Some exact solutions of a new (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation

By using symbolic computation and complex method, we present abundant exact solutions of a new (3 + 1)-dimensional Boiti-Leon-Manna-Pemp-inelli equation, which contain the periodic solitary wave solutions, the interaction solutions between lump wave and solitary waves and the meromorphic solutions


Introduction
It is well acknowledged that the exploration of many significant natural and engineering problems can be intimately related to the study of nonlinear partial differential equations (NLPDEs). For example, KdV equation [1] is regarded as a dynamic model for long wave propagating in a channel, and Laplace equation [2] is applied to the research about the steady state heat conduction problem. For the study of incompressible fluid, i.e. fluids with negligibly density change during the process of flowing, Darvishi [3] introduced the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli (3D-BLMP) equation Since then, the exact solutions to this equation have been discovered through the extensive work of numerous scholars. For example, Ma [4] used Wronskian technology to get the Wronskian Determinant solutions, Liu considered more abundant soliton structures [5], and also obtained the three wave solutions [6]. Their methods of solving equations mainly included Lie symmetry method [7], [8], / -expansion method [9], Bäcklund transformation method [10], etc. Fluid properties such as continuity, fluidity etc. can be uncovered and explored by analyzing their solutions. The laws of nature can be explained more effectively by using NLPDEs.
Considering a more complicated case, Wazwaz [11] added another possible derivative u x to the derivative term u u y z + of the 3D-BLMP equation and got a new equation: where u u x y z t = ( , , , ). Furthermore, he used Painlevé analysis to obtain the compatibility condition and confirm its integrability. This work enlarged the category of integrable equations with timedependent coefficients. In the same article, the bilinear form and multiple soliton solutions of Eq. (1) were given, which greatly promote the study of it. Some breather wave, lump-type solutions and interaction solutions of Eq. (1) were obtained in Yuan's [12] and Liu's [13] papers respectively. After a precise analysis of the mathematical model, Khalid K. Ali [14] used the tanh function method and sine Gordon expansion method to find some new soliton solutions with satisfactory accuracy. Chen [15] found a variety of kink solutions using methods such as the three wave method. In particular, they also obtained the kink-shaped solitary wave solutions with a tail, which can be used to explain some physical phenomena. Recently, Mehwish Rani et al. [16] have obtained the kink and periodic rational solutions of the Eq. (1) through an improved tanh( ) 2 ϕ -expansion method, and have proved the effectiveness of the method.
According to the applicable field of Eq. (1), periodic solitary wave solutions can help to understand the wave fluctuation phenomenon in shallow water, which can occur in both rivers and seas. Moreover, it is well known that lump waves can be regarded as limiting forms of solitons and can propagate higher propagation energy [17], [18], [19]. So in the context of ocean, the appearance of lump waves may bring some disasters. Therefore, the study of lump waves is helpful to better understand and predict the possible extreme conditions in nonlinear systems. In regards to the NLPDEs in the complex plane, the meromorphic solutions are difficult to obtain because we have to consider the factor of singularities. The behavior of solutions near singularities is also complex, so it is very meaningful to investigate meromorphic solutions.
The purpose of this paper is to study Eq. (1) and get some new meromorphic solutions, periodic solitary wave solutions and interaction solutions between lump wave and solitary waves of it. The properties of them are explained by plotting figures. This paper is organized as follows: Section 2 derive the new periodic solitary wave solutions and interaction solutions between lump wave and solitary waves with the homoclinic test approach, Section 3 persents the exact meromorphic solutions obtained by complex method, and the Section 4 summarises the main conclusions.

Periodic Solitary Wave Solutions
The transformation of Eq. (1) is applied as: By the definition of D-operators, Eq. (2) can be rewritten as follows: where [ [ Figure 1.
a c y c z a t b y c z Taking u 1 as an example. The superposition process of solitary waves and periodic wave is shown in Figure 1. We can see that the amplitude of the superposition is from strong to weak and then to strong, and the change of amplitude is affected by the frequency and wavelength of the propagation process. At the same time, the nature of the solitons not being deformed and not being destroyed when interacting can also be seen directly from the figures.

Interaction Solutions between Lump Wave and Solitary Waves
Assume that the form of the solutions of Eq.  ®°°O bserving Figures 2-4, we know that these images are describing the superposition process of lump waves and solitary waves. Taking u 4 as an example, we can see intuitively that the lump wave of Eq. (1) retains its shape, amplitude and some other physical properties after interacting with the solitons. In other words, the images show that the interaction is elastic.

Definition 4.
A meromorphic function f belongs to the class W [20], [21] if f is a rational function of z, or a rational function of e z D D , , » or an elliptic function. The k order Briot-Bouquet equation (BBEq) is defined as ) is a polynomial with constant coefficients. This equation has been studied by Eremenko and his collaborators [21], and their results are widely recognized by scholars. Lemma 1. [22], [23], [24], [25], [26]  where s 0 ∈ » , c 1 is constant, λ = 4 2 2 n k .
We assume the simply periodic solutions of Eq. (11) with pole at s = 0 are ( 1) 1 . future research, we can also use some promising numerical methods, such as Keller Box [27], [28], Shooting method [29] and neural networks [30], [31], to obtain the numerical solution of the equation. The numerical solution obtained by using the approximate method can be used to help solve practical problems in life.
We would like to thank the editor and reviewers for their helpful suggestions.