Exact Solutions of Two Nonlinear Partial Differential Equations by the First Integral Method

In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative algebra. In this paper, exact travelling wave solutions of the Non-Boussinesq wavepacket model and the (2 + 1)-dimensional Zoomeron equation are studied by using the first integral method. From the solving process and results, the first integral method has the characteristics of simplicity, directness and effectiveness about solving the exact travelling wave solutions of nonlinear partial differential equations. In other words, tedious calculations can be avoided by Maple software; the solutions of more accurate and richer travelling wave solutions are obtained. Therefore, this method is an effective method for solving exact solutions of nonlinear partial differential equations.


Introduction
Over the past few decades, finding the exact solutions of nonlinear partial differential equations (PDEs) has become an attractive topic in physical science and nonlinear science. The nonlinear PDE is an important model for describing the problems of fluid mechanics, chemical physics, kinematics, atmosphere and ocean phenomena and so on. In order to find the exact solutions of nonlinear PDEs, pioneers presented the following these methods, such as ( ) Since Feng [9] proposed solving nonlinear PDEs by the first integral method.
The theory has been applied to handle various PDEs by many scholars in science and engineering. such as S. Ibrahim et al. [10] used the first integral method to obtain the exact solutions of three nonlinear schrodinger equations. S. S. Singh [11] employed the first integral method to solve the exact solutions of Kudryashov-Sinelshchikov equation and generalized Radhakrishnan-Kundu-Lakshmanan equation. A. Seadawy et al. [12] obtain the solitary wave solutions of Cubic-Quintic Nonlinear Schrödinger and Variant Boussinesq Equations by the first integral method. N. Taghizadeh et al. [13] used the first integral method to study the exact solutions of three nonlinear PDEs, etc.
For the Non-Boussinesq wavepacket model, H. Wang et al. [14] obtain the so-  [17] applied an enhanced ( ) G G ′ -expansion method to find the traveling wave solutions of the (2 + 1)-dimensional Zoomeron equation. As a result, the hyperbolic and trigonometric functions involving several parameters are derived, and so on.
The rest of this paper is structured as follows. In Section 2, we introduce the basic idea of the first integral method briefly. In Section 3, we use this method to solve two partial differential equations in detail. Finally, a conclusion is given in Section 4.

The First Integral Method
Consider the general nonlinear PDE in the form where P is a polynomial in its arguments.
We use the transformations where c is a constant, then where the subscript denotes the derivation with respect to ξ .
Step 2: Suppose the solution of nonlinear ODE (4) can be translated as .
Step 3: We introduce a new independent variable which leads to the system of ODEs Step 4: According the qualitative theory of ODEs [18], if we can find the first integrals to Equation (7) under the same conditions, then the general solutions to Equation (7) can be found directly. However, there is no systematic theory that can tell us how to find its first integrals, nor we know what these first integrals are. Fortunately, for some equations we can apply the Division Theorem to reduces Equation (4) to a first order integrable ordinary differential equation.
An exact solution to Equation (1)  ,

Application
In this section, we apply the first integral method based on the Division Theorem to solve the two nonlinear PDEs.
Firstly, we consider the following nonlinear Boussinesq wavepacket model The Boussinesq limit in Equation (9) is to let H → ∞ that is let Substituting Equation (12) into Equation (9) and dropping the mark 1, we get Then the Boussinesq limit in Equation (13) Suppose the from of solution to Equation (17) is as following: , . Substituting Equation (18) into Equation (17), and making imaginary and real part are zero respectively: Integrating Equation (19) Secondly, we consider the following (2 + 1)-dimensional Zoomeron equation Suppose the form of solution to Equation (23) is as following: where , , k l λ are real parameters, ( ) φ ξ is real functions. (23), we obtain ( )

Substituting Equation (24) into Equation (23), and integrating the resultant equation twice for Equation
where p is the integration constant.
Taking Equation (6) is an irreducible polynomial in the complex domain C[X,Y] such that In this example, we make two different cases, suppose that n = 1 and n = 2 in Equation (28).
Case I Assume that 1 n = , and then from Equation (29) we have where prime denotes differentiating with respect to the variable X. By comparing with the coefficient of where f is an arbitrary integration constant.
and using Maple solving them, we have Taking the conditions (36) in Equation (28), we get ( ) and combining Equation (37) Case II Suppose that 2 n = , by comparing with the coefficient of ( ) both sides of Equation (28), we obtain and using Maple solving them, we get

Conclusion
In conclusion, we have studied the exact travelling wave solutions of the