6}^{2}\right )\xi _{3}^{4}(F_{\xi _{3}\xi _{3}}+F_{ \xi _{3}})+8a_{2}FF_{\xi _{2}}^{2}+4a_{2}F^{2}F_{\xi _{2}\xi _{2}}+4a_{3}F_{ \xi _{2}\xi _{2}\xi _{2}\xi _{2}} +4a_{1}F_{\xi _{2}\xi _{2}}\\ &\quad +4F_{ \xi _{1}\xi _{2}}=0. \end{aligned} $$

(7)

By setting $a_{6}^{2}=4a_{4}a_{5}$, considering the transformation

$$ F(\xi _{1},\xi _{2},\xi _{3})=G(\epsilon ), ~\epsilon =-\xi _{1}c_{1}+ \xi _{2}c_{2}+\xi _{3}c_{3}, $$

and applying it to Eq. (7), we obtain the following ordinary differential equation (ODE) of higher-order

$$\begin{aligned}& \begin{aligned} &4\,a_{{2}}c_{{2}}^{2} G_{\epsilon \epsilon}G^{2}+ \left ( 4\,a_{{1}}c_{{2}}^{2}-4 \,c_{{1}}c_{{2}} \right )G_{\epsilon \epsilon} +8\,a_{{2}}c_{{2}}^{2} GG_{ \epsilon}^{2}+4\,a_{{3}}c_{{2}}^{4}G_{\epsilon \epsilon \epsilon \epsilon}=0. \end{aligned} \end{aligned}$$

(8)

Now, the ansatz method [41–43] is formally utilized to look for Jacobi elliptic function solutions of Eq. (8).

2.2.1 Periodic snoidal-type waves

To get periodic snoidal-type waves from Eq. (8), the Jacobi elliptic sine function with modulus k

$$ G(\epsilon )=A~\text{SN}(\epsilon ,k),~0< k< 1, $$

is substituted into Eq. (8). Upon organizing the resulting equation, the following nonlinear system is obtained

$$\begin{aligned} &\text{SN}^{5}(\epsilon ,k):24k^{4}a_{3}c_{2}^{3}+4A^{2}k^{2}a_{2}c_{2}=0, \\ &\text{SN}^{3}(\epsilon ,k):-20k^{4}a_{3}c_{2}^{3}-3A^{2}k^{2}a_{2}c_{2}-20k^{2}a_{3}c_{2}^{3}-3A^{2}a_{2}c_{2}+2k^{2}a_{1}c_{2}-2k^{2}c_{1}=0, \\ & \\ &\text{SN}(\epsilon ,k):k^{4}a_{3}c_{2}^{3}+14k^{2}a_{3}c_{2}^{3}+2A^{2}a_{2}c_{2}-k^{2}a_{1}c_{2}+a_{3}c_{2}^{3}+k^{2}c_{1}-a_{1}c_{2}+c_{1}=0, \end{aligned}$$

whose solution gives

$$ A=\pm \sqrt {-6\,{\frac {a_{{3}}}{a_{{2}}}}}c_{{2}}k,~ c_{1} =-(k^{2}a_{3}c_{2}^{2}+a_{3}c_{2}^{2}-a_{1})c_{2}. $$

Hence, the following periodic snoidal-type wave solutions are derived

$$ F(\xi _{1},\xi _{2},\xi _{3})=\pm \sqrt {-6\,{ \frac {a_{{3}}}{a_{{2}}}}}c_{{2}}k~\text{SN}\left (c_{2}(k^{2} a_{3} c_{2}^{2}+a_{3} c_{2}^{2}-a_{1})\xi _{1}+c_{2}\xi _{2}+c_{3}\xi _{3},k\right ). $$

Then, as $k\rightarrow 1$, the exact solutions of the main equation can be derived as

$$\begin{aligned}& \begin{aligned} u(x,y,z,t)=\pm{\sqrt {-6\,{\frac {a_{{3}}}{a_{{2}}}}}c_{{2}}\tanh \left ( {c_{2}}\,x+{ \frac {{c_{3}}}{\sqrt {{y}^{2}a_{{5}}-2\sqrt{a_{{4}}a_{{5}}}yz+{z}^{2}a_{{4}}}}}+\left ( 2c_{2}^{3}a_{3}-{c_{2}}\,a_{{1}} \right ) t \right ) }. \end{aligned} \end{aligned}$$

2.2.2 Periodic dnoidal-type waves

To derive periodic dnoidal-type waves from Eq. (8), the Jacobi elliptic function with modulus k

$$ G(\epsilon )=A~\text{DN}(\epsilon ,k),~0< k< 1, $$

is inserted into Eq. (8). After some calculations, the following nonlinear system can be specified

$$\begin{aligned}& \begin{aligned} &\text{DN}(\epsilon ,k)\text{SN}^{4}(\epsilon ,k):24k^{4}a_{3}c_{2}^{3}+4A^{2}k^{2}a_{2}c_{2}=0, \\ & \\ &\text{DN}(\epsilon ,k)\text{SN}^{2}(\epsilon ,k):-20k^{4}a_{3}c_{2}^{3}-3A^{2}k^{2}a_{2}c_{2}-20k^{2}a_{3}c_{2}^{3}-3A^{2}a_{2}c_{2}+2k^{2}a_{1}c_{2}-2k^{2}c_{1}\\ &\quad =0, \\ & \\ &\text{DN}(\epsilon ,k):k^{4}a_{3}c_{2}^{3}+14k^{2}a_{3}c_{2}^{3}+2A^{2}a_{2}c_{2}-k^{2}a_{1}c_{2}+a_{3}c_{2}^{3}+k^{2}c_{1}-a_{1}c_{2}+c_{1}=0. \end{aligned} \end{aligned}$$

Its solution gives

$$ A=\pm \sqrt {6\,{\frac {a_{{3}}}{a_{{2}}}}}c_{{2}},~c_{1} =-(k^{2}a_{3}c_{2}^{2}-2a_{3}c_{2}^{2}-a_{1})c_{2}. $$

Therefore, the following periodic dnoidal-type wave solutions are constructed

$$ F(\xi _{1},\xi _{2},\xi _{3})=\pm \sqrt {6\,{\frac {a_{{3}}}{a_{{2}}}}}c_{{2}}~ \text{DN}\left (c_{2}(k^{2} a_{3} c_{2}^{2}-2a_{3} c_{2}^{2}-a_{1}) \xi _{1}+c_{2}\xi _{2}+c_{3}\xi _{3},k\right ). $$

So, as $k\rightarrow 1$, the exact solutions of the main equation can be obtained as

$$\begin{aligned}& \begin{aligned} u(x,y,z,t)=\pm{\sqrt {6\,{\frac {a_{{3}}}{a_{{2}}}}}c_{{2}}~ \text{sech} \left ( {c_{2}}\,x+{ \frac {{c_{3}}}{\sqrt {{y}^{2}a_{{5}}-2\sqrt{a_{{4}}a_{{5}}}yz+{z}^{2}a_{{4}}}}}+ \left ( 2c_{2}^{3}a_{{3}}-{c_{2}}\,a_{{1}} \right ) t \right ) }. \end{aligned} \end{aligned}$$

Case 2:

A linear combination of vectors $\mathbf{V}_{1}$, $\mathbf{V}_{2}$, and $\mathbf{V}_{4}$, i.e., $\mathbf{V}=\mathbf{V}_{1}+\mathbf{V}_{2}+\mathbf{V}_{4}$ reduces Eq. (1) to a 3D-variables equation. In this case, we have invariants listed as

$$ \xi _{1}=z,~~~~ \xi _{2}=t-x,~~~~\xi _{3}=y-x,~~~~ u(x,y,z,t)=F(\xi _{1}, \xi _{2},\xi _{3}), $$

where F is dependent, and $\xi _{1}$, $\xi _{2}$, and $\xi _{3}$ are independent variables. According to this invariant solution, the 3D-mNLW equation transform into

$$\begin{aligned}& \begin{aligned} &(2a_{2}F^{2}+2a_{1}-1)F_{\xi _{2}\xi _{3}}+(a_{2}F^{2}+a_{1}-1)F_{ \xi _{2}\xi _{2}}+(a_{2}F^{2}+a_{4}+a_{1})F_{\xi _{3}\xi _{3}} \\ &+2a_{2}F(F_{\xi _{3}}^{2}+2F_{\xi _{2}}F_{\xi _{3}}+F_{\xi _{2}}^{2})+a_{3} \left (F_{\xi _{3}\xi _{3}\xi _{3}\xi _{3}}+4F_{\xi _{2}\xi _{3}\xi _{3} \xi _{3}}+6F_{\xi _{2}\xi _{2}\xi _{3}\xi _{3}}\right . \\ &\left .+4F_{\xi _{2}\xi _{2}\xi _{2}\xi _{3}}+F_{\xi _{2}\xi _{2} \xi _{2}\xi _{2}}\right )+a_{5}F_{\xi _{1}\xi _{1}}+a_{6}F_{\xi _{1} \xi _{3}}=0. \end{aligned} \end{aligned}$$

