Exact solutions of the mKdV equation

Applying the first integral method, and combining the computation software, Maple, we find exact traveling wave solutions of the mKdV equation. The method is based on the ring theory of commutative algebra.

In this paper, we shall study the mKdV equation and find its exact solution. The rest of the Paper is organized as follows. In the following Section 2, the description of the proposed method will be given. The applications of the proposed method to the mKdV equation are illustrated in Section 3. Finally, some conclusions are given.

Description of the first integral method
Consider the general form of the nonlinear partial differential equation on the form: Applying the transformation ( ) x ct    , and the use of some mathematical operations, converts (1) into a second-order nonlinear ordinary differential equation as the following form Where prime denotes the derivative with respect to  . By introducing new variables ( ) ( ) changes into a system of ordinary differential equations as Now, if the solution of the system (3) is obtained, the solution of (2) will be in hand, since ordinary differential (2) is equivalent to the system (3) which is not easy to solve and there is not any systematic theory to find the first integral of this system (3). It is known that division theorem which is based on the ring theory of commutative algebra will be helpful to obtain a first integral of (3), and so the solution of equation (1).
. If ( , ) Q w z vanishes at all zero points of ( , ) P w z , then there exists a polynomial ) , (4)

Applications of method
Let us consider the following mKdV equation: Where , Taking anti-derivative of (6) with integral constant zero yields Using (3), to get the following system Now, the Division Theorem is implemented to seek the first integral to (8). Suppose that ( ) are the nontrivial solutions to (8) and  (9) is also Called the first integral to (8). Due to the Division Theorem, there exists a polynomial In this example, Let's consider special cases, 1 m  , and 2 m  , in (9).
in both sides of (10), leads to are polynomials, then from (11) Where 0 A is an integrating constant. Substituting (14) into (13) and setting all the coefficients of X to zero, the following two sets of solutions will be obtained by Maple.
Since ( ) ( 0,1,2) i a X i  are polynomials, then from (21) one can deduce that 2 ( ) a X is a constant . For convenience, we take 2 ( ) 1 a X  . Now, by balancing the Degrees of 0 ( ) a X , 1 ( ) a X and ( ) g X , we can conclude that and 1 ( ) a X will be computed as 0 0 Where D is an integrating constant. Substituting 0 ( ) a X , 1 ( ) a X and ( ) g X into (24) and setting all the coefficients of X to zero, the following two sets of solutions will be obtained by Maple.

Conclusion
In this manuscript, the first integral method has been applied successfully for solving the mKdV equation. This method has been led to exact solutions. And 1 , u x t       and 2 , u x t       are new solutions not mentioned in other literatures. Therefore, we can conclude that the method is very effective for solving some nonlinear partial differential equations.