Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

In this article, (3+1)-dimensional generalized Shallow Water-like (gSWl) equation is discussed. The infinitesimal generators of the equation are derived by using the Lie symmetry analysis method. The optimal system is obtained based on the adjoint table of the generators of the equation. Exact solutions of the equation are constructed by applying symmetry reduction, Exp − ϕ ( ξ ) expansion method, Exp-function expansion method, Riccati equation method, and G ′ / G expansion method. For analyzing the dynamical behavior of the solutions, we derive the physical structures of dark soliton, kink wave, and periodic solutions via numerical simulations.

The following (3 + 1)-dimensional generalized Shallow Water equation u xxxy − 3u xx u y − 3u x u xy + u yt − u xz 0, (1.1) has been studied by many approaches. Huang and Gao [20] derived the one-, two-and threesoliton solutions of the equation by the Hirota method, and deduced the propagation and interaction of the soliton solutions. In [21], Huang studied the stability of solitons by numerical methods and noticed that the soliton amplitude magnitude is affected by the spectral parameters. In [22], the closed-form solutions of the equation were derived by Lie symmetry, and the soliton solutions were found through the optimal system. Based on the auto-Bäcklund transformation, Li and Liu [23] constructed the multi-periodic solitons of Eq. 1.1 through the variable-coefficient homogeneous balance method and investigated the propagation and interactions of the solutions. In [24], Liu deduced the new periodic solitary solutions of Eq. 1.1 by the direct test function method, and the validity of the direct test function method was shown.
Liu and Zhu [25] investigated the variable coefficients of the gSW equation by the Hirota bilinear method and constructed a large number of breather wave solutions. Tang, Ma and Xu [26] proposed the (3 + 1)-dimensional generalized Shallow Water-like (gSWl) equation u xxxy + 3u xx u y + 3u x u xy − u yt − u xz 0, (1.2) which can be derived by rewriting Eq. 1.1 on the scale x → −x. In [26], the Grammian and Pfaffian solutions of Eq. 1.2 were obtained and the equations were extended with the Pfaffianization method. Kumar et al. [27] derived the multi-stripe and breathing wave solutions of Eq. 1.2 by the bilinear method, combining the quadratic function and hyperbolic cosine method, the behavior between the one-block and multi-stripe solutions were obtained. Sadat et al. [28] applied symbolic calculations to yield lump-type and stripe solutions of Eq. 1.2. Zhang et al. [29] applied the generalized bilinear operator method and obtained the rational and lump solutions of Eq. 1.2. The shallow water wave equation plays an essential role in marine engineering, environmental problems, and ecology, so it is valuable to derive the exact solutions of the shallow water wave equation. Employing the Lie symmetry method to yield exact solutions of the (3 + 1)-dimensional gSWl equation has not been studied. In this paper, the Lie symmetry analysis method is applied to investigate the solutions of Eq. 1.2. Lie symmetry method [30][31][32][33][34] has an important significance for solving partial differential equations (PDEs). Applying the Lie symmetry method, the symmetry group of the equation can be derived, furthermore, the equation can be similarly reduced and the new solutions of the equation can be yielded by the symmetry transformation. The Lie symmetry method can reduce the order of the equation when solving with higher order equations, which is difficult to accomplish by other methods.
The structure of the rest of the paper is as follows: In Sect 2, the infinitesimal generators are obtained by applying the Lie group transformation to the (3 + 1)-dimensional gSWl equation. In Sect 3, the optimal system for Eq. 1.2 is derived under the basis of the adjoint table. The periodic wave, kink wave and soliton solutions of the equation are derived by Exp(−ϕ(ξ)) expansion method, Exp-function expansion method, Riccati equation method, and (G′/G) expansion method in Sect 4. The dynamical behavior of the soliton wave solutions of the gSWl equation are analyzed in Sect 5. The conclusions are given in Sect 6.
2 Lie symmetry analysis for the (3 + 1) gSWl equation The key step for solving non-linear PDEs by Lie symmetry group method is to obtain Lie algebra of the equation. Consider the following one-parameter Lie group transformation: where ε is a parameter, and ε ≪ 1. ξ, η, φ, τ, and ϕ are infinitesimal generators concerning x, y, z, t and u. The one-parameter vector field V of gSWl equation can be written as The vector field V satisfies in which Δ = u xxxy + 3u xx u y + 3u x u xy − u yt − u xz and pr (4) is the fourth prolongation of V. The fourth prolongation of Eq. 1.2 can be derived as The invariant condition can be given as ϕ xxxy + 3ϕ xx u y + 3u x ϕ xy − ϕ yt − ϕ xz + 3u xx ϕ y + 3ϕ x u xy 0. (2.5) Based on Eq. 2.5, the system of determining equations can be given by (2.6) By solving the above equations we can derive where c i and F i (i = 1, 2, 3, 4) are arbitrary constants and functions, respectively. Assume that F 1 (z) 0, F 2 (t) c 5 , F 3 (z) 0, F 4 (z, t) c 6 . The infinitesimal generators have new forms (2.8) [ Frontiers in Physics frontiersin.org 02 3 Optimal systems of one-dimensional subalgebras Based on the Lie brackets, the optimal system of onedimensional subalgebras of the equation can be deduced. By the linear combination of subalgebras, a new form is given by ( 3 .1) By Olver theory [30], using symbolic calculations The adjoint table is shown in Table 2.