(9)

To solve the above equation, we proceed as in the previous case. Using the transformation $F(\xi _{1},\xi _{2},\xi _{3})=G(\epsilon )$, where $\epsilon =\xi _{1}c_{1}-\xi _{2}c_{2}+\xi _{3}c_{3}$, and substituting it into Eq. (9), we can obtain the following high-order ordinary differential equation

$$\begin{aligned}& \begin{aligned} &\left ( a_{{2}}c_{{2}}^{2}-2\,a_{{2}}c_{{2}}c_{{3}}+a_{{2}}c_{{3}}^{2} \right ) G_{\epsilon \epsilon} G^{2}+ \left ( a_{{1}}c_{{2}}^{2}-2\,a_{{1}}c_{{2}}c_{{3}}+a_{{1}}c_{{3}}^{2}+a_{{4}}c_{{3}}^{2}+a_{{5}}c_{{1}}^{2}+a_{{6}}c_{{1}}c_{{3}} \right . \\ &\left . -c_{{2}}^{2}+c_{{2}}c_{{3}} \right ) G_{\epsilon \epsilon} + \left ( 2\,a_{{2}}c_{{2}}^{2}-4\,a_{{2}}c_{{2}}c_{{3}}+2\,a_{{2}}c_{{3}}^{2} \right )G G_{\epsilon}^{2} \\ &+ \left ( a_{{3}}c_{{2}}^{4}-4\,a_{{3}}c_{{2}}^{3}c_{{3}}+6\,a_{{3}}c_{{2}}^{2}c_{{3}}^{2}-4 \,a_{{3}}c_{{2}}c_{{3}}^{3}+a_{{3}}c_{{3}}^{4} \right ) G_{\epsilon \epsilon \epsilon \epsilon} =0. \end{aligned} \end{aligned}$$

(10)

To get periodic snoidal waves, the following Jacobi elliptic function with modulus k

$$ G(\epsilon )=A~\text{SN}(\epsilon ,k),~0< k< 1, $$

is substituted into Eq. (10). After replacing and collecting the resulting equation, the following system can be obtained

$$\begin{aligned} &\text{SN}^{5}(\epsilon ,k):24\,{k}^{4}a_{{3}}c_{{2}}^{4}-96\,{k}^{4}a_{{3}}c_{{2}}^{3}c_{{3}}+144 \,{k}^{4}a_{{3}}c_{{2}}^{2}c_{{3}}^{2}-96\,{k}^{4}a_{{3}}c_{{2}}c_{{3}}^{3}+24 \,{k}^{4}a_{{3}}c_{{3}}^{4}\\ &+4\,{A}^{2}{k}^{2}a_{{2}}c_{{2}}^{2}-8\,{A}^{2}{k}^{2}a_{{2}}c_{{2}}c_{{3}}+4\,{A}^{2}{k}^{2}a_{{2}}c_{{3}}^{2}=0, \\ & \\ &\text{SN}^{3}(\epsilon ,k): -20\,{k}^{4}a_{{3}}c_{{2}}^{4}+80\,{k}^{4}a_{{3}}c_{{2}}^{3}c_{{3}}-120 \,{k}^{4}a_{{3}}c_{{2}}^{2}c_{{3}}^{2}+80\,{k}^{4}a_{{3}}c_{{2}}c_{{3}}^{3}-20 \,{k}^{4}a_{{3}}c_{{3}}^{4}\\ &-3\,{A}^{2}{k}^{2}a_{{2}}c_{{2}}^{2}+6\,{A}^{2}{k}^{2}a_{{2}}c_{{2}}c_{{3}}-3\,{A}^{2}{k}^{2}a_{{2}}c_{{3}}^{2}-20 \,{k}^{2}a_{{3}}c_{{2}}^{4}+80\,{ k}^{2}a_{{3}}c_{{2}}^{3}c_{{3}}-120 \,{k}^{2}a_{{3}}c_{{2}}^{2}c_{{3}}^{2}\\ &+80\,{k}^{2}a_{{3}}c_{{2}}c_{{3}}^{3}-20\,{k}^{2}a_{{3}}c_{{3}}^{4}-3\,{A}^{2}a_{{2}}c_{{2}}^{2}+6\,{A}^{2}a_{{2}}c_{{2}}c_{{3}}-3 \,{A}^{2}a_{{2}}c_{{3}}^{2}+2\,{k}^{2}a_{{1}}c_{{2}}^{2}-4\,{k}^{2}a_{{1}}c_{{2}}c_{{3}}\\ &+2 \,{k}^{2}a_{{1}}c_{{3}}^{2}+2\,{k}^{2}a_{{4}}c_{{3}}^{2}+2\,{k}^{2}a_{{5}}c_{{1}}^{2}+2\,{k}^{2}a_{{6}}c_{{1}}c_{{3}}-2 \,{k}^{2}c_{{2}}^{2}+2\,{k}^{2}c_{{2}}c_{{3}}=0, \\ & \\ &\text{SN}(\epsilon ,k):{k}^{4}a_{{3}}c_{{2}}^{4}-4\,{k}^{4}a_{{3}}c_{{2}}^{3}c_{{3}}+6 \,{k}^{4}a_{{3}}c_{{2}}^{2}c_{{3}}^{2}-4\,{k}^{4}a_{{3}}c_{{2}}c_{{3}}^{3}+{k}^{4}a_{{3}}c_{{3}}^{4}+14 \,{k}^{2}a_{{3}}c_{{2} }^{4}\\ &-56\,{k}^{2}a_{{3}}c_{{2}}^{3}c_{{3}}+84\,{k}^{2}a_{{3}}c_{{2}}^{2}c_{{3}}^{2}-56\,{k}^{2}a_{{3}}c_{{2}}c_{{3}}^{3}+14 \,{k}^{2}a_{{3}}c_{{3}}^{4}+2\,{A}^{2}a_{{2}}c_{{2}}^{2}-4\,{A}^{2}a_{{2}}c_{{2}}c_{{3}}\\ &+2 \,{A}^{2}a_{{2}}c_{{3}}^{2}-{k}^{2}a_{{1}}c_{{2}}^{2}+2\,{k}^{2}a_{{1}}c_{{2}}c_{{3}}-{k}^{2}a_{{1}}c_{{3}}^{2}-{k}^{2}a_{{4}}c_{{3}}^{2}-{k}^{2}a_{{5}}c_{{1}}^{2}-{k}^{2}a_{{6}}c_{{1} }c_{{3}}+a_{{3}}c_{{2}}^{4}\\ &-4\,a_{{3}}c_{{2}}^{3}c_{{3}}+6\,a_{{3}}c_{{2}}^{2}c_{{3}}^{2}-4 \,a_{{3}}c_{{2}}c_{{3}}^{3}+a_{{3}}c_{{3}}^{4}+{k}^{2}c_{{2}}^{2}-{k}^{2}c_{{2}}c_{{3}}-a_{{1}}c_{{2}}^{2}+2 \,a_{{1}}c_{{2}}c_{{3}}-a_{{1}}c_{{3}}^{2}\\ &-a_{{4}}c_{{3}}^{2}-a_{{5}}c_{{1}}^{2}-a_{{6}}c_{{1}}c_{{3}}+c_{{2}}^{2}-c_{{2}}c_{{3}}=0, \end{aligned}$$

whose solution yields the following outcomes

$$ c_{{1}}=\pm \,{ \frac { \left ( -a_{{6}}+\sqrt {-4\,a_{{4}}a_{{5}}+a_{{6 }}^{2}} \right ) c_{{3}}}{2a_{{5}}}},~ c_{2}=c_{3}. $$

Then, the exact solutions of Eq. (9) are

$$ F(\xi _{1},\xi _{2},\xi _{3})=\pm A~\text{SN} \left ( -\,{ \frac { c_{{3}}\left ( -a_{{6}}+\sqrt {-4\,a_{{4}}a_{{5}}+a_{{6} }^{2}} \right )}{2a_{{5}}}}\xi _{1}+c_{{3}}\xi _{2}- c_{{3}}\xi _{3},k \right ). $$

