ABUNDANT NEW NON-TRAVELING WAVE SOLUTIONS FOR THE (3+1)-DIMENSIONAL BOITI-LEON-MANNA-PEMPINELLI EQUATION ∗

Seeking exact solutions of higher-dimensional nonlinear partial differential equations has recently received tremendous attention in mathematics and physics. In this paper, we investigate exact solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation which describes nonlinear wave propagation in incompressible fluid. Firstly, by means of extended homoclinic test approach, we get eight kinds of non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Then, combining the improved tanh function method and new ansatz solutions, we obtain abundant new exact non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. These results include not only many results obtained in other literatures, but also some new exact non-traveling wave solutions. Moreover, the exact kink wave solutions, periodic solitary wave solutions and singular solitary wave solutions are given when arbitrary functions contained in these solutions are taken as some special functions.


Introduction
Many significant phenomena in physics and engineering are represented by nonlinear partial differential equations (NLPDEs). Compared with the low-dimensional systems, the higher-dimensional nonlinear partial differential systems have more complex behaviors. The investigation of the explicit solutions for NLPDEs plays an important role in the study of nonlinear phenomena. Recently, in order to explore exact solutions of NLPDEs and figure out these phenomena in nature, a mass of methods have been established, for instance, bilinear method, Darboux transformations, symmetry reductions, and so on. Using (G ′ /G)-expansion method, the ansatz (positive quadratic and exponential functions) technique, the generalized uni-fied method and Lie symmetry technique, Refs. [5,9,13] obtained nonautonomous complex wave solutions, exact solutions, double-wave solutions, non-traveling wave solutions et al. for (2+1)-dimensional and (3+1)-dimensional nonlinear partial differential equations.
As we known, traveling wave solution, which describes the evolution of physical quantities, is a very special solution of partial differential equations. More solutions of NLPDEs are non-traveling wave solutions. Seeking exact non-traveling wave solutions of nonlinear partial differential equations has recently received tremendous attention in mathematics and physics. Using (G ′ /G)-expansion method, Guo [1] and Liu [6] studied non-traveling wave solutions of (2+1)-dimensional Painlevé integrable Burgers equation, (2+1)-dimensional breaking soliton equation and (3+1)dimensional generalized shallow water equation. Lin [3] got non-traveling wave solutions for (2+1)-dimensional Burgers equation by means of the generalized direct ansatz method and different test functions. Shang [11,15] have studied the non-traveling wave solutions of (3+1)-dimensional potential-YTSF equation and Calogero equation by combining the extended homoclinic test approach with the method of separation of variables. Sheng [17] and Zhang [21] obtained exact nontraveling wave solutions of (2+1)-dimensional KD equation and (3+1)-dimensional KP equation with the aid of symbolic computation. Therefore, it is meaningful to consider the non-traveling wave solutions of higher-dimensional nonlinear partial differential equations.
As one of the most important higher-dimensional NLPDEs, the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation describes nonlinear wave propagation in incompressible fluid, where u is a function depending on spatial variables (x, y, z) and temporal variable t. The researchers have obtained some types of traveling wave solutions to Eq.(1.1) by many methods, such as bilinear method [2,14,16], extended three-wave approach [7], extended homoclinic test technique [4,18], extended three-soliton method [12], Painlevé analysis [19], modified exponential function method [20], respectively. Liu [8] and Luo [10] obtained non-traveling wave solutions of Eq.(1.1) by using (G ′ /G)-expansion method and Bäcklund transformation, respectively. Although many researchers investigated exact traveling wave solutions of Eq.(1.1), there are few results on the non-traveling wave solutions. In this paper, we will investigate non-traveling wave solutions of Eq.(1.1) by utilizing the extended homoclinic test approach [22] and the improved tanh function method [21]. The rest of this paper is organized as follows: In section 2, eight kinds of nontraveling wave solutions are constructed by using the extended homoclinic test approach. When arbitrary functions in these solutions are taken as some special functions, we will get kink solutions, periodic solitary wave solutions, singular solitary wave solutions and single solitary wave solutions, which can be seen in [12,18,19]. In section 3, by constructing new ansatz solutions of equation (1.1), we employ the improved tanh method to get twenty-seven types of non-traveling wave solutions for Eq.(1.1). Furthermore, we can get one hundred and eight types of non-traveling wave solutions for Eq.(1.1) based on the four forms of arbitrary functions η and b. Moreover, we give the graphic analyses of solutions obtained in sections 2. At last, some conclusions and discussions are given.

Extended homoclinic test approach
In this section, we will apply the extended homoclinic test approach to get eight kinds of non-traveling wave solutions of Eq.(1.1).

