Using a New Auxiliary Equation to Construct Abundant Solutions for Nonlinear Evolution Equations

In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As m → 0 and m → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.


Introduction
NLEEs are widely used to describe complex phenomena in natural and social sciences. Many well-known models have been developed to illustrate the dynamics of nonlinear waves in the field of modern science and engineering, such as the Kortewegde Vries (KdV) [1] equation, KDV Burgers equation [2] [3], modified KDV (mKdV) equation [4], modified KDV Kadomtsev Petviashvili (mKdVKP) equation [5], and so on. More and more attention is focused on these nonlinear problems, and much nonlinear identification research can eventually advancement has been produced in recent years and many strong and effective methods have been developed to obtain accurate solutions of NLEEs. For example, homogeneous balance method [12], algebraic method [13], the sine-cosine method [14], tanh-sech method and the extended tanh-coth method [15] [16], F-expansion method [17] [18], Exp-function method [19], Jacobi elliptic function expansion method [20] [21], the modified extended mapping method [22] [23] [24], auxiliary equation method [25] [26] [27], and so on. Based on previous original methods, the auxiliary equation method constructs the exact solution of ELEEs by introducing auxiliary equations. The application of good auxiliary equations can obtain a large number of new exact solutions of ELEEs. Therefore, finding appropriate auxiliary equations is of great significance to enrich the solution of NLEEs. In this paper, a new auxiliary equation is developed to construct new types of periodic wave solutions of NLEEs, which has not been proposed in previous work. With the cooperation of the previous extended mapping method, many new results are obtained.

Method
The Suppose Equation (1) has the following traveling wave solution where ω is a pending wave parameter. Substitute Equation (2) into Equation (1), and Equation (1) becomes the following ordinary differential equation where u' means du/dξ. Suppose Equation (3) has the following formal solution where a i and ν are constants to be determined later. The positive integer n can be obtained by controlling the homogeneous balance between the governing nonlinear term and the highest order derivative of u(ξ) in Equation (3). f (ξ) is determined by the following auxiliary equation: where p, q, r are parameters to be selected. In order to construct different types of periodic wave solutions, different p, q, r are selected to determine the different Jacobi elliptic function solutions of Equation (5). Furthermore, these solutions include hyperbolic function solutions when m → 1 and trigonometric function solutions when m → 0. By using the mapping in Ref. [25], Equation (5) has the Jacobi elliptic function solutions as Table 1.
for a i and ν. Solving the algebraic equations, a i and ν can be obtained expressed by p, q, r. Substituting these solutions into Equation (4) and using the mapping in Table 1, the new type of periodic wave solutions of Equation (3) can be obtained.

The Generalized Klein-Gordon Equation
The following generalized Klein-Gordon equation [28] is considered Journal of Applied Mathematics and Physics where α, β, γ are constants. Substituting the traveling wave solution Equation (2) into Equation (6) yields By controlling the homogeneous balance between u'' and u 3 in Equation (7), 1 n = can be obtained. So the solution of Equation (7) can be expressed as Substituting Equation (8) into Equation (7) and use Equation (5) to yield a set of algebraic equations for a 0 , a 1 , and ν. Solving the algebraic equations, a 0 , a 1 , and ν can be obtained as follows 2 ω α ν ω α β  8) and (9) and Table 1, limited to space, we will not give examples one by one.

= ± −
There are still a large number of new types of periodic wave solutions for Benjamin equation, according to Equations (5), (8) and (9) and Table 1

Conclusion
In this paper, with the use of a new auxiliary Equation (4) and the extended mapping method (Table 1)