Thus, as $k\rightarrow 1$, the exact solutions of the main equations are

$$ u(x,y,z,t)=\pm A~\tanh \left ( -\,{ \frac {z \left ( -a_{{6}}+\sqrt {-4\,a_{ {4}}a_{{5}}+a_{{6}}^{2}} \right ) c_{{3}}}{2a_{{5}}}}+ \left ( t-y \right ) c_{{3}} \right ). $$

Moreover, to obtain periodic dnoidal waves from Eq. (9), by utilizing the Jacobi elliptic function $G(\epsilon )=A~\text{DN}(\epsilon ,k)$ with modulus $0< k<1$ and substituting it into Eq. (10), we have

$$\begin{aligned} &\text{DN}(\epsilon ,k)\text{SN}^{4}(\epsilon ,k):4\,{A}^{2}{k}^{2}a_{{2}}c_{{2}}^{2}-8 \,{A}^{2}{k}^{2}a_{{2}}c_{{2}}c_{{3}}+4\,{A}^{2}{k}^{2}a_{{2}}c_{{3}}^{2}-24 \,{k}^{2}a_{{3}}c_{{2}}^{4}+96\,{k}^{2}a_{{3}}c_{{2}}^{3}c_{{3}} \\ &-144\,{k}^{2}a_{{3}}c_{{2}}^{2}c_{{3}}^{2}+96\,{k}^{2}a_{{3}}c_{{2}}c_{{3}}^{3}-24 \,{k}^{2}a_{{3}}c_{{3}}^{4}=0, \\ & \\ &\text{DN}(\epsilon ,k)\text{SN}^{2}(\epsilon ,k):-3\,{A}^{2}{k}^{2}a_{{2}}c_{{2}}^{2}+6 \,{A}^{2}{k}^{2}a_{{2}}c_{{2}}c_{{3}}-3\,{A}^{2}{k}^{2}a_{{2}}c_{{3}}^{2}+20 \,{k}^{2}a_{{3}}c_{{2}}^{4}\\ &-80\,{k}^{2}a_{{3}}c_{{2}}^{3}c_{{3}}+120\,{k}^{2}a_{{3}}c_{{2}}^{2}c_{{3}}^{2}-80\,{k}^{2}a_{{3}}c_{{2}}c_{{3}}^{3}+20 \,{k}^{2}a_{{3}}c_{{3}}^{4}-2\,{A}^{2}a_{{2}}c_{{2}}^{2}+4\,{A}^{2}a_{{2}}c_{{2}}c_{{3}}\\ &-2 \,{A}^{2}a_{{2}}c_{{3}}^{2}+8\,a_{{3}}c_{{2}}^{4}-32\,a_{{3}}c_{{2}}^{3}c_{{3}}+48\,a_{{3}}c_{{2}}^{2}c_{{3}}^{2}-32 \,a_{{3}}c_{{2}}c_{{3}}^{3}+8\,a_{{3}}c_{{3}}^{4}-2\,a_{{1}}c_{{2}}^{2}+4 \,a_{{1}}c_{{2}}c_{{3}}\\ &-2\,a_{{1}}c_{{3}}^{2}-2\,a_{{4}}c_{{3}}^{2}-2\,a_{{5}}c_{{1}}^{2}-2\,a_{{6}}c_{{1}}c_{{3}}+2\,c_{{2}}^{2}-2\,c_{{2}}c_{{3}}=0, \\ & \\ &\text{DN}(\epsilon ,k):-{k}^{2}a_{{3}}c_{{2}}^{4}+4\,{k}^{2}a_{{3}}c_{{2}}^{3}c_{{3}}-6 \,{k}^{2}a_{{3}}c_{{2}}^{2}c_{{3}}^{2}+4\,{k}^{2}a_{{3}}c_{{2}}c_{{3}}^{3}-{k}^{2}a_{{3}}c_{{3}}^{4}+{A}^{2}a_{{2}}c_{{2}}^{2}+a_{{1}}c_{{3}}^{2} \\ &+{A}^{2}a_{{2}}c_{{3}}^{2}-4\,a_{{3}}c_{{2}}^{4}+16\,a_{{3}}c_{{2}}^{3}c_{{3}}-24 \,a_{{3}}c_{{2}}^{2}c_{{3}}^{2}+16\,a_{{3}}c_{{2}}c_{{3}}^{3}-4\,a_{{3}}c_{{3}}^{4}+a_{{1}}c_{{2}}^{2}-2 \,a_{{1}}c_{{2}}c_{{3}} \\ &-2\,{A}^{2}a_{{2}}c_{{2}}c_{{3}}+a_{{4}}c_{{3}}^{2}+a_{{5}}c_{{1}}^{2}+a_{{6}}c_{{1}}c_{{3}}-c_{{2}}^{2}+c_{{2}}c_{{3}}=0. \end{aligned}$$

With regard to the above system, the coefficients $c_{1}$ and $c_{2}$ are

$$\begin{aligned}& \begin{aligned} &c_{{1}}=\pm \,{ \frac { \left ( -a_{{6}}+\sqrt {-4\,a_{{4}}a_{{5}}+a_{{6 }}^{2}} \right ) c_{{3}}}{2a_{{5}}}}, \\ & c_{2}=c_{3}. \end{aligned} \end{aligned}$$

So, the exact solutions of Eq. (9) are

$$ F(\xi _{1},\xi _{2},\xi _{3})=\pm A~\text{DN} \left ( -\,{ \frac {c_{{3}}\left ( -a_{{6}}+\sqrt {-4\,a_{{4}}a_{{5}}+a_{{6} }^{2}} \right )}{2a_{{5}}}}\xi _{1}+c_{{3}}\xi _{2}-c_{{3}}\xi _{3},k \right ). $$

Then, as $k\rightarrow 1$, the exact solutions of the main equation can be derived as

$$ u(x,y,z,t)=\pm A~\text{sech}\left (- \frac {c_{{3}} \left ( -a_{{6}}+\sqrt {-4\,a_{{4}}a_{{5}}+a_{{6 }}^{2}} \right ) }{2a_{{5}}}z+(t-y)c_{3}\right ). $$

Case 3:

The invariant solution due to the vector field $\mathbf{V}_{7}+\mathbf{V}_{3}$ is

$$ u(x,y,z,t)=F(\xi _{1},\xi _{2},\xi _{3}), $$

where $\xi _{1}=t$, $\xi _{2}=-\,{\dfrac {tya_{{6}}-2\,tza_{{4}}-y}{2a_{{4}}t}}$, and $\xi _{3}=4a_{4}xt+y^{2}$. By substituting it into Eq. (1), the following reduced equation is derived

$$\begin{aligned}& \begin{aligned} &1024a_{3}a_{4}^{5}\xi _{1}^{6}F_{\xi _{3}\xi _{3}\xi _{3}\xi _{3}}+64a_{2}a_{4}^{3} \xi _{1}^{4}F^{2}F_{\xi _{3}\xi _{3}}+128a_{2}a_{4}^{3}\xi _{1}^{4} FF_{ \xi _{3}}^{2}+64a_{1}a_{4}^{3}\xi _{1}^{4}F_{\xi _{3}\xi _{3}} \\ &+16a_{4}^{2}\xi _{1}^{2}\xi _{3}F_{\xi _{3}\xi _{3}}+16a_{4}^{2}\xi _{1}^{3}F_{ \xi _{1}\xi _{3}}+24a_{4}^{2}\xi _{1}^{2}F_{\xi _{3}}+4a_{4}a_{5}\xi _{1}^{2}F_{ \xi _{2}\xi _{2}}-a_{6}^{2}\xi _{1}^{2}F_{\xi _{2}\xi _{2}}+F_{\xi _{2} \xi _{2}}=0. \end{aligned} \end{aligned}$$

According to the above equation, the exact solution of Eq. (1) can be obtained as

$$ u(x,y,z,t)=-\,{ \frac {{\mathit{g}} \left ( t \right ) \left ( a_{{6}}ty-2\,a_{{4 }}tz-y \right ) }{2a_{{4}}t}}+{\mathit{h}} \left ( t \right ), $$

where $g(t)$ and $h(t)$ are arbitrary functions of t.