Construction of group invariants
The exchange and adjoint relations of the six-dimensional Lie algebras are given in Table 1 and Table 2, respectively. Assume that the vectors V in which k k(a 1 , /a 6 , s 1 , . . . , s 6 ) can be derived from Table 1. The values of k were calculated from Table 1 as follows For any s j (j = 1, 2, 3, 4, 5, 6), it have required Gather the coefficients containing s j in the above equation, the following system of differential equations are deduced as After analyzing the above system of PDEs (3.5), it is not difficult to yield that the invariant function as χ(a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) F(a 1 , a 3 ).

One-dimensional optimal system
For J ε n : _ j → _ j defined by l → Ad(exp(ε i l i )s) is a linear map [35], in which n = 1, . . . , 6. The matrix M ε n of J ε n with respect to basis to v 1 , . . . , v 6 { }are deduced below Then, the matrix M can be yielded by The matrix M can be written as The adjoint transformation equation for Eq. 1.2 is Ad Frontiers in Physics frontiersin.org (3.9) By applying the invariants a 1 and a 3 , discuss the situations of the following Lie algebras.
Similarly, the other terms of the optimal system of Eq. 1.2 can be obtained by the above method. All of them are listed below.Single vector fields:

Exact solutions of the gSWl equation
Next, the exact solutions of the gSWl equation are derived by employing the optimal system. The similarity solutions for arbitrary vector field v in the optimal system can be solved by the Lagrange's system. U (x, y, z, t) = F (α, β, δ).in which α = x, β y z , δ = t. Taking the above similarity solution into Eq. 1.2, the reduced NLPDE is given as Similarly, applying the Lie symmetry method, the infinitesimal generators of Eq. 4.3 can be derived which has the similarity solutions from where P = α − δ and Q = 3α − β.
Putting it into Eq. 4.3, the following reduced equation can be yield Repeating the above steps, we get The new similarity solutions from where ϖ = 3P − Q. Replacing (4.10) into Eq. 4.7, we get 3k ϖϖ = 0. The solution of Eq. 1.2 via the above method can be given as in which c 1 and c 2 are constants.

Vector field v 3
The characteristic equation can be composed as The derived similarity solution has the form as.

Vector field
The characteristic equation can be composed as Solving (4.47), we derived the similarity solution u x, y, z, t F α, β, θ , (4.48) in which α = x − t, β = y and θ = z − t are similarity variables. Taking (4.48) into Eq. 1.2, the (2 + 1)-dimensional equation can be yielded Next, applying the Riccati equation method, different forms of solutions of Eq. 4.49 can be deduced. Taking the following traveling wave transform where k, l, m are constants. Substituting (Eq. 4.50) into Eq. 4.49 and integrating once yields where a j (j = 1 p) are constants which can be obtained later and h(υ) satisfies the equation in which ω is an constant. The form of the solutions of Eq. 4.53 are as follows  On the basis of Eq. 4.56, we derive the solution of Eq. 1.2 as follows: For ω < 0, where k, m, a 0 , ω, y, z are constants.

