Lie Symmetry Analysis, Particular Solutions and Conservation Laws of a New Extended (3+1)-Dimensional Shallow Water Wave Equation

Abstract

In this paper, a new extended (3+1)-dimensional shallow water wave equation is discussed via Lie symmetry analysis. Making use of symmetric nodes, we obtain two kinds of symmetrically reduced ODEs. By means of power series, we obtain the two kinds of exact power series solutions. By invoking a new conservation theorem of Ibragimov, the conservation laws are constructed.

1. Introduction

As the initial model of integrable nonlinear partial differential equations (NLPDEs), the Korteweg–de Vries (KdV) equation describes the formation and propagation of medium long waves in shallow water, waves in nonlinear lattices, ionic acoustic waves and magnetoacoustic waves in plasma under the action of gravity, and is an important equation applied to physical phenomena such as fluid mechanics, magnetic systems, plasma physics and nonlinear optics [1,2,3,4,5,6,7]. One of the KdV equation forms is

$$u_t-6u{u}_x+u_{xxx}=0,$$

(1)

which has been studied through various methods, including Darboux transformation and Adomian decomposition [1,2]. With the wave equation becoming a new research hotspot, Equation (1) has evolved a large number of different (1 + 1)-dimensional KdV equations, which have also obtained many significant soliton solutions and periodic solutions [3,4,5,6,7]. The famous (2+1)-dimensional Ablowitz–Kaup–Newell–Segur (AKNS) equation is derived from the KdV equation [8].

Gilson et al. [9] generalized the AKNS equation by a bilinear method to obtain the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation

$$u_{yt}+u_{xxxy}-3{\left(u_y{u}_x\right)}_x=0.$$

(2)

Kaplan constructed the analytical solution of Equation (2) via a transformation rational function, and Kumar and Tiwari solved the explicit exact solution of Equation (2) with the Lie transformation group theory [10,11,12].

In [13], Guo and Xia first proposed the extended Leon–Manna–Pempinelli (eBLMP) equation

$$u_{yt}+u_{xxxy}+3{\left(u_y{u}_x\right)}_x+{\sigma }_1{u}_{xx}+{\sigma }_2{u}_{yy}=0,$$

(3)

which obtains a class of block and block kink soliton solutions by a global solution, with ${\sigma }_1$, ${\sigma }_2$ arbitrary constants. Shen et al. derived the respiratory wave solution, periodic wave solution and several traveling wave solutions of Equation (3) based on an extended Homoclinic test, Riemann-$\theta$ function and polynomial expansion [14].

Zhang and Hu [15] proposed a new (2+1)-dimensional extended shallow water wave equation

$$u_{yt}+u_{xxxy}+3{\left(u_y{u}_x\right)}_x+\alpha u_{xx}+\beta u_{yy}+\gamma u_{xy}=0,$$

(4)

with $\alpha$, $\beta$ and $\gamma$ arbitrary constants. Based on a Hirota bilinear method, the authors obtained the exact solutions of the lumped solution, lumped kink solution, breathing solution and two kink solutions.

In [16], Wazwaz developed a new (3+1)-dimensional KdV equation with constant coefficients

$$u_{yt}+u_{xxxy}+\alpha {\left(u_y{u}_x\right)}_x+\beta u_{xx}+\gamma u_{yy}+\delta u_{zy}=0,$$

(5)

which studies the integrability test of Equation (5) via the Painlevé analysis, and derives the soliton solutions of Equation (5) by the Hirota method. Cheng et al. obtained the bilinear form of Equation (5) and the N-soliton solution, where N is a positive integer. Further, they obtained a higher-order respiratory solution via the N-soliton solution [17].

In [18], Wazwaz studied the extended (3+1)-dimensional shallow water wave equation with constant coefficients,

$$u_{yt}+u_{xxxy}-3{\left(u_y{u}_x\right)}_x+\alpha u_{xx}+\beta u_{yy}+\gamma u_{xy}+\delta u_{zy}=0,$$

(6)

which was used to simulate the dynamic behavior of water wave propagation in oceanography and atmospheric science. Wazwaz proved that the extended Equation (6) maintained integrability, and formally derived the multiple soliton solutions, where $\alpha u_{xx}$, $\beta u_{yy}$, $\gamma u_{xy}$ and $\delta u_{zy}$ were extension terms. Han and Zhang [19] derived two kinds of bilinear self-Bäcklund transformations with the help of the Hirota bilinear method, and thus obtained two different types of solutions: a hyperbolic cosine function solution and a cosine function solution. Through the homoclinic test method, they gave and proved five linear superposition formulas. These results have important practical significance in explaining some important nonlinear systems.

The formation and propagation of waves has important applications in fluid mechanics, nonlinear optics, plasma physics and atmospheric science, which are some of the current research hotspots. Inspired by the evolution process from Equations (1)–(6), here we modify Equation (6) and propose the following extended (3+1)-dimensional shallow water wave equation

$$(u_y+u_x+u_z{)}_t+u_{xxxy}+3{\gamma }_1{\left(u_y{u}_x\right)}_x+{\gamma }_2{u}_{xx}+{\gamma }_3{u}_{yy}+{\gamma }_4{u}_{xy}+{\gamma }_5{u}_{zy}=0,$$

(7)

with ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$, ${\gamma }_4$ and ${\gamma }_5$ arbitrary constants. Equation (7) extends two terms $u_{xt}$ and $u_{zt}$ on the basis of Equation (6), which is still a (3+1)-dimensional shallow water wave equation in essence, but is more complex.

Nonlinear (3+1)-dimensional differential equations vividly describe the real dynamic behavior of water wave propagation in oceanography and the atmosphere. The exact solutions of the equations play an important role in explaining nonlinear physical models and planning further research. According to Wazwaz, we can develop various (3+1)-dimensional integrable models using symmetric methods, which will lay a foundation for understanding more complex fluid mechanics in the future [16,18]. It is consistent with our research direction and interest, especially in mathematics. The paper aims to develop a (3+1)-dimensional integrable shallow water wave equation with constant coefficients. Lie symmetry analysis and power series are used to determine the exact analytical solution of Equation (7), and conservation laws are used to prove the rationality of the solutions.

Through continuous and in-depth research by a large number of scholars, Lie symmetry analysis has made remarkable progress in deriving analytical solutions of NLPDEs, which is one of the most systematic and mature mathematical methods [20,21,22,23,24,25,26]. Conservation laws play an unparalleled role in analyzing the properties of differential equations (DEs) through seeking more accurate numerical solutions and verifying the reliability and stability of numerical solutions. Compared with Noether’s theorem, which cannot be applied to DEs not derived from the variational principle, the new conservation theorem of Ibragimov ingeniously solves the problem and gives an excellent solution [27,28]. Ibragimov proposed a concept of an adjoint equation for DEs: the adjoint equation inherits all the symmetries of the original equation so that the new conservation theorem can find the conservation laws of DEs without relying on Lagrange functions. In general, it is necessary to verify the existence and uniqueness of the exact analytical solutions of the DEs using Lie symmetry analysis [22,23,24,25].

The structure of the paper is as follows: In Section 2, the vector field is given by the Lie symmetry method, and the two combined vector fields are symmetrically reduced. In Section 3, we apply power series to solve the equations after symmetry reduction. In Section 4, we construct the conservation laws of Equation (7) by invoking Ibragimov’s new conservation theorem. In Section 5, we summarize the whole work and draw some conclusions.

2. Symmetries and Reductions

In this section, we give the Lie symmetries of Equation (7) via the calculation software Maple, and then we perform multiple symmetry reductions to obtain the ordinary differential equations (ODEs).

2.1. Lie Symmetry Analysis

In this section, our aim is to determine the symmetric algebra of Equation (7). Assume that Equation (7) satisfies the following one-parameter Lie symmetric transformation system:

$$\begin{array}{cc}\hfill x^{*}=& x+\epsilon \xi (x,y,z,t,u)+O\left({\epsilon }^2\right),\hfill \\ \hfill y^{*}=& y+\epsilon \eta (x,y,z,t,u)+O\left({\epsilon }^2\right),\hfill \\ \hfill z^{*}=& z+\epsilon \zeta (x,y,z,t,u)+O\left({\epsilon }^2\right),\hfill \\ \hfill t^{*}=& t+\epsilon \tau (x,y,z,t,u)+O\left({\epsilon }^2\right),\hfill \\ \hfill u^{*}=& u+\epsilon \varphi (x,y,z,t,u)+O\left({\epsilon }^2\right),\hfill \end{array}$$

(8)

where the expression of the vector field is

$$V=\xi \frac{\partial }{\partial x}+\eta \frac{\partial }{\partial y}+\zeta \frac{\partial }{\partial z}+\tau \frac{\partial }{\partial t}+\varphi \frac{\partial }{\partial u}.$$

(9)

The fourth prolongation of V for the Equation (7) is given by

$$\begin{array}{c}\hfill {Pr}^{\left(4\right)}V\left(F\right){|}_{F=0}=0,\end{array}$$

(10)

where $F=(u_y+u_x+u_z{)}_t+u_{xxxy}+3{\gamma }_1{\left(u_y{u}_x\right)}_x+{\gamma }_2{u}_{xx}+{\gamma }_3{u}_{yy}+{\gamma }_4{u}_{xy}+{\gamma }_5{u}_{zy}$. We can get that the infinitesimal transformation generators of Equation (7) are

$$\begin{array}{cc}\hfill \xi =& C_1+\frac{x+5z}{6}{C}_5+\frac{x-z}{6}{C}_6,\hfill \\ \hfill \eta =& C_2+\frac{y+z-{\gamma }_{5}t}{2}{C}_5+\frac{y-z+{\gamma }_{5}t}{2}{C}_6,\hfill \\ \hfill \zeta =& C_3+C_{5}z,\hfill \\ \hfill \tau =& C_4+C_{6}t,\hfill \\ \hfill \varphi =& f_1\left(t\right)+f_2\left(z\right)-\frac{1}{6}(C_5+C_6)(u+\frac{2}{3{\gamma }_1}({\gamma }_4-{\gamma }_5)x+\frac{4{\gamma }_2}{3{\gamma }_1}y),\hfill \end{array}$$