2.3 Computing nonclassical symmetries to the main equation

To investigate the nonclassical state of the Lie group, in addition to $Pr ^{4} (\mathbf{V}) =0$, we must also satisfy the following invariant condition

$$\begin{aligned} \Omega _{1}&\equiv \xi (x,y,z,t,u)u_{x}+\zeta (x,y,z,t,u)u_{y}+ \varsigma (x,y,z,t,u)u_{z}+\tau (x,y,z,t,u)u_{t}\\ &\quad -\eta (x,y,z,t,u)=0. \end{aligned}$$

By assuming $\xi =1$ and $\zeta =\varsigma =\tau =0$, the invariant surface condition can be obtained as follows

$$\begin{aligned}& \begin{aligned} &u_{x}=\eta , \end{aligned} \end{aligned}$$

then

$$\begin{aligned}& \begin{aligned} u_{xx}&=\eta _{x}+\eta _{u}\eta , \\ u_{xt} &=\eta _{t}+\eta _{u} u_{t}, \\ u_{xxx}&=\eta ^{2}\eta _{uu}+\eta \eta _{u}^{2}+2\eta \eta _{xu}+ \eta _{x}\eta _{u}+\eta _{xx}, \\ u_{xxxx}&=\eta ^{3}\eta _{uuu}+4\eta ^{2}\eta _{u}\eta _{uu}+\eta \eta _{u}^{3}+3\eta ^{2}\eta _{xuu}+5\eta \eta _{u}\eta _{xu} \\ &\,\,\,\,\,+3\eta \eta _{uu}\eta _{x}+\eta _{u}^{2}\eta _{x}+3\eta \eta _{xxu}+\eta _{u}\eta _{xx}+3\eta _{x}\eta _{xu}+\eta _{xxx}. \end{aligned} \end{aligned}$$

Substituting the above relations into Eq. (5) and collecting the terms with same power yields $\eta =0$, we have the infinitesimal symmetry ${W}_{1}=\dfrac{\partial}{\partial x}$. This vector field is derived from the classical mode. In the second case, when $\xi =1$ and $\zeta =\varsigma =\tau \neq 0$, three infinitesimal symmetries are examined. For the first case, the vector field is introduced as

$$\begin{aligned}& \begin{aligned} &{W}_{2}=\dfrac{\partial}{\partial x}+z\dfrac{\partial}{\partial y}+z \dfrac{\partial}{\partial z}+\dfrac{\partial}{\partial t}, \end{aligned} \end{aligned}$$

which generates the invariant solution of Eq. (1) as $u ( x,y,z,t ) ={F} (\xi _{1},\xi _{2},\xi _{3})$, where $\xi _{1}=z{{\mathrm{e}}^{-x}}$, $\xi _{2}=t-x$, and $\xi _{3}=y-z$. Now, the reduced system related to the main equation is

$$\begin{aligned} \begin{aligned} &a_{{5}}\xi _{1}^{2}F_{\xi _{1}\xi _{1}}=0, \\ &a_{{6}}\xi _{1}F_{\xi _{1}\xi _{3}} -2a_{{5}}\xi _{1}F_{\xi _{1}\xi _{3}}=0 , \\ &4a_{2}\xi _{1}FF_{\xi _{1}}F_{\xi _{2}}+a_{3}\xi _{1}^{4}F_{\xi _{1} \xi _{1}\xi _{1}\xi _{1}}+a_{2}F^{2}F_{\xi _{2}\xi _{2}}+2a_{1}\xi _{1}F_{ \xi _{1}\xi _{2}}+12a_{3}\xi _{1}^{2}F_{\xi _{1}\xi _{1}\xi _{2}}+4a_{3} \xi _{1}F_{\xi _{1}\xi _{2}\xi _{2}\xi _{2}} \\ &+6a_{3}\xi _{1}F_{\xi _{1}\xi _{2}\xi _{2}}+4a_{3}\xi _{1}F_{\xi _{1} \xi _{2}}+7a_{3}\xi _{1}^{2}F_{\xi _{1}\xi _{1}}+4a_{3}\xi _{1}^{3}F_{ \xi _{1}\xi _{1}\xi _{1}\xi _{2}}+6a_{3}\xi _{1}^{3}F_{\xi _{1}\xi _{1} \xi _{1}}\\ &+6a_{3}\xi _{1}^{2}F_{\xi _{1}\xi _{1}\xi _{2}\xi _{2}}+a_{1}\xi _{1}^{2}F_{\xi _{1}\xi _{1}}-F_{\xi _{2}\xi _{2}}+a_{1}F_{ \xi _{2}\xi _{2}}+a_{3}F_{\xi _{2}\xi _{2}\xi _{2}\xi _{2}}+a_{5}F_{ \xi _{3}\xi _{3}}+a_{4}F_{\xi _{3}\xi _{3}}-\xi _{1}F_{\xi _{1}\xi _{2}}\\ &-a_{6}F_{ \xi _{3}\xi _{3}}+a_{2}\xi _{1}F^{2}F_{\xi _{1}} +2a_{2} \xi _{1}^{2}FF_{\xi _{1}}^{2}+2a_{2} \xi _{1}F^{2}F_{\xi _{1}\xi _{2}}+a_{2}\xi _{1}^{2}F^{2}F_{\xi _{1} \xi _{1}}+2a_{2}FF_{\xi _{2}}^{2} \\ &+a_{3}\xi _{1}F_{\xi _{1}}+a_{1}\xi _{1}F_{\xi _{1}}=0. \end{aligned} \end{aligned}$$

(11)

In a similar manner to Case 1 and Case 2 of classical symmetries, let $a_{1}=a_{3}$ and

$$ F(\xi _{1},\xi _{2},\xi _{3})=G(\epsilon), $$

where $\epsilon =c_{1}\xi _{1}-c_{2}\xi _{2}+c_{3}\xi _{3}$. Due to this transformation, Eq. (11) reduces to the following ODE system

$$\begin{aligned} &a_{{5}}\xi _{1}^{2}c_{{1}}^{2} G_{\epsilon \epsilon}=0, \\ &\left ( -2\,\xi _{1} a_{{5}}c_{{1}}c_{{3}}+\xi _{1} a_{{6}}c_{{1}}c_{{3}} \right ) G_{\epsilon \epsilon}=0, \end{aligned}$$

(12)

$$\begin{aligned} &\left ( \xi _{1}^{4}a_{{3}}c_{{1}}^{4}-4\xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}+6 \xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2}-4\,\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3}+a_{{3}}c_{{2}}^{4} \right ) G_{\epsilon \epsilon \epsilon \epsilon} \\ &+ \left ( 6\xi _{1}^{3}a_{{3}}c_{{1}}^{3}-12\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}+6 \xi _{1} a_{{3}}c_{{1}}c_{{2}}^{2} \right ) G_{ \epsilon \epsilon \epsilon}+ \left ( \xi _{1}^{2}a_{{2}}c_{{1}}^{2}-2\xi _{1} a_{{2}}c_{{1}}c_{{2}}+a_{{2}}c_{{2}}^{2} \right ) G_{\epsilon \epsilon} G^{2} \\ &+ \big( \xi _{1}^{2}a_{{1}}c_{{1}}^{2}+7\xi _{1}^{2}a_{{3}}c_{{1}}^{2}-2 \xi _{1} a_{{1}}c_{{1}}c_{{2}}-4\xi _{1} a_{{3}}c_{{1}}c_{{2}}+ \xi _{1} c_{{1}}c_{{2}}+a_{{1}}c_{{2}}^{2}+a_{{4}}c_{{3}}^{2}+a_{{5}}c_{{3}}^{2}\\ &-a_{{6}}c_{{3}}^{2}-c_{{2}}^{2} \big)G _{\epsilon \epsilon}+\left ( 2\xi _{1}^{2}a_{{2}}c_{{1}}^{2}-4\xi _{1} a_{{2}}c_{{1}}c_{{2}}+2a_{{2}}c_{{2}}^{2} \right ) G_{\epsilon}^{2}G +a_{{2}}c_{{1}}\xi _{1} G^{2}G_{\epsilon} \\ &+\left ( \xi _{1} a_{{1}}c_{{1}}+\xi _{1} a_{{3}}c_{{1}} \right )G_{ \epsilon} =0. \end{aligned}$$

(13)

