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This paper is concerned with the variable coefficients mKdV (VC-mKdV) equation. First, through some transformation we convert VC-mKdV equation into the constant coefficient mKdV equation. Then, using the first integral method we obtain the exact solutions of VC-mKdV equation, such as rational function solutions, periodic wave solutions of triangle function, bell-shape solitary wave solution, kink-shape solitary wave solution, Jacobi elliptic function solutions, and Weierstrass elliptic function solution. Furthermore, with the aid of Mathematica, the extended hyperbolic functions method is used to establish abundant exact explicit solution of VC-mKdV equation. By the results of the equation, the first integral method and the extended hyperbolic function method are extended from the constant coefficient nonlinear evolution equations to the variable coefficients nonlinear partial differential equation.

It is well known that the KdV equation plays an important role in the soliton theory. Many properties of the KdV equation, such as symmetry, Bäcklund transformation, infinite conservation laws, Lax pairs, and Painleve analysis, have been studied. Miura transformation links the KdV equation with the mKdV equation. Therefore, as the KdV equation, mKdV equation is also important in mathematical physics field. In recent years, some authors considered the constant coefficients mKdV equation [

This paper will discuss the variable coefficients mKdV equation (VC-mKdV):

The rest of this paper is organized as follows. In Section

The first-integral method, which is based on the ring theory of commutative algebra, was first proposed by Professor Feng Zhaosheng [

In the recent years, many authors employed this method to solve different types of nonlinear partial differential equations in physical mathematics. More information about these applications can be found in [

The main steps of this method are summarized as follows.

Given a system of nonlinear partial differential equations, for example, in two independent variables,

According to the Division theorem from ring theory of commutative algebra, there exists polynomials

We determine polynomials

Then substituting

In order to transfer (

We firstly obtain explicit and exact solutions of the constant coefficients mKdV equation (

In the case of

In this case, (

For

For

For

For

In the case of

In this case, we assume that

By analyzing all kinds of possibilities, we have the following.

While

While

While

We obtain various of explicit and exact solutions of (

Combining (

In summary, motivated by [

This work is supported by the NSF of China (40890150, 40890153), the Science and Technology Program (2008B080701042) of Guangdong Province, and the Natural Science foundation of Guangdong Province (S2012010010121). The authors would like to thank Professor Feng Zhaosheng for his helpful suggestions.