(11)

with $C_1$, $C_2$, $C_3$, $C_4$, $C_5$ and $C_6$ arbitrary constants, and $f_1\left(t\right)$ and $f_2\left(z\right)$ arbitrary functions about t, z, respectively. Therefore, the Lie symmetries of Equation (7) are

$$\begin{array}{cc}\hfill & V_1=\frac{\partial }{\partial x},V_2=\frac{\partial }{\partial y},V_3=\frac{\partial }{\partial z},V_4=\frac{\partial }{\partial t},\hfill \\ \hfill & V_5=\frac{x+5z}{6}\frac{\partial }{\partial x}+\frac{y+z-{\gamma }_{5}t}{2}\frac{\partial }{\partial y}+z\frac{\partial }{\partial z}-\frac{1}{6}(u+\frac{2}{3{\gamma }_1}({\gamma }_4-{\gamma }_5)x+\frac{4{\gamma }_2}{3{\gamma }_1}y)\frac{\partial }{\partial u},\hfill \\ \hfill & V_6=\frac{x-z}{6}\frac{\partial }{\partial x}+\frac{y-z+{\gamma }_{5}t}{2}\frac{\partial }{\partial y}+t\frac{\partial }{\partial t}-\frac{1}{6}(u+\frac{2}{3{\gamma }_1}({\gamma }_4-{\gamma }_5)x+\frac{4{\gamma }_2}{3{\gamma }_1}y)\frac{\partial }{\partial u},\hfill \\ \hfill & V_7\left(f_1\left(t\right)\right)=f_1\left(t\right)\frac{\partial }{\partial u},V_8\left(f_2\left(t\right)\right)=f_2\left(z\right)\frac{\partial }{\partial u}.\hfill \end{array}$$

(12)

2.2. Symmetry Reductions

Through the Lie symmetries Equation (12), we choose two kinds of symmetries $\alpha V_1+\beta V_2+\lambda V_3+V_4$, $V_5-V_6$ as examples to illustrate the superiority and reliability of the Lie symmetry method, where $\alpha$, $\beta$ and $\lambda$ are arbitrary constants.

2.2.1. Symmetry $\alpha V_1+\beta V_2+\lambda V_3+V_4=\alpha \frac{\partial }{\partial x}+\beta \frac{\partial }{\partial y}+\lambda \frac{\partial }{\partial z}+\frac{\partial }{\partial t}$

The characteristic equation of Equation (7) gives the invariants

$$u(x,y,z,t)=\phi (X,Y,Z),X=x-\alpha t,Y=y-\beta t,Z=z-\lambda t,$$

(13)

which reduces Equation (7) to

$$\begin{array}{cc}\hfill & ({\gamma }_2-\alpha ){\phi }_{XX}+({\gamma }_3-\beta ){\phi }_{YY}-\lambda {\phi }_{ZZ}+({\gamma }_4-\alpha -\beta ){\phi }_{XY}-(\alpha +\lambda ){\phi }_{XZ}\hfill \\ \hfill & +({\gamma }_5-\beta -\lambda ){\phi }_{YZ}+{\phi }_{XXXY}+3{\gamma }_1{\left({\phi }_Y{\phi }_X\right)}_X=0.\hfill \end{array}$$

(14)

Equation (14) has three translation symmetries ${\mathrm{Z}}_1=\frac{\partial }{\partial X},{\mathrm{Z}}_2=\frac{\partial }{\partial Y},{\mathrm{Z}}_3=\frac{\partial }{\partial Z}$. Symmetry $s_1{\mathrm{Z}}_1+s_2{\mathrm{Z}}_2+{\mathrm{Z}}_3$ produces the invariants

$$\phi =h(p,q),p=X-s_{1}Z,q=Y-s_{2}Z,$$

(15)

with $s_1$, $s_2$ as arbitrary constants. Substituting Equation (15) into Equation (14), we get

$$h_{pppq}+3{\gamma }_1{\left(h_p{h}_q\right)}_p+A{h}_{pq}+B{h}_{qq}+C{h}_{pp}=0,$$

(16)

with $A=\alpha s_2+\lambda s_2+\beta s_1+\lambda s_1-{\gamma }_5{s}_1+{\gamma }_4-\alpha -\beta -2\lambda s_1{s}_2$, $B=\beta s_2+\lambda s_2-{\gamma }_5{s}_2+{\gamma }_3-\beta -\lambda s_2^2$ and $C={\gamma }_2-\alpha -\lambda s_1^2+\alpha s_1+\lambda s_1$. The infinitesimal generators of Equation (16) are

$${\xi }_p=\frac{c_1}{3}p+c_3,{\xi }_q=c_{1}q+c_2,{\eta }_h=-\frac{c_1}{3}h-\frac{2c_{1}A}{9{\gamma }_1}p-\frac{4c_{1}C}{9{\gamma }_1}q+c_4,$$

(17)

with $c_1$, $c_2$, $c_3$ and $c_4$ arbitrary constants. Therefore, the Lagrange system of Equation (16) is

$$\frac{dp}{\frac{c_1}{3}p+c_3}=\frac{dq}{c_{1}q+c_2}=\frac{dh}{-\frac{c_1}{3}h-\frac{2c_{1}A}{9{\gamma }_1}p-\frac{4c_{1}C}{9{\gamma }_1}q+c_4}.$$

(18)

When $c_1\ne 0$ and $c_2=c_3=c_4=0$, the group invariant solution of Equation (18) is

$$h(p,q)=\frac{\psi \left(r\right)}{p}-\frac{A}{3{\gamma }_1}p-\frac{C}{3{\gamma }_1}q,r=\frac{q}{p^3}.$$

(19)

Substitute Equation (19) into Equation (16) to obtain

$$B{\psi }^{{}^{″}}-3(28{\psi }^{{}^{\prime }}+104r{\psi }^{{}^{″}}+66r^2{\psi }^{{}^{‴}}+9r^3{\psi }^{{}^{⁗}})+9{\gamma }_1(2\psi {\psi }^{{}^{\prime }}+r\psi {\psi }^{{}^{″}}+8r{{\psi }^{{}^{\prime }}}^2+6r^2{\psi }^{{}^{\prime }}{\psi }^{{}^{″}})=0.$$

(20)

2.2.2. Symmetry $V_5-V_6=z\frac{\partial }{\partial x}+(z-{\gamma }_{5}t)\frac{\partial }{\partial y}+z\frac{\partial }{\partial z}-t\frac{\partial }{\partial t}$

The characteristic equation of Equation (7) gives the invariants

$$u(x,y,z,t)=\phi (X,Y,T),X=z-x,Y=y-x-{\gamma }_{5}t,T=zt.$$

(21)

Equation (7) can be reduced to the following form

$$\begin{array}{cc}\hfill & -{\phi }_{XXXY}-3{\phi }_{XXYY}-3{\phi }_{XYYY}-{\phi }_{YYYY}+3{\gamma }_1({\phi }_{XX}{\phi }_Y+{\phi }_X{\phi }_{XY}+{\phi }_X{\phi }_{YY}+3{\phi }_{XY}{\phi }_Y+2{\phi }_{YY}{\phi }_Y)\hfill \\ \hfill & +{\phi }_T+T{\phi }_{TT}+{\gamma }_2{\phi }_{XX}+({\gamma }_2+{\gamma }_3-{\gamma }_4){\phi }_{YY}+(2{\gamma }_2-{\gamma }_4+{\gamma }_5){\phi }_{XY}=0.\hfill \end{array}$$

(22)

The symmetries of Equation (22) are

$$\begin{array}{cc}\hfill & {\mathrm{Z}}_1=\frac{\partial }{\partial X},{\mathrm{Z}}_2=\frac{\partial }{\partial Y},{\mathrm{Z}}_3=\frac{\partial }{\partial \phi },{\mathrm{Z}}_4=ln\left(T\right)\frac{\partial }{\partial \phi },\hfill \\ \hfill & {\mathrm{Z}}_5=\frac{X}{6}\frac{\partial }{\partial X}+\frac{-2X+3Y}{6}\frac{\partial }{\partial Y}+T\frac{\partial }{\partial T}+(-\frac{\phi }{6}-\frac{2{\gamma }_2}{9{\gamma }_1}Y+\frac{2{\gamma }_2+{\gamma }_4-{\gamma }_5}{9{\gamma }_1}X)\frac{\partial }{\partial \phi }.\hfill \end{array}$$

(23)

By utilizing the linear combination ${\mathrm{Z}}_1-{\mathrm{Z}}_2$, the invariant solution of Equation (22) is

$$\phi =h(p,q),p=X+Y,q=T.$$

(24)

Substituting Equation (24) into Equation (22), we obtain the equation in the form

$$-8h_{pppp}+24{\gamma }_1{h}_{pp}{h}_p+h_q+q{h}_{qq}+(4{\gamma }_2+{\gamma }_3-2{\gamma }_4+{\gamma }_5)h_{pp}=0.$$

(25)

The symmetries of Equation (25) are

$$\begin{array}{cc}\hfill & W_1=\frac{1}{4}p\frac{\partial }{\partial p}+q\frac{\partial }{\partial q}+(-\frac{1}{4}h-\frac{4{\gamma }_2+{\gamma }_3-2{\gamma }_4+{\gamma }_5}{48{\gamma }_1}p)\frac{\partial }{\partial h}\hfill \\ \hfill & W_2=ln\left(q\right)\frac{\partial }{\partial h},W_3=\frac{\partial }{\partial h},W_4=\frac{\partial }{\partial p}.\hfill \end{array}$$

(26)

Consider the symmetry $W_1$, the invariant solution of Equation (25) is

$$h(p,q)=\frac{\psi \left(r\right)}{p}-\frac{4{\gamma }_2+{\gamma }_3-2{\gamma }_4+{\gamma }_5}{24{\gamma }_1}p,r=\frac{q}{p^4}.$$

Equation (25) can be written as

$$\begin{array}{cc}\hfill & {\psi }^{{}^{\prime }}+r{\psi }^{{}^{″}}-64(3\psi +219r{\psi }^{{}^{\prime }}+540r^2{{\psi }^{{}^{\prime }}}^2+272r^3{\psi }^{{}^{‴}}+32r^4{\psi }^4)\hfill \\ \hfill & -48{\gamma }_1({\psi }^2+18r\psi {\psi }^{{}^{\prime }}+8r^2\psi {\psi }^{{}^{″}}+56r^2{{\psi }^{{}^{\prime }}}^2+32r^3{\psi }^{{}^{\prime }}{\psi }^{{}^{″}})=0.\hfill \end{array}$$

(27)

3. Power Series Solutions

Analyzing Equations (20) and (27), we understand that they are nonlinear ordinary differential equations (NLODEs), which cannot be solved by basic functions and integrals. However, power series can solve some complex NLODEs [25,26]. Thus, this section gives the power series solutions of the equations.