To derive periodic snoidal waves, substituting the following Jacobi elliptic function

$$ G(\epsilon )=A~\text{SN}(\epsilon ,k),~~0< k< 1, $$

into Eq. (12) gives the following system

$$\begin{aligned} &\text{SN}^{5}(\epsilon ,k):24\,{k}^{4}\xi _{1}^{4}a_{{3}}c_{{1}}^{4}-96 \,{k}^{4}\xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}+144\,{k}^{4}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2}-96 \,{k}^{4}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3}\\ &+4\,{A}^{2}{k}^{2}\xi _{1}^{2}a_{{2}}c_{{1}}^{2}+24\,{k}^{4}a_{{3}}c_{{2}}^{4}-8\,{A}^{2}{k}^{2}\xi _{1} a_{{2}}c_{{1}}c_{{2}}+4{A}^{2}{k}^{2}a_{{2}}c_{{2}}^{2}=0, \\ & \\ &\text{SN}^{3}(\epsilon ,k): -20\,{k}^{4}\xi _{1}^{4}a_{{3}}c_{{1}}^{4}+80 \,{k}^{4}\xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}-120\,{k}^{4}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2}-20 \,{k}^{2}\xi _{1}^{4}a_{{3}}c_{{1}}^{4}\\ &+80\,{k}^{4}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3}+80\,{k}^{2}\xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}-3\,{A}^{2}{k}^{2} \xi _{1}^{2}a_{{2}}c_{{1}}^{2}-20\,{k}^{4}a_{{3}}c_{{2}}^{4}-120\,{k}^{2} \xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2}\\ &+2\,{k}^{2}\xi _{1}^{2}a_{{1}}c_{{1}}^{2}+2 \,{k}^{2}a_{{1}}c_{{2}}^{2}-20\,{k}^{2}a_{{3}}c_{{2}}^{4}+6\,{A}^{2}\xi _{1} a_{{2}}c_{{1}}c_{{2}}-4 \,{k}^{2}\xi _{1} a_{{1}}c_{{1}}c_{{2}}-8\,{k}^{2}\xi _{1} a_{{3}}c_{{1}} c_{2}\\ &-3\,{A}^{2}a_{{2}}c_{{2}}^{2}+2\,{k}^{2}\xi _{1} c_{{1}}c_{{2}}-2 \,{k}^{2}c_{{2}}^{2}+2\,{k}^{2}a_{{4}}c_{{3}}^{2}+2\,{k}^{2} a_{{5}}c_{{3}}^{2}+6\,{A}^{2}{k}^{2} \xi _{1} a_{{2}}c_{{1}}c_{{2}} \\ &+80\,{k}^{2}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3}-3 \,{A}^{2}{k}^{2}a_{{2}}c_{{2}}^{2}-3\,{A}^{2}\xi _{1}^{2}a_{{2}}c_{{1}}^{2}-2\,{k}^{2}a_{{6}}c_{{3}}^{2}+14\,{k}^{2}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}=0,\\ &k^{2}\xi _{1}^{2}a_{5}c_{1}^{2}=0,~-\xi _{1} c_{{1}}c_{{3}} \left ( 4 \,{k}^{2}a_{{5}}-2\,{k}^{2}a_{{6}} \right )=0, \end{aligned}$$

$$\begin{aligned} &\text{SN}(\epsilon ,k):{k}^{4}\xi _{1}^{4}a_{{3}}c_{{1}}^{4}-4\,{k}^{4} \xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}+6\,{k}^{4}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2}+14 \,{k}^{2}\xi _{1}^{4}a_{{3}}c_{{1}}^{4}-4\,{k}^{4}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3}\\ &+2 \,{k}^{2}\xi _{1} a_{{1}}c_{{1}}c_{{2}}-56\,{k}^{2}\xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}+{k}^{4}a_{{3}}c_{{2}}^{4}+84 \,{k}^{2}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2}+\xi _{1}^{4}a_{{3}}c_{{1}}^{4}-56 \,{k}^{2}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3} \\ &-4\,\xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}-{k}^{2}a_{{5}}c_{{3}}^{2}+2\,{A}^{2}\xi _{1}^{2}a_{{2}}c_{{1}}^{2}-{k}^{2}\xi _{1}^{2}a_{{1}}c_{{1}}^{2}-7 \,{k}^{2}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}+14\,{k}^{2}a_{{3}}c_{{2}}^{4}+6 \,\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2} \\ &-4\,{A}^{2}\xi _{1} a_{{2}}c_{{1}}c_{{2}}-{k}^{2}a_{{4}}c_{{3}}^{2}+4\,{k}^{2}\xi _{1} a_{{3}}c_{{1}}c_{{2}}-4\,\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3}+2 \,{A}^{2}a_{{2}}c_{{2}}^{2}-{k}^{2}\xi _{1} c_{{1}}c_{{2}}-{k}^{2}a_{{1}}c_{{2}}^{2} \\ &+{k}^{2}a_{{6}}c_{{3}}^{2}-\xi _{1}^{2}a_{{1}}c_{{1}}^{2}-7\,\xi _{1}^{2}a_{{3}}c_{{1}}^{2} +{k}^{2}c_{{2}}^{2}+2\,\xi _{1} a_{{1}}c_{{1}}c_{{2}}+4\,\xi _{1} a_{{3}}c_{{1}}c_{{2}}- \xi _{1} c_{{1}}c_{{2}}-a_{{1}}c_{{2}}^{2}-a_{{4}}c_{{3}}^{2} \\ &-a_{{5}}c_{{3}}^{2}+a_{{3}}c_{{2}}^{4}+a_{{6}}c_{{3}}^{2}+c_{{2}}^{2}=0,\\ & -a_{{5}}\xi _{1}^{2}c_{{1}}^{2} \left ({k}^{2}+1 \right ) =0,~ \xi _{1} c_{{1}}c_{{3}} \left ( a_{{6}}-2\,{k}^{2}a_{{5}}+{k}^{2}a_{{6}}-2\,a_{{5}} \right )=0, \\ & \\ &\text{DN}(\epsilon ,k)\text{CN}(\epsilon ,k): -6\,{k}^{2}\xi _{1}^{3}a_{{3}}c_{{1}}^{3}+12 \,{k}^{2}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}-6\,{k}^{2}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{2}+12 \,\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}\\ &+\xi _{1} a_{{3}}c_{{1}}+\xi _{1} a_{{1}}c_{{1}}-6\,\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{2}-6\,\xi _{1}^{3}a_{{3}}c_{{1}}^{3}=0, \\ & \\ &\text{SN}(\epsilon ,k)^{2}\text{DN}(\epsilon ,k)\text{CN}(\epsilon ,k): 36 \,{k}^{2}\xi _{1}^{3}a_{{3}}c_{{1}}^{3}-72\,{k}^{2}\xi _{1}^{2}a_{{3}} c_{{1}}^{2}c_{{2}}+36\,{k}^{2}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{2}\\ &+{A}^{2} \xi _{1} a_{{2}}c_{{1}}=0. \end{aligned}$$

Solving this system yields

$$\begin{aligned}& \begin{aligned} &A=\pm \sqrt {-6\,{\frac {a_{{3}}}{a_{{2}}}}}c_{{2}}k, \\ &c_{1}=0, \\ &c_{3}=\pm \sqrt {-{ \frac {-{k}^{2}a_{{3}}c_{{2}}^{2}-a_{{3}}c_{{2}}^{2}+a_{{ 1}}-1}{a_{{4}}+a_{{5}}-a_{{6}}}}}c_{{2}}. \end{aligned} \end{aligned}$$

So, the exact solutions of Eq. (11) are

$$ F(\xi _{1},\xi _{2},\xi _{3})=\pm \sqrt {-6\,{ \frac {a_{{3}}}{a_{{2}}}}}c_{{2}}k~{\text{SN} } \left ( c_{{2}}\xi _{2}-c_{{2}} \sqrt {-{ \frac {-{k}^{2}a_{{3}}c_{{2}}^{2}-a_{{3 }}c_{{2}}^{2}+a_{{1}}-1}{a_{{4}}+a_{{5}}-a_{{6}}}}}\xi _{3},k\right ). $$

Then, as $k\rightarrow 1$, the exact solutions of main equation (1) can be obtained as

$$\begin{aligned}& \begin{aligned} u(x,y,z,t)=\pm \sqrt {-6{\frac {a_{{3}}}{a_{{2}}}}}c_{{2}}~{\tanh} \left ( \left ( t-x \right ) c_{{2}}- \left ( y-z \right ) \sqrt {-{ \frac {a_{{ 3}}c_{{2}}^{2}+a_{{3}}-1}{a_{{4}}+a_{{5}}-a_{{6}}}}}c_{{2}}\right ). \end{aligned} \end{aligned}$$

