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 , the simplest equation method , 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,
Eq.(1) is derived from (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation by Darvishi et al. . They have submitted multisoliton solutions of both equations. There are several studies on the Boiti–Leon–Manna–Pempinelli equation in the literature. Authors have investigated the Lax pair of Eq.(1) by singular manifolds method . Employing Hirota’s bilinear method different types of lump solitons of Eq.(1) have been submitted in . They have disscused Nth-order soliton solutions, rational solutions, periodic wave solutions using Pffafian tecnique, the ansatz method and the Hirota-Riemann method respectively . Further researchs can be seen [?, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 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.
2 The Manner of the Method
Let’s consider the following general form of nonlinear partial differential equation;
Step 1. Regarding travelling wave transformation as follows;
Step 2: We think the solution function Uin Eq. (4) as follows;
Family 1: When µ ≠ 0, λ2 − 4µ > 0,
Family 2: When µ ≠ 0, λ2 − 4µ < 0,
Family 3: When µ = 0, λ ≠ 0, and λ2 − 4µ > 0,
Family 4: When µ ≠ 0, λ ≠ 0, and λ2 − 4µ = 0,
Family 5: When µ = 0, λ = 0, and λ2 − 4µ = 0,
The equation systems obtained are equalized to zero and A0,A1,...,AN,B0,B1,...,BM,E,λ, µ are obtained. These coefficients are written in Eq. (5) instead of Eq. (2) to provide the travelling wave solutions.
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).
When the balancing principle is applied to the equation (14), the following relation is found between M and N,
Using the coefficients given above, the following solution families are obtained.
Equations (20) the following solution families are obtained from equations.
According to the coefficients, the following solution families are get.
Since the above cases do not meet the requirements of the conditions, no suitable solution has been found for the families.
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 .
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