3.1. Solutions of Equation (20)

We define that Equation (20) has a power series solution of the following form

$$\psi \left(r\right)=\sum _{n=0}^{\infty }{a}_n{r}^n,$$

(28)

so that

$${\psi }^{{}^{\prime }}\left(r\right)=\sum _{n=0}^{\infty }n{a}_n{r}^n,{\psi }^{{}^{⁗}}\left(r\right)=\sum _{n=0}^{\infty }(n+1)(n+2)(n+3)(n+4)a_{n+4}{r}^n.$$

(29)

Equation (20) can be converted into the following form

$$\begin{array}{cc}\hfill & B\sum _{n=0}^{\infty }(n+1)(n+2)a_{n+2}{r}^n-3(28\sum _{n=0}^{\infty }(n+1)a_{n+1}{r}^n+104\sum _{n=0}^{\infty }n(n+1)a_{n+1}{r}^n\hfill \\ \hfill & +66\sum _{n=0}^{\infty }(n-1)n(n+1)a_{n+1}{r}^n+9\sum _{n=0}^{\infty }(n-2)(n-1)n(n+1)a_{n+1}{r}^n)\hfill \\ \hfill & +9{\gamma }_1(2\sum _{n=0}^{\infty }\sum _{i=0}^n(n-i+1)a_i{a}_{n-i+1}{r}^n+\sum _{n=0}^{\infty }\sum _{i=0}^n(n-i)(n-i+1)a_i{a}_{n-i+1}{r}^n\hfill \\ \hfill & +8\sum _{n=0}^{\infty }\sum _{i=0}^{n}i(n-i+1)a_i{a}_{n-i+1}{r}^n+6\sum _{n=0}^{\infty }\sum _{i=0}^{n}i(n-i)(n-i+1)a_i{a}_{n-i+1}{r}^n)=0.\hfill \end{array}$$

(30)

According to Equation (30), we get, for $n=0$,

$$a_2=\frac{42a_1-9{\gamma }_1{a}_0{a}_1}{\beta s_2+\lambda s_2-{\gamma }_5{s}_2+{\gamma }_3-\beta -\lambda s_2^2},$$

(31)

and for $n⩾1$

$$\begin{array}{cc}\hfill a_{n+2}=& \frac{1}{(\beta s_2+\lambda s_2-{\gamma }_5{s}_2+{\gamma }_3-\beta -\lambda s_2^2)(n+1)(n+2)}\left(3a_{n+1}\right(28(n+1)\hfill \\ \hfill & +104n(n+1)+66(n-1)n(n+1)+9(n-2)(n-1)n(n+1))\hfill \\ \hfill & -9{\gamma }_1(2\sum _{i=0}^n(n-i+1)a_i{a}_{n-i+1}+\sum _{i=0}^n(n-i)(n-i+1)a_i{a}_{n-i+1}\hfill \\ \hfill & +8\sum _{i=0}^{n}i(n-i+1)a_i{a}_{n-i+1}+6\sum _{i=0}^{n}i(n-i)(n-i+1)a_i{a}_{n-i+1}\left)\right).\hfill \end{array}$$

(32)

The power series solution of Equation (20) is

$$\begin{array}{cc}\hfill \psi \left(r\right)=& a_0+a_{1}r+a_2{r}^2+\sum _{n=1}^{\infty }{a}_{n+2}{r}^{n+2}\hfill \\ \hfill =& a_0+a_{1}r+\frac{42a_1-9{\gamma }_1{a}_0{a}_1}{\beta s_2+\lambda s_2-{\gamma }_5{s}_2+{\gamma }_3-\beta -\lambda s_2^2}{r}^2\hfill \\ \hfill & +\sum _{n=1}^{\infty }\frac{1}{(\beta s_2+\lambda s_2-{\gamma }_5{s}_2+{\gamma }_3-\beta -\lambda s_2^2)(n+1)(n+2)}\left(3\right(28(n+1)\hfill \\ \hfill & +104n(n+1)+66(n-1)n(n+1)+9(n-2)(n-1)n(n+1))a_{n+1}\hfill \\ \hfill & -9{\gamma }_1(2\sum _{i=0}^n(n-i+1)a_i{a}_{n-i+1}+\sum _{i=0}^n(n-i)(n-i+1)a_i{a}_{n-i+1}\hfill \\ \hfill & +8\sum _{i=0}^{n}i(n-i+1)a_i{a}_{n-i+1}+6\sum _{i=0}^{n}i(n-i)(n-i+1)a_i{a}_{n-i+1}\left)\right)r^{n+2}.\hfill \end{array}$$

(33)

Thus, the particular solution of Equation (7) is

$$\begin{array}{cc}\hfill u(x,y,z,t)=& -\frac{A}{3{\gamma }_1}p-\frac{C}{3{\gamma }_1}q+\frac{1}{p}\psi \left(r\right)\hfill \\ \hfill =& -\frac{(\alpha +\lambda )s_2+(\beta +\lambda -{\gamma }_5)s_1+{\gamma }_4-\alpha -\beta -2\lambda s_1{s}_2}{3{\gamma }_1}(x-s_{1}z+(s_1\lambda -\alpha )t)\hfill \\ \hfill & -\frac{{\gamma }_2-\alpha -\lambda s_1^2+\alpha s_1+\lambda s_1}{3{\gamma }_1}(y-s_{2}z+(s_2\lambda -\beta )t)\hfill \\ \hfill & +\frac{1}{x-s_{1}z+(s_1\lambda -\alpha )t}\{a_0+a_1\frac{y-s_{2}z+(s_2\lambda -\beta )t}{{(x-s_{1}z+(s_1\lambda -\alpha )t)}^3}\hfill \\ \hfill & +\frac{42a_1-9{\gamma }_1{a}_0{a}_1}{\beta s_2+\lambda s_2-{\gamma }_5{s}_2+{\gamma }_3-\beta -\lambda s_2^2}{\left(\frac{y-s_{2}z+(s_2\lambda -\beta )t}{{(x-s_{1}z+(s_1\lambda -\alpha )t)}^3}\right)}^2\hfill \\ \hfill & +\sum _{n=1}^{\infty }\frac{1}{(\beta s_2+\lambda s_2-{\gamma }_5{s}_2+{\gamma }_3-\beta -\lambda s_2^2)(n+1)(n+2)}\left(3\right(28(n+1)\hfill \\ \hfill & +104n(n+1)+66(n-1)n(n+1)+9(n-2)(n-1)n(n+1))a_{n+1}\hfill \\ \hfill & -9{\gamma }_1(2\sum _{i=0}^n(n-i+1)a_i{a}_{n-i+1}+\sum _{i=0}^n(n-i)(n-i+1)a_i{a}_{n-i+1}\hfill \\ \hfill & +8\sum _{i=0}^{n}i(n-i+1)a_i{a}_{n-i+1}\hfill \\ \hfill & +6\sum _{i=0}^{n}i(n-i)(n-i+1)a_i{a}_{n-i+1}\left)\right){\left(\frac{y-s_{2}z+(s_2\lambda -\beta )t}{{(x-s_{1}z+(s_1\lambda -\alpha )t)}^3}\right)}^{n+2}\}.\hfill \end{array}$$

(34)

3.2. Solutions of Equation (27)

Similarly, Equation (27) can be rewritten

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }(n+1)a_{n+1}{r}^n+\sum _{n=0}^{\infty }n(n+1)a_{n+1}{r}^n-64(3\sum _{n=0}^{\infty }{a}_n{r}^n+219\sum _{n=0}^{\infty }n{a}_n{r}^n\hfill \\ \hfill & +540\sum _{n=0}^{\infty }n(n-1)a_n{r}^n+272\sum _{n=0}^{\infty }n(n-1)(n-2)a_n{r}^n\hfill \\ \hfill & +32\sum _{n=0}^{\infty }n(n-1)(n-2)(n-3)a_n{r}^n)-48{\gamma }_1(\sum _{n=0}^{\infty }\sum _{i=0}^n{a}_i{a}_{n-i}{r}^n\hfill \\ \hfill & +18\sum _{n=0}^{\infty }\sum _{i=0}^n(n-i)a_i{a}_{n-i}{r}^n+8\sum _{n=0}^{\infty }\sum _{i=0}^n(n-i)(n-1-i)a_i{a}_{n-i}{r}^n\hfill \\ \hfill & +56\sum _{n=0}^{\infty }\sum _{i=0}^{n}i(n-i)a_i{a}_{n-i}{r}^n+32\sum _{n=0}^{\infty }\sum _{i=0}^{n}i(n-i)(n-1-i)a_i{a}_{n-i}{r}^n)=0.\hfill \end{array}$$