In addition, to obtain periodic dnoidal waves, we utilize the Jacobi elliptic function

$$ G(\epsilon )=A~\text{DN}(\epsilon ,k), ~~0< k< 1, $$

and substitute in Eq. (12) to get the following system

$$\begin{aligned} &\text{DN}(\epsilon ,k)\text{SN}^{4}(\epsilon ,k):-24\,{k}^{2}\xi _{1}^{4}a_{{3}}c_{{1}}^{4}+96 \,{k}^{2}\xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}+4\,{A}^{2}{k}^{2}\xi _{1}^{2}a_{{2}}c_{{1}}^{2}-144 \,{k}^{2}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2}\\ &-24\,{k}^{2}a_{{3}}c_{{2}}^{4}+96\,{k}^{2}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3}+4\,{A}^{2}{k}^{2}a_{{2}}c_{{2}}^{2}-8 \,{A}^{2}{k}^{2}\xi _{1} a_{{2}}c_{{1}}c_{{2}}=0, \\ & \\ &\text{CN}(\epsilon ,k)\text{SN}^{3}(\epsilon ,k):36\,{k}^{2}\xi _{1}^{3}a_{{3}}c_{{1}}^{3}-72 \,{k}^{2}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}-{A}^{2}{k}^{2}\xi _{1} a_{{2}}c_{{1}}+36 \,{k}^{2}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{2}=0, \\ & \\ &\text{DN}(\epsilon ,k)\text{SN}^{2}(\epsilon ,k):20\,{k}^{2}\xi _{1}^{4}a_{{3}}c_{{1}}^{4}-80 \,{k}^{2}\xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}-3\,{A}^{2}{k}^{2}\xi _{1}^{2}a_{{2}}c_{{1}}^{2}+120 \,{k}^{2}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2}\\ &+8\,\xi _{1}^{4}a_{{3}}c_{{1}}^{4}+6\,{A}^{2}{k}^{2}\xi _{1} a_{{2}}c_{{1}}c_{{2}}-80\,{k}^{2}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3}-32 \,\xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}-3\,{A}^{2}{k}^{2}a_{{2}}c_{{2}}^{2}-2 \,{A}^{2}\xi _{1}^{2}a_{{2}}c_{{1}}^{2} \\ &+20\,{k}^{2}a_{{3}}c_{{2}}^{4}+2 \,c_{{2}}^{2}+4\,{A}^{2}\xi _{1} a_{{2}}c_{{1}}c_{{2}}-32\,\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3}-2 \,{A}^{2}a_{{2}}c_{{2}}^{2}-2\,\xi _{1}^{2}a_{{1}}c_{{1}}^{2}-14\, \xi _{1}^{2}a_{{3}}c_{{1}}^{2}\\ &+8\,a_{{3}}c_{{2}}^{4}+4\,\xi _{1} a_{{1}}c_{{1}}c_{{2}}-2 \,a_{{1}}c_{{2}}^{2}+48\,\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2}-2\,a_{{4}}c_{{3}}^{2}-2 \,a_{{5}}c_{{3}}^{2}+2\,a_{{6}}c_{{3}}^{2}+8\,\xi _{1} a_{{3}}c_{{1}}c_{{2}} \\ &-2 \,\xi _{1} c_{{1}}c_{{2}}=0,~{k}^{2}\xi _{1}^{2}a_{{5}}c_{{1}}^{2}=0,~\xi _{1} c_{{1}}c_{{3}}{k}^{2} \left ( 4\,a_{{5}}-2\,a_{{6}} \right ) =0, \\ & \\ &\text{CN}(\epsilon ,k)\text{SN}(\epsilon ,k):-6\,{k}^{2}\xi _{1}^{3}a_{{3}}c_{{1}}^{3}+12 \,{k}^{2}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}-6\,{k}^{2}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{2}-24 \,\xi _{1}^{3}a_{{3}}c_{{1}}^{3}\\ &+48\,\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}-24\,\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{2}+\xi _{1} a_{{1}}c_{{1}}+\xi _{1} a_{{3}}c_{{1}}+{A}^{2}\xi _{1} a_{{2}}c_{{1}}=0, \end{aligned}$$

$$\begin{aligned}& \begin{aligned} &\text{DN}(\epsilon ,k):-{k}^{2}\xi _{1}^{4}a_{{3}}c_{{1}}^{4}+4\,{k}^{2} \xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}-6\,{k}^{2}\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2} -4\,\xi _{1}^{4}a_{{3}}c_{{1}}^{4}+4\,{k}^{2}\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3}\\ &+16 \,\xi _{1}^{3}a_{{3}}c_{{1}}^{3}c_{{2}}+{A}^{2}\xi _{1}^{2}a_{{2}}c_{{1}}^{2}-{k}^{2}a_{{3}}c_{{2}}^{4}-24 \,\xi _{1}^{2}a_{{3}}c_{{1}}^{2}c_{{2}}^{2}-2\,{A}^{2}\xi _{1} a_{{2}}c_{{1}}c_{{2}}+16 \,\xi _{1} a_{{3}}c_{{1}}c_{{2}}^{3} \\ &+{A}^{2}a_{{2}}c_{{2}}^{2}+\xi _{1}^{2}a_{{1}}c_{{1}}^{2}+\xi _{1} c_{{1}}c_{{2}}+7\,\xi _{1}^{2}a_{{3}}c_{{1}}^{2}-4\,a_{{3}}c_{{2}}^{4}-2\,\xi _{1} a_{{1}}c_{{1}}c_{{2}}-4 \,\xi _{1} a_{{3}}c_{{1}}c_{{2}}+a_{{1}}c_{{2}}^{2}+a_{{4}}c_{{3}}^{2}\\ &+a_{{5}}c_{{3}}^{2}-a_{{6}}c_{{3}}^{2}-c_{{2}}^{2}=0,~{k}^{2}\xi _{1}^{2}a_{{5}}c_{{1}}^{2}=0,~\xi _{1} c_{{1}}c_{{3}}{k}^{2} \left ( a_{{6}}-2\,a_{{5}} \right ) =0. \end{aligned} \end{aligned}$$

With regard to the above relations, the following coefficients are obtained

$$\begin{aligned}& \begin{aligned} &A=\pm \sqrt {6\,{\frac {a_{{3}}}{a_{{2}}}}}c_{{2}}k, \\ &c_{1}=0,~c_{3}=\pm \sqrt {-{ \frac {-{k}^{2}a_{{3}}c_{{2}}^{2}+2\,a_{{3}}c_{{2}}^{2}+a _{{1}}-1}{a_{{4}}+a_{{5}}-a_{{6}}}}}c_{{2}}. \end{aligned} \end{aligned}$$

Thus, the exact solutions of Eq. (11) are

$$ F(\xi _{1},\xi _{2},\xi _{3})=\pm \sqrt {6\,{\frac {a_{{3}}}{a_{{2}}}}}c_{{2}}k~ \text{DN} \left ( c_{{2}}\xi _{2}+c_{{2} }\sqrt {-{ \frac {-{k}^{2}a_{{3}}c_{{2}}^{2}+2\,a_{{3}}c_{{2}}^{2}+a_{{1}}-1}{a_{{4}}+a_{{5}}-a_{{6}}}}} \xi _{3},k \right ). $$

Then, as $k\rightarrow 1$, the exact solutions of the main equation can be derived as

$$\begin{aligned}& \begin{aligned} u(x,y,z,t)=\pm \sqrt {6\,{\frac {a_{{3}}}{a_{{2}}}}}c_{{2}}~{\mathrm{sech}} \left ( \left ( t-x \right ) c_{{2}}+ \left ( y-z \right ) \sqrt {-{ \frac {a_{{3}}c_{{2}}^{2}+a_{{3}}-1}{a_{{4}}+a_{{5}}-a_{{6}}}}}c_{{2}} \right ). \end{aligned} \end{aligned}$$

Furthermore, we have two other vector fields as

$$\begin{aligned}& \begin{aligned} &{W}_{3}=\dfrac{\partial}{\partial x}+z\dfrac{\partial}{\partial y}+y \dfrac{\partial}{\partial z}+t\dfrac{\partial}{\partial t}, \\ &{W}_{4}=\dfrac{\partial}{\partial x}+z\dfrac{\partial}{\partial y}+y \dfrac{\partial}{\partial z}+\dfrac{\partial}{\partial t}+ \dfrac{\partial}{\partial u}, \end{aligned} \end{aligned}$$

where extracting their exact solutions gives

$$\begin{aligned}& \begin{aligned} &u(x,y,z,t)={ \frac {\sqrt {3}\sqrt {-a_{{2}} \left ( a_{{1}}-1 \right ) }}{a_{{2}}} \tanh \left ( \,{ \frac {\sqrt {2}\sqrt {a_{{3}} \left ( a_{{1}}-1 \right ) }(t-x)+ 2c_{1}\,a_{{3}}}{2a_{{3}}}} \right ) } , \\ &u(x,y,z,t)=-c_{1}{{\mathrm{e}}^{-t}}\, \left ( y+z \right ) +c_{2}{{\mathrm{e}}^{t}} \, \left ( z-y \right ) +t+c_{3}, \end{aligned} \end{aligned}$$

respectively, where $c_{1}$, $c_{2}$, and $c_{3}$ are constants.

