A New Approach to (3+1) Dimensional Boiti–Leon–Manna–Pempinelli Equation

In this article, some new travelling wave solutions of the (3+1) dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation are obtained using the modiﬁed exponential function method. When the solution functions obtained are examined, it is seen that functions with periodic functions are obtained. Two and three dimensional graphs of the travelling wave solutions of the BLMP equation are drawn by selecting the appropriate parameters


Introduction
Nonlinear partial differential equations (NPDE) have an important role to describe natural phenomenon from biology to engineering. Especially in engineering, acoustic waves, water waves, electromagnetic waves have been model via NPDE. Physics and engineering applications have concentrated to the behavior of waves, for this reason solutions of such equations have attracted the attention of many scientists for many years. Hence, there are various analytical methods in the literature used by researchers to obtain solutions for such equations. Some of these are the multipliers method [1],the simplest equation method [2], the (G /G)-expansion method [3][4][5][6], the Sine-Gordon expansion method [7][8][9][10][11],the extended trial equation method [12,13], the new function method [14,15]. In this study, we used the Modified Exponential Function Method (MEFM) to the (3+1) dimensional Boiti-Leon-Manna-Pempinelli equation (BLMP) which is used to describe incompressible liquid in fluid mechanics. The equation is given as,  [17]. Employing Hirota's bilinear method different types of lump solitons of Eq.(1) have been submitted in [18]. They have disscused Nth-order soliton solutions, rational solutions, periodic wave solutions using Pffafian tecnique, the ansatz method and the Hirota-Riemann method respectively [19]. Further researchs can be seen [?, 2, 20-33]. This paper is rested as manner; we give steps of the modified exponential function method in section 2, then an application of the mention method is given in section 3. In last section 4, we give some conclusions on the obtained wave solutions.

The Manner of the Method
In this section, we give the manner of the MEFM [34][35][36].
Let's consider the following general form of nonlinear partial differential equation; where u = u(x, y, z,t)is unknown solution function.
Step 1. Regarding travelling wave transformation as follows; where c is a non-zero real value, required derivative terms are substituted into Eq. (2). By this way, the following nonlinear ordinary differential equation is obtained, Step 2: We think the solution function Uin Eq. (4) as follows; When the Eq. (6) is solved, the following solution families are obtained [37].
If the equation (12) is integrated, It is described in a simpler way as follows by applying the transformation υ = ω to the nonlinear ordinary differential equation (13).

CASE 2:
Equations (20) the following solution families are obtained from equations.

Conclusion
In this study, new travelling wave solutions of Boiti-Leon-Manna-Pempinelli (BLMP) equation have been successfully obtained by using modified exponential function method. When we compare our results with the solutions obtained for this equation in the literature, we see that all solutions are completely different. We have drawn two and three dimensional graphs of all travelling wave solutions by selecting the suitable constants. The solutions obtained can be said to be an effective method for obtaining analytical solutions of such nonlinear differential equations. The solutions found include trigonometric and hyperbolic functions. Such functions are also periodic functions. The advantage of such functions is that it allows us to comfortably comment on the physical behavior of the wave, regardless of the range of the graph of the resulting solution function. The hyperbolic functions and trigonometric functions are arisen in both mathematics and physics. For example, the hyperbolic cosine functions are shape of catenary, the hyperbolic tangent functions arise in calculate to magnetic moment and rapidity of special relativity, the hyperbolic secant functions arise in the profile of a laminar jet, the hyperbolic cotangent functions arise in the Langevin function for magnetic polarization [38].