(35)

Here, instead of repeating the calculation of $n=0$ and $n⩾1$, the power series solution of Equation (27) is directly given

$$\begin{array}{cc}\hfill \psi \left(r\right)=& a_0+a_{1}r+\sum _{n=1}^{\infty }{a}_{n+1}{r}^{n+1}\hfill \\ \hfill =& a_0+(192a_0+48{\gamma }_1{a}_0^2)r+\sum _{n=1}^{\infty }\frac{1}{{(n+1)}^2}\left\{64a_n\right(3+219n+540n(n-1)\hfill \\ \hfill & +272n(n-1)(n-2)+32n(n-1)(n-2)(n-3))\hfill \\ \hfill & +48{\gamma }_1(\sum _{i=0}^n{a}_i{a}_{n-i}+18\sum _{i=0}^n(n-i)a_i{a}_{n-i}+8\sum _{i=0}^n(n-i)(n-1-i)a_i{a}_{n-i}\hfill \\ \hfill & +56\sum _{i=0}^{n}i(n-i)a_i{a}_{n-i}+32\sum _{i=0}^{n}i(n-i)(n-1-i)a_i{a}_{n-i}\left)\right\}{r}^{n+1}.\hfill \end{array}$$

(36)

The particular solution of Equation (7) is received

$$\begin{array}{cc}\hfill u(x,y,z,t)=& -\frac{4{\gamma }_2+{\gamma }_3-2{\gamma }_4+{\gamma }_5}{24{\gamma }_1}p+\frac{1}{p}\{a_0+a_{1}r+\sum _{n=1}^{\infty }{a}_{n+1}{r}^{n+1}\}\hfill \\ \hfill =& -\frac{4{\gamma }_2+{\gamma }_3-2{\gamma }_4+{\gamma }_5}{24{\gamma }_1}(z-2x+y-{\gamma }_{5}t)+\frac{1}{z-2x+y-{\gamma }_{5}t}\{a_0\hfill \\ \hfill & +(192a_0+48{\gamma }_1{a}_0^2)\frac{zt}{{(z-2x+y-{\gamma }_{5}t)}^4}\hfill \\ \hfill & +\sum _{n=1}^{\infty }\frac{1}{{(n+1)}^2}\left(64a_n\right(3+219n+540n(n-1)+272n(n-1)(n-2)\hfill \\ \hfill & +32n(n-1)(n-2)(n-3))+48{\gamma }_1(\sum _{i=0}^n{a}_i{a}_{n-i}+18\sum _{i=0}^n(n-i)a_i{a}_{n-i}\hfill \\ \hfill & +8\sum _{i=0}^n(n-i)(n-1-i)a_i{a}_{n-i}+56\sum _{i=0}^{n}i(n-i)a_i{a}_{n-i}\hfill \\ \hfill & +32\sum _{i=0}^{n}i(n-i)(n-1-i)a_i{a}_{n-i}\left)\right){\left(\frac{zt}{{(z-2x+y-{\gamma }_{5}t)}^4}\right)}^{n+1}\}.\hfill \end{array}$$

(37)

4. Conservation Laws

In this section, we derive the conservation laws of Equation (7) with Ibragimov’s new conservation theorem [27,28]. We can apply the theorem to any DEs, and it is not necessary to determine whether a Lagrange exists. The conservation laws can be connected with any symmetric systems of DEs to determine the conservation vector of DEs. The adjoint equation of Equation (7) is

$$\begin{array}{cc}\hfill F^{*}=& \frac{\delta }{\delta u}\left(v\right({(u_x+u_y+u_z)}_t+u_{xxxy}+3{\gamma }_1{\left(u_y{u}_x\right)}_x\hfill \\ \hfill & +{\gamma }_2{u}_{xx}+{\gamma }_3{u}_{yy}+{\gamma }_4{u}_{xy}+{\gamma }_5{u}_{zy}\left)\right)=0,\hfill \end{array}$$

(38)

with $v=v(x,y,z,t)$ an arbitrary function about x, y, z, t. The Euler operator $\frac{\delta }{\delta u}$ is defined as

$$\begin{array}{cc}\hfill \frac{\delta }{\delta u}=& -D_x\frac{\partial }{\partial u_x}-D_y\frac{\partial }{\partial u_y}+D_x^2\frac{\partial }{\partial u_{xx}}+D_y^2\frac{\partial }{\partial u_{yy}}+D_x{D}_y\frac{\partial }{\partial u_{xy}}+D_z{D}_y\frac{\partial }{\partial u_{zy}}\hfill \\ \hfill & +D_x{D}_t\frac{\partial }{\partial u_{xt}}+D_y{D}_t\frac{\partial }{\partial u_{yt}}+D_z{D}_t\frac{\partial }{\partial u_{zt}}+D_x^3{D}_y\frac{\partial }{\partial u_{xxxy}},\hfill \end{array}$$

(39)

where $(D_t,D_x,D_y,D_z)$ are the total derivative, defined as

$$\begin{array}{cc}\hfill D_t=& \frac{\partial }{\partial t}+u_t\frac{\partial }{\partial u}+u_{tt}\frac{\partial }{\partial u_t}+u_{tx}\frac{\partial }{\partial u_x}+u_{ty}\frac{\partial }{\partial u_y}+u_{tz}\frac{\partial }{\partial u_z}+\cdots ,\hfill \\ \hfill D_x=& \frac{\partial }{\partial x}+u_x\frac{\partial }{\partial u}+u_{xx}\frac{\partial }{\partial u_x}+u_{xt}\frac{\partial }{\partial u_t}+u_{xy}\frac{\partial }{\partial u_y}+u_{xz}\frac{\partial }{\partial u_z}+\cdots ,\hfill \\ \hfill D_y=& \frac{\partial }{\partial y}+u_y\frac{\partial }{\partial u}+u_{yy}\frac{\partial }{\partial u_y}+u_{yx}\frac{\partial }{\partial u_x}+u_{yt}\frac{\partial }{\partial u_t}+u_{yz}\frac{\partial }{\partial u_z}+\cdots ,\hfill \\ \hfill D_z=& \frac{\partial }{\partial z}+u_z\frac{\partial }{\partial u}+u_{zt}\frac{\partial }{\partial u_t}+u_{zx}\frac{\partial }{\partial u_x}+u_{zy}\frac{\partial }{\partial u_y}+u_{zz}\frac{\partial }{\partial u_z}+\cdots .\hfill \end{array}$$

(40)

Expand Equation (38) to obtain

$$\begin{array}{cc}\hfill F^{*}=& v_{yt}+v_{xt}+v_{zt}+v_{xxxy}+3{\gamma }_1{u}_y{v}_{xx}+6{\gamma }_1{u}_{xy}{v}_x+3{\gamma }_1{v}_{xy}{u}_x\hfill \\ \hfill & +{\gamma }_2{v}_{xx}+{\gamma }_3{v}_{yy}+{\gamma }_4{v}_{xy}+{\gamma }_5{v}_{zy}=0.\hfill \end{array}$$

(41)

Therefore, the Lagrange of Equation (7) and its adjoint Equation (41) have a second-order Lagrange

$$\begin{array}{cc}\hfill L=& v(u_{xt}+u_{yt}+u_{zt}+3{\gamma }_1{u}_y{u}_{xx}+3{\gamma }_1{u}_{xy}{u}_x+{\gamma }_2{u}_{xx}\hfill \\ \hfill & +{\gamma }_3{u}_{yy}+{\gamma }_4{u}_{xy}+{\gamma }_5{u}_{zy})+u_{xx}{v}_{xy}.\hfill \end{array}$$

(42)

According to the eight Lie point symmetries given by Equation (7) in Equation (10), we call

$$C^i={\xi }^{i}L+W^{\alpha }[\frac{\partial L}{\partial u_i^{\alpha }}-D_j\frac{\partial L}{\partial u_{ij}^{\alpha }}]+D_j\left(W^{\alpha }\right)\frac{\partial L}{\partial u_{ij}^{\alpha }},$$

(43)

to construct the second-order Lagrangian conservative vectors, with Lie eigenfunction $W^{\alpha }={\varphi }^{\alpha }-{\xi }^j{u}_j^{\alpha }$, $\alpha =1,2$, $j=1,2,3,4$.

