The Rational Solutions and the Interactions of the N-Soliton Solutions for Boiti-Leon-Manna-Pempinelli-Like Equation

These rational solutions which can be described a kind of algebraic solitary waves which have great potential in applied value in atmosphere and ocean. It has attracted more and more attention recently. In this paper, the generalized bilinear method instead of the Hirota bilinear method is used to obtain the rational solutions to the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (hereinafter referred to as BLMP equation). Meanwhile, the (2 + 1)-dimensional BLMP-like equation is derived on the basis of the generalized bilinear operators D3,x D3,y and D3,t. And the rational solutions to the (2 + 1)-dimensional BLMP-like equation are obtained successively. Finally, with the help of the N-soliton solutions of the (2 + 1)-dimensional BLMP equation, the interactions of the N-soliton solutions can be derived. The results show that the two soliton still maintained the original waveform after happened collision.


Introduction
In recent years, with the emergence of high-powered computer and the development of mathematical research, various nonlinear problems increasingly cause attentions of scientists. Especially some key problems in engineering, science and modern physics are ultimately dependent on the specific solution of the nonlinear equations. So the solution of the nonlinear equations [1] [2] [3] [4] [5] occupies very important position no matter on theory research or practical application. But the diverse nonlinear equations of mathematical physics was descried with Hirota bilinear equations [6] [7] [8] and generalized bilinear equations [9] [11] [12] [13], such as the KdV equations [10] [11] [14] [15], the BLMP equations [16] [17] [18], the NLS equation, the Boussinesq equation [19], the KP equations [9] [20] [21] and so on. There has been a growing attention on rational solutions [2] [3] [9] [10] [11] [22] to nonlinear differential equations in recent years. One kind of particular rational solutions are rogue wave solutions [24] [25] [26], which describe significant nonlinear wave phenomena in oceanography. So the rational solutions of the nonlinear equations become crucial in atmosphere and ocean.
As we know some rational solutions to integrable equations have been considered systematically on the basis of the Wronskian formulation, the Casoratian formulation and the Pfaffian formulation [9]. Rational solutions to the nonlinear differential equation are also considered by different approaches, such as the G G ′ expansion methods [9] [10] [27] and the tanh-coth function method [28]. Moreover, the Hirota method is also used to construct rational solutions to the nonlinear differential equations. And it is very necessary and interesting for us to study rational solutions and generate the generalized bilinear form to nonlinear differential equations.
In this paper, the rational solutions to the (2 + 1)-dimensional BLMP equation can be derived by the Hirota bilinear method. And the (2 + 1)-dimensional BLMP-like nonlinear differential equation on the basic of existing BLMP bilinear differential equation can be derived via applying three generalized bilinear operators 3,

Hirota Bilinear D-Operators
The D-operators are defined in Refs. [6] [7] [8] as following: where , m n are all non-negative integer. Assume

The Rational Solutions to the (2 + 1)-Dimensional BLMP Equation
Consider a (2 + 1)-dimensional BLMP equation according to the formula Equation (1) and the bilinear form with D-operators Apply the direct Maple symbolic computation to obtain the polynomial solutions to Equation (5). where the ijk c are constants.
The twelve classes of polynomial solutions to Equation (4) are obtained, as follows.

The (2 + 1)-Dimensions BLMP-Like Equation
Consider a generalized bilinear differential equation of (2 + 1)-dimensions BLMP type: Which has the same bilinear form as the BLMP equation. The above differential operators are a kind of generalized bilinear differential operators that are put forward by professor Ma in [11].
The formula reads.
y t x y y t y t xxy x xx xy Through the transformation ( ) 2 ln , (25) can be reduced to the (2 + 1)-dimensional BLMP-like nonlinear differential equation 3

The Rational Solutions to the (2 + 1)-Dimensional BLMP-Like Equation
Apply the maple symbolic computing Equation (7), twelve classes of rational solutions to the (2 + 1)-dimensional BLMP-like equation Equation (26) as follows: The first class of rational solutions to Equation (26): The forth class:

The N-Soliton Solutions of the (2 + 1)-Dimensional BLMP Equation
Consider the N-soliton solutions of Equation (4) by using the perturbation ap-  proach and the Hirota direct method. Equation (6) is the bilinear form of Equation (4) by using transformation . Expand the F into small parameter exponential as follows 1 , and 0 λ = .
Substituting Equation (39) into Equation (6)   When the coefficient 0, A ij e = the soliton solution is changed into the resonance solution. Its propagation will be showed in Figure 5.
Let the coefficient 0, A ij e ≠ the two-soliton happen in the collision because of the different soliton speed. The pursue collision between the two soliton will be showed in Figure 6.

Conclusion
With the help of the generalized bilinear methods [9] [11] [12] [15], we present the bilinear form to the (2 + 1)-dimensional BLMP equation and the (2 + 1)-dimensional BLMP-like equation. Afterwards, the twelve classes of rational solutions are obtained respectively. The 3d graphics to the rational solutions shows some properties about these rational solutions, such as symmetry. These rational solutions which can be described as a kind of algebraic solitary waves have great potential in applied value in atmosphere and ocean. Every solitary wave velocity is different (the faster speed wave will overtake the slower speed wave), so it comes to a conclusion that these solutions are changed into the resonance solution when the coefficient 0, A ij e = and the soliton collision will happen when the coefficient 0 A ij e ≠ . After the collision, the two solitons will continue to spread as the pre-vious speed and the direction. The generalized bilinear method could be applied to more high dimensional equation to obtain its rational solutions and deduce new equation. This method also could be applied to the half a discrete nonlinear differential equation and discrete nonlinear differential equation. Continuing to study this generalized bilinear method in depth is meaningful and interesting.