3 3D-mNLW equation: its bifurcation analysis and Jacobi elliptic function solutions

Serving the transformation $u(x,y,z,t)=U(\epsilon )$, $\epsilon =x+y+z-\omega t$, where ω is the speed of the wave, the governing Eq. (1) changes into

$$ a_{3}U_{\epsilon \epsilon \epsilon \epsilon}+\left (a_{1}+a_{4}+a_{5}+a_{6}- \omega \right )U_{\epsilon \epsilon}+a_{2}U^{2}U_{\epsilon \epsilon}+2a_{2}UU_{ \epsilon}^{2}=0. $$

(14)

Integrating Eq. (14) twice with respect to ϵ leads to the following well-known ODE

$$ a_{3}U_{\epsilon \epsilon}+\left (a_{1}+a_{4}+a_{5}+a_{6}-\omega \right )U+\frac{1}{3}a_{2}U^{3}=0. $$

Here, the Galilean transformation can be applied to transform the aforementioned equation into the following dynamical system

$$\begin{aligned} \textstyle\begin{cases} & \frac{dU(\varepsilon )}{d\varepsilon }=\Omega (U,y)=y(\epsilon ), \\ & \\ & \frac{dy(\varepsilon )}{d\varepsilon }=\Psi (U,y)=-{{\phi }_{1}}{{U}^{3}}( \epsilon )+\phi _{2}U(\epsilon ), \end{cases}\displaystyle \end{aligned}$$

(15)

where $\phi _{1}=\,{\dfrac {a_{{2}}}{3a_{{3}}}}$ and $\phi _{2}={\dfrac {\omega -a_{{1}}-a_{{4}}-a_{{5}}-a_{{6}}}{a_{{3}}}}$.

3.1 Bifurcation analysis

The bifurcation analysis of the dynamical system (15) is presented herein. The Hamiltonian function can be shown as

$$ H(U,y)=\dfrac{1}{2}y^{2}+\dfrac{1}{4}\phi _{1}U^{4}(\epsilon )- \dfrac{1}{2}\phi _{2}U^{2}(\epsilon )=h, $$

where h represents the Hamiltonian constant. It is not hard to demonstrate that the equilibrium points are

$$ E_{1}=(0,0),\,\,\, E_{2}=\left (\sqrt{\dfrac{\phi _{2}}{\phi _{1}}},0 \right ),\,\,\, E_{3}=\left (-\sqrt{\dfrac{\phi _{2}}{\phi _{1}}},0 \right ). $$

For the Jacobian matrix of system (15), the determinant is

$$D(U,y)=\left|\begin{array}{cc}0& 1\\ -3{\varphi }_1{U}^2+{\varphi }_2& 0\end{array}\right|=3{\varphi }_1{U}^2-{\varphi }_2.$$

If $D(E_{j})$ is negative, then $E_{j}$ is called a saddle point; if $D(E_{j})$ is positive, then $E_{j}$ is referred to as a center point; and if $D(E_{j})$ equals zero, then $E_{j}$ is named a cuspidal point. Varying the parameters yields the following potential outcomes.

Case 1: $\big( \phi _{1}>0 \text{ and } \phi _{2}>0 \big)$

By considering $a_{1}=1$, $a_{2}=3$, $a_{3}=1$, $a_{4}=1$, $a_{5}=1$, $a_{6}=1$, and $\omega =5$, the three equilibrium points are $E_{1}=(0,0)$, $E_{2}=(1,0)$, and $E_{3}=(-1,0)$. $E_{1}$ is a saddle point because $D(E_{1})<0$, while $E_{2}$ and $E_{3}$ are center points since $D(E_{2})>0$ and $D(E_{3})>0$.

Case 2: $\big( \phi _{1}<0 \text{ and } \phi _{2}>0 \big)$

By considering $a_{1}=1$, $a_{2}=-3$, $a_{3}=1$, $a_{4}=1$, $a_{5}=1$, $a_{6}=1$, and $\omega =5$, the equilibrium point is $E_{1}=(0,0)$. $E_{1}$ is a saddle point because $D(E_{1})<0$.

Case 3: $\big( \phi _{1}>0 \text{ and } \phi _{2}<0 \big)$

By considering $a_{1}=1$, $a_{2}=3$, $a_{3}=1$, $a_{4}=-1$, $a_{5}=1$, $a_{6}=-1$, and $\omega =-1$, the equilibrium point is $E_{1}=(0,0)$. $E_{1}$ is a center point as $D(E_{1})>0$.

Case 4: $\big( \phi _{1}<0 \text{ and } \phi _{2}<0 \big)$

By considering $a_{1}=1$, $a_{2}=-3$, $a_{3}=1$, $a_{4}=-1$, $a_{5}=1$, $a_{6}=-1$, and $\omega =-1$, the three equilibrium points are $E_{1}=(0,0)$, $E_{2}=(1,0)$, and $E_{3}=(-1,0)$. $E_{1}$ is a center point because $D(E_{1})>0$, while $E_{2}$ and $E_{3}$ are saddle points since $D(E_{2})<0$ and $D(E_{3})<0$.

3.2 Jacobi elliptic function solutions

In the current section, the authors employ the dynamical system method to construct exact solutions of the 3D-mNLW equation in terms of Jacobi elliptic functions. To this end, suppose that

$$ h^{*}=|H(E_{3},y)|=|H(E_{2},y)|=\dfrac{\phi _{2}^{2}}{4|\phi _{1} |}. $$

Case 1: $\big( \phi _{1}>0 \text{ and } \phi _{2}>0 \big)$

I. Assume that $-h^{*}< h<0$. Consequently, we derive the following Jacobi elliptic solutions to the 3D-mNLW equation

$$ u_{1,2}(x,y,z,t)=\pm \delta _{2}~\text{DN}\left (\delta _{2}\sqrt{ \dfrac{\phi _{1}}{2}}(x+y+z-\omega t-\epsilon _{0}), \dfrac{\sqrt{\delta _{2}^{2}-\delta _{1}^{2}}}{\delta _{2}}\right ), $$

where

$$ \delta _{1,2}^{2}= \dfrac{\phi _{2}\mp \sqrt{\phi _{2}^{2}+4h\phi _{1}}}{\phi _{1}},\,\, \, \phi _{1}=\dfrac{a_{2}}{3a_{3}},\,\,\, \phi _{2}={ \dfrac {\omega -a_{{1}}-a_{{4}}-a_{{5}}-a_{{6}}}{a_{{3}}}}. $$

II. Let $h=0$, then

$$ \delta _{1}^{2}=0,~ \delta _{2}^{2}=\dfrac{2\phi _{2}}{\phi _{1}}. $$

So, we can construct the following bright solitary wave solutions to the 3D-mNLW equation

$$ u_{3,4}(x,y,z,t)=\pm \sqrt{\dfrac{2\phi _{2}}{\phi _{1}}} ~ \text{sech}\left (\sqrt{\phi _{2}}(x+y+z-\omega t-\epsilon _{0}) \right ), $$

where

$$\begin{aligned}& \begin{aligned} & \phi _{1}=\dfrac{a_{2}}{3a_{3}},~ \phi _{2}={ \dfrac {\omega -a_{{1}}-a_{{4}}-a_{{5}}-a_{{6}}}{a_{{3}}}}. \end{aligned} \end{aligned}$$

III. Assume that $0< h<\infty$. Therefore, we can extract Jacobi elliptic solutions to the 3D-mNLW equation as follows

$$ u_{5,6}(x,y,z,t)=\pm \delta _{4}~\text{CN}\left (\sqrt{ \dfrac{\phi _{1}(\delta _{3}^{2}+\delta _{4}^{2})}{2}}(x+y+z-\omega t- \epsilon _{0}), \dfrac{\delta _{4}}{\sqrt{\delta _{3}^{2}+\delta _{4}^{2}}}\right ), $$

where

$$\begin{aligned}& \begin{aligned} \delta _{3,4}^{2}= \dfrac{\mp \phi _{2}+\sqrt{\phi _{2}^{2}+4h\phi _{1}}}{\phi _{1}},~ \phi _{1}=\dfrac{a_{2}}{3a_{3}}, ~\phi _{2}={ \dfrac {\omega -a_{{1}}-a_{{4}}-a_{{5}}-a_{{6}}}{a_{{3}}}}. \end{aligned} \end{aligned}$$