We first consider Lie point symmetry $V_1=\frac{\partial }{\partial x}$, so the corresponding values of $Y_1$ are $\frac{\partial }{\partial x}$, $W^1=-u_x$ and $W^2=-v_x$. By means of Equations (43) and (42), we obtain the vector field with Lie point symmetry $V_1=\frac{\partial }{\partial x}$ as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_1^x=& v{u}_{yt}+v{u}_{zt}+\frac{3}{2}{\gamma }_{1}v{u}_{xy}{u}_x+{\gamma }_{3}v{u}_{yy}+{\gamma }_{5}v{u}_{zy}+v_t{u}_x+3{\gamma }_1{v}_x{u}_x{u}_y+{\gamma }_2{v}_x{u}_x+u_x{v}_{xxy}\hfill \\ & +\frac{3}{2}{\gamma }_1{v}_y{u}_x^2+{\gamma }_4{v}_y{u}_x-u_{xx}{v}_{xy},\hfill \\ \hfill C_1^y=& \frac{1}{2}{v}_t{u}_x+\frac{3}{2}{\gamma }_1{v}_x{u}_x^2+\frac{1}{2}{\gamma }_4{v}_x{u}_x+{\gamma }_3{v}_y{u}_x+\frac{1}{2}{\gamma }_5{v}_z{u}_x-\frac{1}{2}v{u}_{xt}-\frac{3}{2}{\gamma }_{1}v{u}_x{u}_{xx}-\frac{1}{2}{\gamma }_{4}v{u}_{xx}\hfill \\ & -{\gamma }_{3}v{u}_{xy}-\frac{1}{2}{\gamma }_{5}v{u}_{xz}-u_{xx}{v}_{xx},\hfill \\ \hfill C_1^z=& \frac{1}{2}{v}_t{u}_x+\frac{1}{2}{\gamma }_5{v}_y{u}_x-\frac{1}{2}v{u}_{xt}-\frac{1}{2}{\gamma }_{5}v{u}_{xy},\hfill \\ \hfill C_1^t=& \frac{1}{2}{v}_x{u}_x+\frac{1}{2}{v}_y{u}_x+\frac{1}{2}{v}_z{u}_x-\frac{1}{2}v{u}_{xx}-\frac{1}{2}v{u}_{xy}-\frac{1}{2}v{u}_{xz}.\hfill \end{array}\end{array}$$