Improved tanh function method
In this section, we will obtain many non-traveling wave solutions of Eq.(1.1) by combining the improved tanh function method with a generalized Riccati equation.
Balancing the u xxxy and u x u xy , we construct the solution of the following new form where k, r, p and q are all real constants, a(y, z, t), b(y, z, t), c(y, z, t) and η(y, z, t) are all differentiable functions. As we known, Eq.(3.2) is the generalized Riccati equation and has 27 solutions which were given by Zhang et al. [21](see Appendix 1). Substituting (3.1) and (3.2) into (1.1), combining similar terms of ϕ(kx + η(y, z, t)), and setting each coefficient to zero, we have the following system of over-determined partial differential equations for a(y, z, t), b(y, z, t), c(y, z, t) and η(y, z, t) Solving the system of (3.3), we can get the following non-trivial results.
For simplification, in the rest of this paper, we take y, z, t).
Family 16: When q ̸ = 0, r = p = 0, one gets the solution where d is an arbitrary constant.

Graphic analyses
In section 2, we employ the extended homoclinic test approach to get eight types of solutions. These solutions have a tail. The tails in these solutions maybe give a prediction of physical phenomenon and the free parameters in these solutions of Eq. Here, through 3D graphic, we draw the cross sections of some solutions with a linear tail or a parabolic tail. Figure 1 and Figure 2 show the kink type solitary solution u 1 with a linear tail and kink-like type solution u 1 with a parabolic tail, respectively. Figure 1 and Figure 2 are kinking, but Figure 2 has a parabolic slot. Figure 3 and Figure 4 express the singular solitary wave solution u 2 with a linear tail and singular solitary wave-like solution u 2 with a parabolic tail. The white parts of Figure 3 and Figure  4 represent singularities. Figure 5 shows the periodic solitary wave-like solution u 3 with a parabolic tail. Figure 6 expresses the singular solitary wave-like solution u 7 with a parabolic tail.  Kink-like solution with a parabolic tail for u1 as a2 = −1, k2 = m = 1, n = 3, c = 0, θ(z) = z 2 , x = y = 0.    In a word, the figures of solutions will change greatly under the influence of a tail, such as u 1 , u 2 and u 7 , which have parabolic characteristics under the action of parabolic tail.

Conclusion and discussion
In conclusion, though there are various effective methods to solve the solutions of nonlinear partial differential equations during the past several decades, such as homogeneous balance method, tanh-function method, Hirota's bilinear operators method, F-expansion method, auxiliary equation method, Exp-function method and so on, these methods may only obtain one form or several forms of solutions. The extended homoclinic test approach (EHTA) used in this manuscript, which is based on the bilinear form of the nonlinear partial differential equations, is a fairly effective method to seek solutions. Moreover, applying extended homoclinic test approach, some new types of special solutions including breather type of soliton and two soliton, and periodic type of soliton can be obtained.
In this paper, we firstly employ the multi-linear variable separation approach to reduce (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (1.1) to (1+1)dimensional nonlinear equation with variable coefficients (2.2). Then, we introduce an appropriate transformation to simplify (2.2) to a constant coefficients equation. Furthermore, by the extended homoclinic test approach, we solve the simplified equation and obtain eight kinds of non-traveling wave solutions of (3+1)dimensional Boiti-Leon-Manna-Pempinelli equation in conditions of a 1 ∈ R or a 1 = k 3 i, k 3 ∈ R. Finally, combining the improved tanh function method with a generalized Riccati equation, we obtain abundant new exact non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. These solutions include kink-like type solutions, periodic solitary-wave-like type solutions, singular solitary wave-like type solutions, and so on. As arbitrary functions included in these solutions are taken as some special functions, kink-type solitons, singular solitary wave solutions, periodic solitary wave solutions and so on are presented. Also, the arbitrary functions in these solutions imply that these solutions have rich local structures. Therefore, the results obtained in this paper are the supplement and extension of the results of the existing literature [2,4,7,8,10,12,14,16,[18][19][20]. From our abundant results obtained in this paper, the methods employed here have been proved to be fairly effective methods for seeking non-traveling wave solutions of higher-dimensional nonlinear partial differential equations. It is expected that our results are helpful for theoretical study of the associated higher-dimensional nonlinear partial differential equations in mathematics and physics. Moreover, in practical applications, most of real nonlinear partial differential equations possess variable coefficients. The exact solutions of the variable coefficients nonlinear partial differential equations have a greater application value. As we known, there is few research on solutions for (3+1)-dimensional variable coefficients Boiti-Leon-Manna-Pempinelli equation. Next, we will set out to study the traveling wave solutions and non-traveling wave solutions of (3+1)-dimensional variable coefficients Boiti-Leon-Manna-Pempinelli(VC-BLMP) equation. As the special cases of (3+1)-VC-BLMP equation, several generalized BLMP equations with different form and the corresponding results of them will be given.