Case 2: $\big( \phi _{1}<0 \text{ and } \phi _{2}<0 \big)$

I. Assume that $0< h< h^{*}$. Therefore, we can establish the following Jacobi elliptic solutions to the 3D-mNLW equation

$$ u_{7,8}(x,y,z,t)=\pm \delta _{1}~\text{SN}\left (\delta _{2}\sqrt{- \dfrac{\phi _{1}}{2}}(x+y+z-\omega t-\epsilon _{0}), \dfrac{\delta _{1}}{\delta _{2}}\right ), $$

where

$$\begin{aligned}& \begin{aligned} \phi _{1}=\dfrac{a_{2}}{3a_{3}}, ~\phi _{2}={ \dfrac {\omega -a_{{1}}-a_{{4}}-a_{{5}}-a_{{6}}}{a_{{3}}}}. \end{aligned} \end{aligned}$$

II. Let $h=-h^{*}$. Then $\delta _{1,2}^{2}=\dfrac{\phi _{2}}{\phi _{1}}$ and the following kink solutions to the 3D-mNLW equation are constructed

$$ u_{9,10}(x,y,z,t)=\pm \sqrt{\dfrac{\phi _{2}}{\phi _{1}}}~\text{tanh} \left (\sqrt{\dfrac{-\phi _{2}}{2}}(x+y+z-\omega t-\epsilon _{0}) \right ), $$

where

$$\begin{aligned}& \begin{aligned} \phi _{1}=\dfrac{a_{2}}{3a_{3}},~\phi _{2}={ \dfrac {\omega -a_{{1}}-a_{{4}}-a_{{5}}-a_{{6}}}{a_{{3}}}}. \end{aligned} \end{aligned}$$

4 Results and discussion

In the study of dynamical systems and wave propagation, visualizing the behavior of systems through plots and phase portraits is a critical tool for understanding complex phenomena. This section explores a series of plots that illustrate the dynamical behavior of specific systems under various conditions. Additionally, the changes in the nonlinear coefficient of the equation are investigated in detail. All graphs in this study have been plotted using Maple software. The topics covered include:

4.1 Figure 1:

Figure 1 signifies the plots related to the dynamical system (15) and its phase portraits when (a) $\phi _{1}$ and $\phi _{2}$ are positive, (b) $\phi _{1}$ is negative and $\phi _{2}$ is positive, (c) $\phi _{1}$ is positive and $\phi _{2}$ is negative, (d) $\phi _{1}$ and $\phi _{2}$ are negative. These scenarios include Cases (a) and (d), which show the existence of both hyperbolic and periodic solutions, while Cases (b) and (c) represent the existence of hyperbolic and periodic solutions, respectively.

Figure 1

The plots related to the dynamical system (15) and its phase portraits when (a) $\phi _{1}$ and $\phi _{2}$ are positive, (b) $\phi _{1}$ is negative, and $\phi _{2}$ is positive, (c) $\phi _{1}$ is positive, and $\phi _{2}$ is negative, (d) $\phi _{1}$ and $\phi _{2}$ are negative

4.2 Figure 2:

Figure 2 represents the plots of $u_{1} (x,y,z,t)$ for $a_{1}=1$, $a_{3}=0.5$, $a_{4}=1$, $a_{5}=0.5$, $a_{6}=1$, $\omega =5$, $\epsilon _{0}=0$, $y=1$, and $t=1$ when (a) $a_{2}=1$; (b) $a_{2}=2$; (c) $x=1$, $a_{2}=1$ and $x=1$, $a_{2}=2$. By analyzing Fig. 2, it can be concluded that the height and width of the wave decrease as the nonlinear parameter increases from 1 to 2.

Figure 2

The plots of $u_{1} (x,y,z,t)$ for $a_{1}=1$, $a_{3}=0.5$, $a_{4}=1$, $a_{5}=0.5$, $a_{6}=1$, $\omega =5$, $\epsilon _{0}=0$, $y=1$, and $t=1$ when (a) $a_{2}=1$; (b) $a_{2}=2$; (c) $x=1$, $a_{2}=1$ and $x=1$, $a_{2}=2$

4.3 Figure 3:

Similarly, the behavior of the $u_{3}(x,y,z,t)$ under the same parameter settings as above is explored. The analysis focuses on how variations in $a_{2}$ influence the height and width of the bright soliton wave, enhancing our understanding of how the nonlinear parameter influences wave dynamics. As can be seen, the height and width of the wave decrease with the increase of the parameter $a_{2}$.

4.4 Figure 4:

Figure 4 shows the plots of $u_{5} (x,y,z,t)$ for $a_{1}=1$, $a_{3}=0.5$, $a_{4}=1$, $a_{5}=0.5$, $a_{6}=1$, $\omega =5$, $\epsilon _{0}=0$, $y=1$, and $t=1$ when (a) $a_{2}=1$; (b) $a_{2}=2$; (c) $x=1$, $a_{2}=1$ and $x=1$, $a_{2}=2$. By analyzing Fig. 4, it can be concluded that the height and width of the wave decrease as the nonlinear parameter increases from 1 to 2.

Figure 3

The plots of $u_{3} (x,y,z,t)$ for $a_{1}=1$, $a_{3}=0.5$, $a_{4}=1$, $a_{5}=0.5$, $a_{6}=1$, $\omega =5$, $\epsilon _{0}=0$, $y=1$, and $t=1$ when (a) $a_{2}=1$; (b) $a_{2}=2$; (c) $x=1$, $a_{2}=1$ and $x=1$, $a_{2}=2$

Figure 4

The plots of $u_{5} (x,y,z,t)$ for $a_{1}=1$, $a_{3}=0.5$, $a_{4}=1$, $a_{5}=0.5$, $a_{6}=1$, $\omega =5$, $\epsilon _{0}=0$, $y=1$, and $t=1$ when (a) $a_{2}=1$; (b) $a_{2}=2$; (c) $x=1$, $a_{2}=1$ and $x=1$, $a_{2}=2$

4.5 Figure 5:

Figure 5 signifies the plots of $u_{7} (x,y,z,t)$ for $a_{1}=1$, $a_{3}=0.5$, $a_{4}=1$, $a_{5}=0.5$, $a_{6}=1$, $\omega =5$, $\epsilon _{0}=0$, $y=1$, and $t=1$ when (a) $a_{2}=1$; (b) $a_{2}=2$; (c) $x=1$, $a_{2}=1$ and $x=1$, $a_{2}=2$. Based on Fig. 5, as the nonlinear parameter increases from 1 to 2, the height and width of the wave decline.

Figure 5

The plots of $u_{7} (x,y,z,t)$ for $a_{1}=1$, $a_{3}=0.5$, $a_{4}=1$, $a_{5}=0.5$, $a_{6}=1$, $\omega =5$, $\epsilon _{0}=0$, $y=1$, and $t=1$ when (a) $a_{2}=1$; (b) $a_{2}=2$; (c) $x=1$, $a_{2}=1$ and $x=1$, $a_{2}=2$

4.6 Figure 6:

Figure 6 illustrates the plots of $u_{9} (x,y,z,t)$ for $a_{1}=1$, $a_{3}=-1$, $a_{4}=1$, $a_{5}=0.5$, $a_{6}=1$, $\omega =5$, $\epsilon _{0}=0$, $y=1$, and $t=1$ when (a) $a_{2}=1$; (b) $a_{2}=2$; (c) $x=1$, $a_{2}=1$ and $x=1$, $a_{2}=2$. By analyzing Fig. 6, it can be concluded that the height and width of the wave decrease as the nonlinear parameter increases from 1 to 2.

Figure 6

The plots of $u_{9} (x,y,z,t)$ for $a_{1}=1$, $a_{3}=-1$, $a_{4}=1$, $a_{5}=0.5$, $a_{6}=1$, $\omega =5$, $\epsilon _{0}=0$, $y=1$, and $t=1$ when (a) $a_{2}=1$; (b) $a_{2}=2$; (c) $x=1$, $a_{2}=1$ and $x=1$, $a_{2}=2$

5 Conclusion

The current paper examined the dynamics of nonlinear waves for a 3D-modified nonlinear wave equation with applications in fluid. More precisely, formally constructed invariant solutions of the governing equation were obtained by solving the PDEs resulting from classical and nonclassical Lie symmetries. Furthermore, the Jacobi elliptic function solutions to the 3D-mNLW equation were derived from symbolic computations as a result of the bifurcation analysis of the planar dynamical system. In the end, an in-depth analysis of the effect of the nonlinear parameter on the characteristics of bright and kink solitary waves and continuous periodic waves was carried out. The results of such a comprehensive analysis can be utilized in various fields of applied science and contribute to a deeper understanding of the dynamics inherent in nonlinear phenomena. The authors, in the future, examine the efficiency of the methods used in the present paper in handling some other well-known PDEs [44–51].