By repeating the previous calculation method, we calculate the remaining seven Lie point symmetric conservation vectors. The results are as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_2^x=& \frac{1}{2}{v}_t{u}_y+3{\gamma }_1{v}_x{u}_y^2+{\gamma }_2{v}_x{u}_y+u_y{v}_{xxy}+3{\gamma }_1{v}_y{u}_x{u}_y+\frac{1}{2}{\gamma }_4{v}_y{u}_y-\frac{1}{2}v{u}_{yt}-{\gamma }_{2}v{u}_{xy}\hfill \\ & -v_{xy}{u}_{xy}-3{\gamma }_{1}v{u}_x{u}_{yy}-\frac{1}{2}{\gamma }_{4}v{u}_{yy}-u_{xx}{v}_{yy},\hfill \\ \hfill C_2^y=& v{u}_{xt}+v{u}_{zt}+3{\gamma }_{1}v{u}_y{u}_{xx}+{\gamma }_{2}v{u}_{xx}+v_t{u}_y+3{\gamma }_1{v}_x{u}_x{u}_y+{\gamma }_4{v}_x{u}_y+{\gamma }_3{v}_y{u}_y+{\gamma }_5{v}_z{u}_y,\hfill \\ \hfill C_2^z=& \frac{1}{2}{v}_t{u}_y+\frac{1}{2}{\gamma }_5{v}_y{u}_y-\frac{1}{2}v{u}_{yt}-\frac{1}{2}{\gamma }_{5}v{u}_{yy},\hfill \\ \hfill C_2^t=& \frac{1}{2}{v}_x{u}_y+\frac{1}{2}{v}_y{u}_y+\frac{1}{2}{v}_z{u}_y-\frac{1}{2}v{u}_{xy}-\frac{1}{2}v{u}_{yy}-\frac{1}{2}v{u}_{zy},\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_3^x=& \frac{1}{2}{v}_t{u}_z+3{\gamma }_1{v}_x{u}_y{u}_z+{\gamma }_2{v}_x{u}_z+u_z{v}_{xxy}+\frac{3}{2}{\gamma }_{1}v{u}_{xy}{u}_z+\frac{3}{2}{\gamma }_1{v}_y{u}_x{u}_z+\frac{1}{2}{\gamma }_4{v}_y{u}_z\hfill \\ & -\frac{1}{2}v{u}_{zt}-3{\gamma }_{1}v{u}_y{u}_{xz}-{\gamma }_{2}v{u}_{xz}-v_{xy}{u}_{xz}-\frac{3}{2}{\gamma }_{1}v{u}_x{u}_{zy}-\frac{1}{2}{\gamma }_{4}v{u}_{zy}-u_{xx}{v}_{zy},\hfill \\ \hfill C_3^y=& \frac{1}{2}{v}_t{u}_z+3{\gamma }_1{v}_x{u}_x{u}_z+\frac{1}{2}{\gamma }_4{v}_x{u}_z+{\gamma }_3{v}_y{u}_z+\frac{1}{2}{\gamma }_5{v}_z{u}_z-\frac{1}{2}v{u}_{zt}-3{\gamma }_{1}v{u}_x{u}_{xz}\hfill \\ & -\frac{1}{2}{\gamma }_{4}v{u}_{xz}-{\gamma }_{3}v{u}_{zy}-\frac{1}{2}{\gamma }_{5}v{u}_{zz}-u_{xx}{v}_{zx},\hfill \\ \hfill C_3^z=& v{u}_{xt}+v{u}_{yt}+3{\gamma }_{1}v{u}_y{u}_{xx}+3{\gamma }_{1}v{u}_{xy}{u}_x+{\gamma }_{2}v{u}_{xx}+{\gamma }_{3}v{u}_{yy}+{\gamma }_{4}v{u}_{xy}+u_{xx}{v}_{xy}\hfill \\ & +v_t{u}_z+{\gamma }_5{v}_y{u}_z,\hfill \\ \hfill C_3^t=& \frac{1}{2}{v}_x{u}_z+\frac{1}{2}{v}_y{u}_z+\frac{1}{2}{v}_z{u}_z-\frac{1}{2}v{u}_{xz}-\frac{1}{2}v{u}_{zy}-\frac{1}{2}v{u}_{zz},\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_4^x=& \frac{1}{2}{v}_t{u}_t+3{\gamma }_1{v}_x{u}_y{u}_t+{\gamma }_2{v}_x{u}_t+u_t{v}_{xxy}+\frac{3}{2}{\gamma }_{1}v{u}_{xy}{u}_t+\frac{3}{2}{\gamma }_1{v}_y{u}_x{u}_t+\frac{1}{2}{\gamma }_4{v}_y{u}_t\hfill \\ & -\frac{1}{2}v{u}_{tt}-3{\gamma }_{1}v{u}_y{u}_{xt}-{\gamma }_{2}v{u}_{xt}-v_{xy}{u}_{xt}-\frac{3}{2}{\gamma }_{1}v{u}_x{u}_{yt}-\frac{1}{2}{\gamma }_{4}v{u}_{yt}-u_{xx}{v}_{ty},\hfill \\ \hfill C_4^y=& \frac{1}{2}{v}_t{u}_t+3{\gamma }_1{v}_x{u}_x{u}_t+\frac{1}{2}{\gamma }_4{v}_x{u}_t+{\gamma }_3{v}_y{u}_t+\frac{1}{2}{\gamma }_5{v}_z{u}_t-\frac{1}{2}v{u}_{tt}-3{\gamma }_{1}v{u}_x{u}_{xt}\hfill \\ & -\frac{1}{2}{\gamma }_{4}v{u}_{xt}-{\gamma }_{3}v{u}_{yt}-\frac{1}{2}{\gamma }_{5}v{u}_{zt}-u_{xx}{v}_{xt},\hfill \\ \hfill C_4^z=& \frac{1}{2}{v}_t{u}_t+\frac{1}{2}{\gamma }_5{v}_y{u}_t-\frac{1}{2}v{u}_{tt}-\frac{1}{2}{\gamma }_{5}v{u}_{yt},\hfill \\ \hfill C_4^t=& 3{\gamma }_{1}v{u}_y{u}_{xx}+3{\gamma }_{1}v{u}_{xy}{u}_x+{\gamma }_{2}v{u}_{xx}+{\gamma }_{3}v{u}_{yy}+{\gamma }_{4}v{u}_{xy}+{\gamma }_{5}v{u}_{zy}+u_{xx}{v}_{xy}+v_x{u}_t\hfill \\ & +v_y{u}_t+v_z{u}_t,\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_5^x=& \frac{x+5z}{6}v{u}_{yt}+\frac{x+5z}{6}v{u}_{zt}+\frac{{\gamma }_3(x+5z)}{6}v{u}_{yy}+\frac{{\gamma }_5(x+5z)}{6}v{u}_{zy}+\frac{1}{12}u{v}_t+\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_t\hfill \\ & +\frac{({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_t+\frac{x+5z}{6}{u}_x{v}_t+\frac{y+z-{\gamma }_{5}t}{4}{u}_y{v}_t+\frac{1}{2}z{u}_z{v}_t+\frac{{\gamma }_1}{2}u{u}_y{v}_x+\frac{2{\gamma }_2}{3}y{u}_y{v}_x\hfill \\ & +\frac{({\gamma }_4-{\gamma }_5)}{3}x{u}_y{v}_x+\frac{{\gamma }_2}{6}u{v}_x+\frac{{\gamma }_1(x+5z)}{2}{u}_x{u}_y{v}_x+\frac{3{\gamma }_1(y+z-{\gamma }_{5}t)}{2}{u}_y^2{v}_x+\frac{2{\gamma }_2^2}{9{\gamma }_1}y{v}_x\hfill \\ & +3{\gamma }_{1}z{u}_z{u}_y{v}_x+\frac{{\gamma }_2({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_x+\frac{{\gamma }_2(x+5z)}{6}{u}_x{v}_x+\frac{{\gamma }_2(y+z-{\gamma }_{5}t)}{2}{u}_y{v}_x+\frac{1}{6}u{v}_{xxy}\hfill \\ & +{\gamma }_{2}z{u}_z{v}_x+\frac{({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_{xxy}+\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_{xxy}+\frac{x+5z}{6}{u}_x{v}_{xxy}+\frac{y+z-{\gamma }_{5}t}{2}{u}_y{v}_{xxy}\hfill \\ & +z{u}_z{v}_{xxy}+\frac{{\gamma }_1}{2}uv{u}_{xy}+\frac{({\gamma }_4-{\gamma }_5)}{6}xv{u}_{xy}+\frac{{\gamma }_2}{3}yv{u}_{xy}+\frac{{\gamma }_1(x+5z)}{2}v{u}_x{u}_{xy}+\frac{{\gamma }_1}{2}u{u}_x{v}_y\hfill \\ & +\frac{3{\gamma }_1(y+z-{\gamma }_{5}t)}{4}v{u}_y{u}_{xy}+\frac{3{\gamma }_1}{2}zv{u}_z{u}_{xy}+\frac{({\gamma }_4-{\gamma }_5)}{6}x{u}_x{v}_y+\frac{{\gamma }_2}{3}y{u}_x{v}_y\hfill \\ & +\frac{{\gamma }_1(x+5z)}{2}{u}_x^2{v}_y+\frac{3{\gamma }_1(y+z-{\gamma }_{5}t)}{4}{u}_x{u}_y{v}_y+\frac{3{\gamma }_1}{2}z{u}_x{u}_z{v}_y+\frac{1}{6}{\gamma }_{4}u{v}_y\hfill \\ & +\frac{({\gamma }_4-{\gamma }_5){\gamma }_4}{9{\gamma }_1}x{v}_y+\frac{{\gamma }_2{\gamma }_4}{9{\gamma }_1}y{v}_y+\frac{{\gamma }_4(x+5z)}{6}{u}_x{v}_y+\frac{{\gamma }_4(y+z-{\gamma }_{5}t)}{4}{u}_y{v}_y\hfill \\ & +\frac{{\gamma }_4}{2}z{u}_z{v}_y-\frac{1}{12}v{u}_t+\frac{{\gamma }_5}{4}v{u}_y-\frac{y+z-{\gamma }_{5}t}{4}v{u}_{yt}-\frac{1}{2}zv{u}_{zt}-{\gamma }_{1}v{u}_x{u}_y\hfill \\ & -\frac{({\gamma }_4-{\gamma }_5)}{3}v{u}_y-\frac{3(y+z-{\gamma }_{5}t)}{2}{\gamma }_{1}v{u}_y{u}_{xy}-3{\gamma }_{1}zv{u}_y{u}_{xz}-\frac{1}{3}{\gamma }_{2}v{u}_x-\frac{{\gamma }_2({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}v\hfill \\ & -\frac{(y+z-{\gamma }_{5}t){\gamma }_2}{2}v{u}_{xy}-{\gamma }_{2}zv{u}_{xz}-\frac{1}{3}{u}_x{v}_{xy}-\frac{1}{9{\gamma }_1}({\gamma }_4-{\gamma }_5)v_{xy}-{\gamma }_{1}v{u}_x{u}_y-\frac{{\gamma }_2}{3}v{u}_x\hfill \\ & -\frac{y+z-{\gamma }_{5}t}{2}{u}_{xy}{v}_{xy}-z{u}_{xz}{v}_{xy}-\frac{3(y+z-{\gamma }_{5}t){\gamma }_1}{4}v{u}_x{u}_{yy}-\frac{3{\gamma }_1}{2}zv{u}_x{u}_{zy}-\frac{{\gamma }_4}{3}v{u}_y\hfill \\ & -\frac{{\gamma }_2{\gamma }_4}{9{\gamma }_1}v-\frac{(y+z-{\gamma }_{5}t){\gamma }_4}{4}v{u}_{yy}-\frac{{\gamma }_4}{2}zv{u}_{zy}-v_y{u}_{xx}-\frac{x+5z}{6}{v}_{xy}{u}_{xx}\hfill \\ & -\frac{y+z-{\gamma }_{5}t}{2}{u}_{xx}{v}_{yy}-z{u}_{xx}{v}_{zy},\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_5^y=& \frac{y+z-{\gamma }_{5}t}{2}v{u}_{xt}+\frac{y+z-{\gamma }_{5}t}{2}v{u}_{zt}+\frac{3{\gamma }_1(y+z-{\gamma }_{5}t)}{2}v{u}_y{u}_{xx}+\frac{{\gamma }_2(y+z-{\gamma }_{5}t)}{2}v{u}_{xx}\hfill \\ & +\frac{({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_t+\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_t+\frac{x+5z}{12}{u}_x{v}_t+\frac{y+z-{\gamma }_{5}t}{2}{u}_y{v}_t+\frac{1}{2}z{u}_z{v}_t+\frac{{\gamma }_1}{2}u{u}_x{v}_x\hfill \\ & +\frac{1}{12}u{v}_t+\frac{({\gamma }_4-{\gamma }_5)}{3}x{u}_x{v}_x+\frac{2{\gamma }_2}{3}y{u}_x{v}_x+\frac{{\gamma }_1(x+5z)}{4}{u}_x^2{v}_x+3{\gamma }_{1}z{u}_x{u}_z{v}_x+\frac{{\gamma }_4}{6}u{v}_x\hfill \\ & +\frac{3{\gamma }_1(y+z-{\gamma }_{5}t)}{2}{u}_x{u}_y{v}_x+\frac{{\gamma }_4({\gamma }_4-{\gamma }_5)}{18{\gamma }_1}x{v}_x+\frac{2{\gamma }_2{\gamma }_4}{9{\gamma }_1}y{v}_x+\frac{{\gamma }_4(x+5z)}{12}{u}_x{v}_x\hfill \\ & +\frac{{\gamma }_4(y+z-{\gamma }_{5}t)}{2}{u}_y{v}_x+\frac{{\gamma }_4}{2}z{u}_z{v}_x+\frac{{\gamma }_3}{6}u{v}_y+\frac{{\gamma }_3({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_y+\frac{2{\gamma }_2{\gamma }_3}{9{\gamma }_1}y{v}_y+\frac{{\gamma }_5}{6}u{v}_z\hfill \\ & +\frac{{\gamma }_3(x+5z)}{6}{u}_x{v}_y+{\gamma }_{3}z{u}_z{v}_y+\frac{{\gamma }_3(y+z-{\gamma }_{5}t)}{2}{u}_y{v}_y+\frac{{\gamma }_5({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_z+\frac{2{\gamma }_2{\gamma }_5}{9{\gamma }_1}y{v}_z\hfill \\ & +\frac{{\gamma }_5(x+5z)}{12}{u}_x{v}_z+\frac{{\gamma }_5}{2}z{u}_z{v}_z+\frac{{\gamma }_5(y+z-{\gamma }_{5}t)}{2}{u}_y{v}_z-\frac{1}{12}v{u}_t+\frac{{\gamma }_5}{2}v{u}_y-\frac{1}{2}zv{u}_{zt}\hfill \\ & -\frac{x+5z}{12}v{u}_{xt}-\frac{{\gamma }_4}{6}v{u}_x-\frac{{\gamma }_4({\gamma }_4-{\gamma }_5)}{18{\gamma }_1}v-\frac{{\gamma }_4(x+5z)}{6}v{u}_{xx}-\frac{{\gamma }_4}{2}zv{u}_{xz}-\frac{{\gamma }_1}{2}v{u}_x^2\hfill \\ & -\frac{({\gamma }_4-{\gamma }_5)}{3}{u}_x-\frac{{\gamma }_1(x+5z)}{4}v{u}_x{u}_{xx}-3{\gamma }_{1}z{u}_x{u}_{xz}-\frac{2{\gamma }_3}{3}v{u}_y-\frac{2{\gamma }_2{\gamma }_3}{9{\gamma }_1}v-\frac{7{\gamma }_5}{12}v{u}_z\hfill \\ & -\frac{{\gamma }_3(x+5z)}{6}v{u}_{xy}-{\gamma }_{3}vz{u}_{zy}-\frac{5{\gamma }_5}{12}v{u}_x-\frac{{\gamma }_5}{2}v{u}_y-\frac{{\gamma }_5(x+5z)}{12}v{u}_{xz}-\frac{{\gamma }_5}{2}vz{u}_{zz}\hfill \\ & -\frac{2}{3}{v}_x{u}_{xx}-\frac{x+5z}{6}{u}_{xx}{v}_{xx}-z{u}_{xx}{v}_{xz},\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_5^z=& \frac{1}{6}zv{u}_{xt}+\frac{1}{2}zv{u}_{yt}+3{\gamma }_{1}zv{u}_y{u}_{xx}+3{\gamma }_{1}zv{u}_{xy}{u}_x+{\gamma }_{2}zv{u}_{xx}+z({\gamma }_{3}v-{\gamma }_{5}v)u_{yy}\hfill \\ & +{\gamma }_{4}zv{u}_{xy}+z{u}_{xx}{v}_{xy}+\frac{1}{12}{\gamma }_5{v}_{y}u+\frac{({\gamma }_4-{\gamma }_5){\gamma }_5}{9{\gamma }_1}{v}_{y}x+\frac{{\gamma }_2{\gamma }_5}{9{\gamma }_1}{v}_{y}y+\frac{x+5z}{12}{\gamma }_5{v}_y{u}_x\hfill \\ & +\frac{y+z-{\gamma }_{5}t}{2}{\gamma }_5{v}_y{u}_y+{\gamma }_5{v}_{y}z{u}_z+\frac{1}{12}u{v}_t+\frac{1}{9{\gamma }_1}({\gamma }_4-{\gamma }_5)x{v}_t+\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_t\hfill \\ & +\frac{x+5z}{12}{v}_t{u}_x+\frac{y+z-{\gamma }_{5}t}{2}{v}_t{u}_y+z{v}_t{u}_z-\frac{1}{12}v{u}_t+\frac{{\gamma }_5}{2}v{u}_y-\frac{x}{12}v{u}_{xt}\hfill \\ & -\frac{y-{\gamma }_{5}t}{2}v{u}_{yt}-\frac{1}{3}{\gamma }_{5}v{u}_y-\frac{{\gamma }_2{\gamma }_5}{9{\gamma }_1}v-\frac{x+5z}{12}{\gamma }_{5}v{u}_{xy}-\frac{y-{\gamma }_{5}t}{2}{\gamma }_{5}v{u}_{yy},\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_5^t=& \frac{1}{6}u{v}_x+\frac{1}{18{\gamma }_1}({\gamma }_4-{\gamma }_5)x{v}_x+\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_x+\frac{x+5z}{12}{u}_x{v}_x+\frac{y+z-{\gamma }_{5}t}{4}{u}_y{v}_x+\frac{1}{2}z{u}_z{v}_x\hfill \\ & +\frac{1}{6}u{v}_y+\frac{1}{9{\gamma }_1}({\gamma }_4-{\gamma }_5)x{v}_y+\frac{{\gamma }_2}{9{\gamma }_1}y{v}_y+\frac{x+5z}{12}{u}_x{v}_y+\frac{y+z-{\gamma }_{5}t}{4}{u}_y{v}_y+\frac{1}{2}z{u}_z{v}_y\hfill \\ & +\frac{1}{6}u{v}_z+\frac{1}{9{\gamma }_1}({\gamma }_4-{\gamma }_5)x{v}_z+\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_z+\frac{x+5z}{12}{u}_x{v}_z+\frac{y+z-{\gamma }_{5}t}{4}{u}_y{v}_z+\frac{1}{2}z{u}_z{v}_z\hfill \\ & -\frac{({\gamma }_4-{\gamma }_5)}{18{\gamma }_1}v-\frac{x+5z}{12}v{u}_{xx}-\frac{y+z-{\gamma }_{5}t}{4}v{u}_{xy}-\frac{1}{2}zv{u}_{xz}-\frac{{\gamma }_2}{9{\gamma }_1}v-\frac{x+5z}{12}v{u}_{xy}\hfill \\ & -\frac{y+z-{\gamma }_{5}t}{4}v{u}_{yy}-\frac{1}{2}zv{u}_{zy}-\frac{7}{12}v{u}_x-\frac{7}{12}v{u}_y-\frac{7}{12}v{u}_z-\frac{x+5z}{12}v{u}_{xz}\hfill \\ & -\frac{y+z-{\gamma }_{5}t}{4}v{u}_{yz}-\frac{1}{2}zv{u}_{zz},\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_6^x=& \frac{x-z}{6}v{u}_{yt}+\frac{x-z}{6}v{u}_{zt}+\frac{x-z}{6}{\gamma }_{3}v{u}_{yy}+\frac{x-z}{6}{\gamma }_{5}v{u}_{zy}+\frac{1}{12}u{v}_t+\frac{({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_t\hfill \\ & +\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_t+\frac{x-z}{6}{u}_x{v}_t+\frac{y-z+{\gamma }_{5}t}{4}{u}_y{v}_t+\frac{1}{2}t{u}_t{v}_t+\frac{1}{2}{\gamma }_{1}u{v}_x{u}_y+\frac{({\gamma }_4-{\gamma }_5)}{3}x{v}_x{u}_y\hfill \\ & +\frac{2{\gamma }_2}{3}y{v}_x{u}_y+\frac{{\gamma }_1(x-z)}{2}{v}_x{u}_y{u}_x+\frac{3{\gamma }_1(y-z+{\gamma }_{5}t)}{2}{v}_x{u}_y^2+3{\gamma }_{1}t{v}_x{u}_y{u}_t+\frac{{\gamma }_2}{6}u{v}_x\hfill \\ & +\frac{({\gamma }_4-{\gamma }_5){\gamma }_2}{9{\gamma }_1}x{v}_x+\frac{2{\gamma }_2^2}{9{\gamma }_1}y{v}_x+\frac{{\gamma }_2(x-z)}{6}{u}_x{v}_x+\frac{{\gamma }_2(y-z+{\gamma }_{5}t)}{2}{u}_y{v}_x+{\gamma }_{2}t{u}_t{v}_x\hfill \\ & +\frac{1}{6}u{v}_{xxy}+\frac{({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_{xxy}+\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_{xxy}+\frac{x-z}{6}{u}_x{v}_{xxy}+\frac{y-z+{\gamma }_{5}t}{2}{u}_y{v}_{xxy}\hfill \\ & +\frac{{\gamma }_1}{2}uv{u}_{xy}+\frac{({\gamma }_4-{\gamma }_5)}{6}xv{u}_{xy}+\frac{{\gamma }_2}{3}yv{u}_{xy}+\frac{{\gamma }_1(x-z)}{2}v{u}_{xy}{u}_x+\frac{3}{2}{\gamma }_{1}tv{u}_{xy}{u}_t\hfill \\ & +\frac{3{\gamma }_1(y-z+{\gamma }_{5}t)}{4}v{u}_{xy}{u}_y+\frac{({\gamma }_4-{\gamma }_5)}{6}x{v}_y{u}_x+\frac{{\gamma }_2}{3}y{v}_y{u}_x+\frac{{\gamma }_1(x-z)}{2}{v}_y{u}_x^2\hfill \\ & +\frac{3{\gamma }_1(y-z+{\gamma }_{5}t)}{4}{v}_y{u}_x{u}_y+\frac{3}{2}{\gamma }_{1}t{v}_y{u}_x{u}_t+\frac{{\gamma }_4}{6}u{v}_y+\frac{{\gamma }_4({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_y+\frac{{\gamma }_2{\gamma }_4}{9{\gamma }_1}y{v}_y\hfill \\ & +\frac{{\gamma }_4(x-z)}{6}{v}_y{u}_x+\frac{{\gamma }_4(y-z+{\gamma }_{5}t)}{4}{v}_y{u}_y+\frac{{\gamma }_4}{2}t{v}_y{u}_t-\frac{7}{12}v{u}_t-\frac{{\gamma }_5}{4}v{u}_y-{\gamma }_{1}v{u}_y{u}_x\hfill \\ & -\frac{y-z+{\gamma }_{5}t}{4}v{u}_{yt}-\frac{1}{2}tv{u}_{tt}-\frac{({\gamma }_4-{\gamma }_5)}{3}v{u}_y-3{\gamma }_{1}tv{u}_y{u}_{xt}-\frac{3{\gamma }_1(y-z+{\gamma }_{5}t)}{2}v{u}_y{u}_{xy}\hfill \\ & -\frac{{\gamma }_2}{3}v{u}_x-\frac{{\gamma }_2({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}v-\frac{{\gamma }_2(y-z+{\gamma }_{5}t)}{2}v{u}_{xy}-{\gamma }_{2}tv{u}_{xt}-v_y{u}_{xx}-\frac{x-z}{6}{u}_{xx}{v}_{xy}\hfill \\ & -\frac{y-z+{\gamma }_{5}t}{2}{u}_{xx}{v}_{yy}-t{u}_{xx}{v}_{yt}-\frac{1}{3}{u}_x{v}_{xy}-\frac{({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}{v}_{xy}-\frac{y-z+{\gamma }_{5}t}{2}{u}_{xy}{v}_{xy}\hfill \\ & -{\gamma }_{1}v{u}_x{u}_y-\frac{{\gamma }_2}{3}v{u}_x-\frac{3{\gamma }_1(y-z+{\gamma }_{5}t)}{4}v{u}_x{u}_{yy}-\frac{3{\gamma }_1}{2}tv{u}_x{u}_{yt}-\frac{{\gamma }_4}{3}v{u}_y-\frac{{\gamma }_2{\gamma }_4}{9{\gamma }_1}v\hfill \\ & -\frac{{\gamma }_4(y-z+{\gamma }_{5}t)}{4}v{u}_{yy}-\frac{{\gamma }_4}{2}vt{u}_{yt}+t{u}_t{v}_{xxy}+\frac{{\gamma }_1}{2}u{v}_y{u}_x-t{u}_{xt}{v}_{xy},\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_6^y=& \frac{y-z+{\gamma }_{5}t}{2}v{u}_{xt}+\frac{3{\gamma }_1(y-z+{\gamma }_{5}t)}{2}v{u}_y{u}_{xx}+\frac{{\gamma }_2(y-z+{\gamma }_{5}t)}{2}v{u}_{xx}+\frac{{\gamma }_4-{\gamma }_5}{9{\gamma }_1}x{v}_t\hfill \\ & +\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_t+\frac{x-z}{12}{u}_x{v}_t+\frac{y-z+{\gamma }_{5}t}{2}{u}_y{v}_t+\frac{1}{2}t{u}_t{v}_t+\frac{{\gamma }_1}{2}u{u}_x{v}_x+\frac{({\gamma }_4-{\gamma }_5)}{3}x{u}_x{v}_x\hfill \\ & +\frac{{\gamma }_1(x-z)}{4}{u}_x^2{v}_x+\frac{3{\gamma }_1(y-z+{\gamma }_{5}t)}{2}{u}_x{u}_y{v}_x+3{\gamma }_{1}t{u}_x{u}_t{v}_x+\frac{{\gamma }_4}{6}u{v}_x+\frac{{\gamma }_4({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_x\hfill \\ & +\frac{2{\gamma }_2{\gamma }_4}{9{\gamma }_1}y{v}_x+\frac{{\gamma }_4(x-z)}{12}{u}_x{v}_x+\frac{{\gamma }_4(y-z+{\gamma }_{5}t)}{2}{u}_y{v}_x+\frac{{\gamma }_4}{2}t{u}_t{v}_x+\frac{{\gamma }_3({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_y\hfill \\ & +\frac{{\gamma }_3}{6}u{v}_y+\frac{2{\gamma }_2{\gamma }_3}{9{\gamma }_1}y{v}_y+\frac{{\gamma }_3(x-z)}{6}{u}_x{v}_y+\frac{{\gamma }_3(y-z+{\gamma }_{5}t)}{2}{u}_y{v}_y+{\gamma }_{3}t{u}_t{v}_y+\frac{{\gamma }_5}{6}u{v}_z\hfill \\ & +\frac{{\gamma }_5({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_z+\frac{2{\gamma }_2{\gamma }_5}{9{\gamma }_1}y{v}_z+\frac{{\gamma }_5(x-z)}{12}{u}_x{v}_z+\frac{{\gamma }_5(y-z+{\gamma }_{5}t)}{2}{u}_y{v}_z+\frac{{\gamma }_5}{2}tu{v_z}_t\hfill \\ & -\frac{7}{12}v{u}_t-\frac{{\gamma }_5}{2}v{u}_y-\frac{1}{2}tv{u}_{tt}-\frac{x-z}{12}v{u}_{xt}-\frac{{\gamma }_1}{2}v{u}_x^2-\frac{({\gamma }_4-{\gamma }_5)}{3}v{u}_x-\frac{{\gamma }_1(x-z)}{4}v{u}_x{u}_{xx}\hfill \\ & -3{\gamma }_{1}tv{u}_x{u}_{xt}-\frac{{\gamma }_4({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}v-\frac{{\gamma }_4}{6}v{u}_x-\frac{{\gamma }_4(x-z)}{12}v{u}_{xx}-\frac{{\gamma }_4}{2}tv{u}_{xt}-\frac{x-z}{6}{u}_{xx}{v}_{xx}\hfill \\ & -\frac{2}{3}{u}_{xx}{v}_x-t{u}_{xx}{v}_{xt}-\frac{{\gamma }_3}{3}v{u}_y-\frac{2{\gamma }_2{\gamma }_3}{9{\gamma }_1}v-\frac{{\gamma }_3(x-z)}{6}v{u}_{xy}-{\gamma }_{3}tv{u}_{yt}+\frac{{\gamma }_5}{12}v{u}_x+\frac{{\gamma }_5}{2}v{u}_y\hfill \\ & -\frac{{\gamma }_5}{6}v{u}_z-\frac{{\gamma }_5(x-z)}{12}v{u}_{xz}-\frac{{\gamma }_5}{2}vt{u}_{zt}+\frac{2{\gamma }_2}{3}y{u}_x{v}_x+\frac{1}{12}u{v}_t,\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_6^z=& \frac{1}{6}u{v}_t+\frac{({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_t+\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_t+\frac{x-z}{12}{u}_x{v}_t+\frac{y-z+{\gamma }_{5}t}{4}{u}_y{v}_t+\frac{1}{2}t{u}_t{v}_t\hfill \\ & +\frac{{\gamma }_5({\gamma }_4-{\gamma }_5)}{9{\gamma }_1}x{v}_y+\frac{{\gamma }_2{\gamma }_5}{9{\gamma }_1}y{v}_y+\frac{x-z}{12}{\gamma }_5{v}_y{u}_x+\frac{y-z+{\gamma }_{5}t}{4}{\gamma }_5{v}_y{u}_y\hfill \\ & -\frac{7}{12}v{u}_t-\frac{x-z}{12}v{u}_{xt}-\frac{y-z+{\gamma }_{5}t}{4}v{u}_{yt}-\frac{1}{2}vt{u}_{tt}-\frac{7}{12}{\gamma }_{5}v{u}_y-\frac{{\gamma }_2{\gamma }_{5}v}{9{\gamma }_1}\hfill \\ & -\frac{x-z}{12}{\gamma }_{5}v{u}_{xy}-\frac{y-z+{\gamma }_{5}t}{4}{\gamma }_{5}v{u}_{yy}-\frac{1}{2}{\gamma }_{5}tv{u}_{yt}+\frac{1}{6}{\gamma }_5{v}_{y}u+\frac{{\gamma }_{5}t}{2}{v}_y{u}_t,\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_6^t=& 3{\gamma }_{1}tv{u}_y{u}_{xx}+3{\gamma }_{1}tv{u}_{xy}{u}_x+{\gamma }_{2}tv{u}_{xx}+{\gamma }_{3}tv{u}_{yy}+{\gamma }_{4}tv{u}_{xy}+{\gamma }_{5}tv{u}_{zy}+t{u}_{xx}{v}_{xy}\hfill \\ & +\frac{({\gamma }_4-{\gamma }_5)x}{18{\gamma }_1}{v}_x+\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_x+\frac{x-z}{12}{u}_x{v}_x+\frac{y-z+{\gamma }_{5}t}{4}{u}_y{v}_x+t{u}_t{v}_x+\frac{1}{6}u{v}_y\hfill \\ & +\frac{({\gamma }_4-{\gamma }_5)x}{9{\gamma }_1}{v}_y+\frac{{\gamma }_2}{9{\gamma }_1}y{v}_y+\frac{x-z}{12}{u}_x{v}_y+\frac{y-z+{\gamma }_{5}t}{4}{u}_y{v}_y+t{u}_t{v}_y+\frac{1}{12}u{v}_z\hfill \\ & +\frac{({\gamma }_4-{\gamma }_5)x}{9{\gamma }_1}{v}_z+\frac{2{\gamma }_2}{9{\gamma }_1}y{v}_z+\frac{x-z}{6}{u}_x{v}_z+\frac{y-z+{\gamma }_{5}t}{2}{u}_y{v}_z+t{u}_t{v}_z-\frac{1}{12}v{u}_x\hfill \\ & -\frac{({\gamma }_4-{\gamma }_5)}{18{\gamma }_1}v-\frac{x-z}{12}v{u}_{xx}-\frac{y-z+{\gamma }_{5}t}{4}v{u}_{xy}-\frac{1}{12}v{u}_y-\frac{{\gamma }_2}{9{\gamma }_1}v-\frac{x-z}{12}v{u}_{xy}\hfill \\ & -\frac{y-z+{\gamma }_{5}t}{4}v{u}_{yy}-\frac{1}{12}v{u}_z-\frac{x-z}{6}v{u}_{xz}-\frac{y-z+{\gamma }_{5}t}{2}v{u}_{zy}+\frac{1}{6}u{v}_x,\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_7^x=& \frac{1}{2}v{f}_1^{{}^{\prime }}\left(t\right)-\frac{1}{2}{v}_t{f}_1\left(t\right)-3{\gamma }_1{v}_x{u}_y{f}_1\left(t\right)-{\gamma }_2{v}_x{f}_1\left(t\right)-v_{xxy}{f}_1\left(t\right)-3{\gamma }_{1}v{u}_{xy}{f}_1\left(t\right)\hfill \\ & -3{\gamma }_1{v}_y{u}_x{f}_1\left(t\right)-{\gamma }_4{v}_y{f}_1\left(t\right),\hfill \\ \hfill C_7^y=& \frac{1}{2}v{f}_1^{{}^{\prime }}\left(t\right)-\frac{1}{2}{v}_t{f}_1\left(t\right)-3{\gamma }_1{v}_x{u}_x{f}_1\left(t\right)-{\gamma }_4{v}_x{f}_1\left(t\right)-{\gamma }_3{v}_y{f}_1\left(t\right)-{\gamma }_5{v}_z{f}_1\left(t\right),\hfill \\ \hfill C_7^z=& \frac{1}{2}v{f}_1^{{}^{\prime }}\left(t\right)-\frac{1}{2}{v}_t{f}_1\left(t\right)-{\gamma }_5{v}_y{f}_1\left(t\right),\hfill \\ \hfill C_7^t=& -v_x{f}_1\left(t\right)-v_y{f}_1\left(t\right)-v_z{f}_1\left(t\right),\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_8^x=& -v_t{f}_2\left(z\right)-3{\gamma }_1{v}_x{u}_y{f}_2\left(z\right)-{\gamma }_2{v}_x{f}_2\left(z\right)-v_{xxy}{f}_2\left(z\right)-3{\gamma }_{1}v{u}_{xy}{f}_2\left(z\right)\hfill \\ & -3{\gamma }_1{v}_y{u}_x{f}_2\left(z\right)-{\gamma }_4{v}_y{f}_2\left(z\right),\hfill \\ \hfill C_8^y=& \frac{1}{2}{\gamma }_{5}v{f}_2^{{}^{\prime }}\left(z\right)-v_t{f}_2\left(z\right)-3{\gamma }_1{v}_x{u}_x{f}_2\left(z\right)-{\gamma }_4{v}_x{f}_2\left(z\right)-{\gamma }_3{v}_y{f}_2\left(z\right)-\frac{1}{2}{\gamma }_5{v}_z{f}_2\left(z\right),\hfill \\ \hfill C_8^z=& -v_t{f}_2\left(z\right)-{\gamma }_5{v}_y{f}_2\left(z\right),\hfill \\ \hfill C_8^t=& \frac{1}{2}v{f}_2^{{}^{\prime }}\left(z\right)-v_x{f}_2\left(z\right)-v_y{f}_2\left(z\right)-\frac{1}{2}{v}_z{f}_2\left(z\right).\hfill \end{array}\end{array}$$