FIGURE 2
Annihilation of the kink wave solution of (4.34) at y =1.

FIGURE 6
The symmetric two-periodic solution of (4.68).

Vector field
The characteristic equation can be composed as where α = x, β = y and θ = z − t are similarity variables. Taking (Eq. 4.60) into Eq. 1.2, the (2 + 1)-dimensional equation can be obtained by Taking the traveling wave transform    where m, s 0 , s 1 , s 2 , r 0 , r 1 and r 2 are constants. Based on Eq. 4.67, replacing the parameter k = ik, l = il, m = im and picking the real part, the following periodic wave solution can be given ) . (4.68)

Analysis and discussion
In this part, the geometric representation of the solution of Eq. 1.2 is discussed by employing graphical description. The physical phenomena of the solutions can be seen more obviously via numerical simulation. The solutions of the gSWl equation yielded from the above process include periodic, dark soliton, kink wave and annihilation structures of solutions. The dynamic structure of the solutions is investigated below. Figure 1 depicts the physical structure of the singular solution when the parameter c 1 = 1, = 1, x = 1, y = 1. (B) Indicates the density plot of the corresponding solution. Figure 2 describes the physical structure of the kink solution when t = 1, and the rest of the parameters take the value of y = 1, = 3, = 1, k = 1, l = 1, m = 1, = 1, = 1. When the time increases from t = 1 FIGURE 7 Dark soliton solution of (4.67).
Frontiers in Physics frontiersin.org to t = 28, the energy of the wave is gradually depleted and eventually becomes a plane wave. The physical structure of the antisymmetric periodic solution (4.35) is shown in Figure 3. The 3-D plot of the antisymmetric periodic solution is described when the parameter is taken as z = 0, y = 0, = 1, = 1, k = 1, l = 1, m = 1, = 1, = −1. (B) show the density plot of the solution.
The dynamics structure of the kink wave solution at z = 0 is plotted in Figure 4. When k = −10, c = 10, = 1, = −10, y = 1. (A) shows the 3-D plot of the solution and (B) depicts the spread route of the solution along the x-axis when t = 0, t = 1, t = 2 and t = 3, respectively.
It is shown in Figure 5 and Figure 6 that the physical structure of the periodic wave solutions (4.58) and (4.68). (A) Is the corresponding 3D structure, (B) is the track of the solution along the x-axis, which is given when the parameterk = 1, = −1, = 1, r = 1, y = 0, z = 0 (4.68) shows the 3-D structure of the symmetric twoperiod wave solution, with the corresponding parameter a 0 = 1, = 1, = 1, = 5, = 1, = 1, m = 1. (B) Depicts the spread route of the solution along the z-axis at t = 0.
A structure of the dynamics of the dark soliton (4.67) is depicted in Figure 7. The 3-D plot of the dark soliton is obtained when the parameter is selected as a 0 = 1, = 1, = 1, = 1, = 2, = 1, m = 1. The spread route behavior of the dark soliton along the z-axis can be derived by choosing t = 0, t = 1, t = 2 and t = 3.

Conclusion
In summary, the (3 + 1)-dimensional generalized Shallow Water-like wave equation is shown in this paper which is studied based on the Lie symmetry method and the symbolic calculation. By the adjoint table of the infinitesimal generators, a one-dimensional optimal system is formulated. In terms of the optimal system, some new solutions of the gSWl equation are derived by Exp(−ϕ(ξ)) expansion method, Riccati equation method, Exp-function expansion method, and (G′/G) expansion method. In particular, the physical structures of the detected dark soliton, kink wave, and periodic solutions are investigated to make this study more credible.
In this work, a situation of the (3 + 1)-dimensional gSWl equation has been investigated based on the Lie symmetry method, and the rest of the latter cases are presented in other subsequent papers. More work needs to be done in the future. Firstly, in this paper, the exact solutions of the equation are derived richly with the Lie symmetry method, and other methods can be employed for the solutions of the equation, such as the numerical analysis method [36][37][38]. Secondly, the natural properties of the solutions to the equation can be investigated further in subsequent studies through the generalized multi-symplectic method and the structure-preserving method [39][40][41][42].

Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.