The conservation vectors obtained via symmetrics $V_1$, $V_2$, $V_3$ and $V_4$ in this paper are similar to those obtained by the same symmetrics in [22,23]. This not only demonstrates that we have successfully applied the new conservation laws of Ibragimov and verified the rationality of the obtained exact analytical solutions, but it also shows that Equation (7) has some similar properties to the main equations in [22,23]. More enlightening, in the future study of Equation (7), we can also use the simplified Hirota method to calculate its soliton solutions and analyze its integrability.

5. Conclusions

In the paper, we generalize a new (3+1)-dimensional shallow water wave equation of Wazwaz, in which power series exact solutions and conservation laws are discussed. The Lie symmetry method transforms the equation into nodes that are easy to solve. Owing to the symmetry reduction, the independent variables of the equation can be reduced, so we transform the equation into NLODEs. Power series can be used to solve complex NLODEs with non-constant coefficients, which is a very convenient and effective method. In addition, the accuracy of the numerical solution can be verified by comparing the images of the same types of solutions. Based on Lie symmetries and the new conservation laws of Ibragimov, we establish the conservation vectors of the equation. We observe that the exact analytical solutions contain constant coefficients, so we guess that the coefficients of the equation are related to translation invariance. We only provide the exact analytical solutions obtained by Lie symmetry analysis and power series, which are only a small part of all solutions of the equation. Future research can consider whether it is a Painlevé integrable equation, and the soliton solutions and dynamic behavior of various waves, etc.