A Modified Approach to the New Solutions of Generalized mKdV Equation Using ( G / G )-Expansion

The modified -expansion method is applied for finding new solutions of the generalized mKdV equation. By taking an appropriate transformation, the generalized mKdV equation is solved in different cases and hyperbolic, trigonometric, and rational function solutions are obtained.


Introduction
The evolutions of the physical, engineering, and other systems always behave nonlinearly; hence many nonlinear evolution equations have been introduced to interpret the phenomena.Many kinds of mathematical methods have been established to investigate the solutions of those nonlinear evolution equations both numerically and asymptotically, while the exact solutions are of particular interests.In recent decades, with the rapid progress of computation methods, many effective calculating approaches have been developed, for example, the tanh-coth expansion [1,2], -expansion [3,4], Painlevé expansion [5], Jacobi elliptic function method [6], Hirota bilinear transformation [7], Backlund/Darboux transformation [8,9], variational method [10], the homogeneous balance method [11], exp-function expansion [12], and so on.However, a unified approach to obtain the complete solutions of the nonlinear evolution equations has not been revealed.
The mKdV equation, a modified version of the Kortewegde Vries (KdV) equation, has been investigated extensively since Zabusky showed how this equation depict the oscillations of a lattice of particles connected by nonlinear springs as the Fermi-Pasta-Ulam (FPU) model [22][23][24][25].Afterwards, this equation has been used to describe the evolution of internal waves at the interface of two layers of equal depth [26].Generally, the KdV theory describes the weak nonlinearity and weak dispersion while, in the study of nonlinear optics, the complex mKdV equation has even been used to describe the propagation of optical pulses in nematic optical fibers when we go beyond the usual weakly nonlinear limit of Kerr medium [27].In some cases, the exponential order may be not a positive integer, but just a real number.After this kind of generalized mKdV equation [28] has been introduced, interests of investigating the solutions of it [29] have been inspired; then the standard expansion methods cannot be applied, and some kinds of transformation are needed.
In this paper, we modify the standard (  /)-expansion method and use it to solve the generalized mKdV equation.In next section, we briefly introduce the modified (  /)expansion method while in Section 3, we apply it to find some types of new solutions of mKdV equation, and the last section gives the summary and conclusion.

An Introduction to the Modified (𝐺 󸀠 /𝐺)-Expansion Method
Recently, a new approach called (  /)-expansion has been proposed dealing with the problems of finding solutions of nonlinear evolution equations [13] and some modifications to this method have been developed.Here we briefly outline the main steps of the modified (  /)-expansion method in the following.
Step 1.We consider a given  (,   ,   ,   ,   ,   , . ..) = 0, where  is a polynomial for its arguments and  = (, ) is the unknown function.Introducing the new variable  and supposing that () = (, ) and  =  − V, hence, the partial differential equation (PDE) (1) is reduced to an ordinary differential equation (ODE) for  = () as Step 2. For ODE (2) above, the solution could be expressed by a polynomial in   / as where  = () is the solution of a second order linear ODE   () +   () +  () = 0 with constants ,  to be determined later.Positive integer  is an index yet undetermined which should be calculated by the balance between the highest order derivatives and the nonlinear terms from ODE (2).By solving (4), it is apparent that the form of   / in three different cases read as follows.
Step 3. Substituting solution (3) into ODE (2) using ( 4), we have a set of differential equations.Collecting all terms together according to the same order of   /, the left-hand side of ODE (2) becomes a long expression in the form of polynomial of   /.Making all coefficients of each order of   / equal to zero, the solution sets of the parameters , ,   , and V will be got after solving the algebra equations.
Step 4. Based on the last step, we now have the solutions of the coefficient algebra equations and after substituting the parameters   , V, and so forth into solution (3), we could reach different types of travelling wave solutions of the PDE (1).

Application to the Generalized mKdV Equation
Now, we consider the generalized mKdV equation with the form [28]   +     +   = 0, while parameters  and  are real constants.Denoting the travelling wave solution as with  =  − V, then the PDE (8) becomes an ODE.After integrating with single variable  and setting the integrate constant to zero, we have It can be easily seen that the standard (  /)-expansion cannot be applied directly to this situation because of the arbitrary power index , which results in the noninteger power index of   /.So it is necessary to introduce a transformation to deal with it via assuming that then (10) becomes The balancing between the highest order nonlinear term and the highest order derivative term leads to the balance parameter  = 2; hence the solution (3) could be expressed as with () =   ()/() and  0 ,  1 ,  2 ,  −1 , and  −2 being constants to be determined later.Substituting ( 13) into ( 12), making use of ( 4), a polynomial of () is obtained; a set of nonlinear algebra equations about undetermined constants  0 ,  1 ,  2 ,  −1 ,  −2 , , , and V are reached through setting the coefficients of each order of  to zero.These equations are expressed as follows.
It is straight forward to give the solution sets of the algebraic equations in different cases as follows.
Case i.All the coefficients in (13) are equal to 0, and , , and V are arbitrary.
Case ii.All the coefficients in (13) are equal to 0 except for  0 = (1 + )V/, and , , V are arbitrary.

Case iv. Consider
Case v. Consider Case vi.Consider Case vii.Consider Case viii.Consider Case ix.Consider Case x.Consider Cases i to iii are trivial and of no interest, hence not discussed here.We focus our attention to cases from iv to x.Using solutions ( 21) to (27), solution ( 13) can be expressed in different forms corresponding to different cases listed above.
For Case iv, there are two solution types with  =  + 4/ 2 .
(iv-1) When  < 0, we obtain the hyperbolic function solution: where  1 and  2 are integration constants.Recalling (11) we get For simplicity, we only show expression for  rather than  in the following cases.
(iv-2) When  > 0, we have the trigonometric function solution: For Case v, there are also two types of solution with  =  −  2 / 2 .
(v-1) When  ̸ = 0: For Case vi, there are three types of solution with For Case vii, we only have one type of solution: where  = .For Case viii, there are three types of solution with  =  + 16/ 2 .
( (43) As an example, we consider solutions of Case (iv-1) when  1 = 0 and  = −1; then the solution reduces to the soliton form as for  = 1 as the standard KdV case and for the standard mKdV case when  = 2.We show the diagrams of Case (iv-1) in Figure 1 to illustrate the behaviors of the solutions for different power index .We choose coefficients of the generalized mKdV equation as  = 6 and  = 1; the constants of the solutions  1 and  2 equal 0.5 and 2, respectively, while the coefficient  in the equation equals −0.2.It is found that (i) this kind of solution is a type of soliton solution; the amplitude decreases with the increase of power index ; (ii) it gives that the width of the wave packet broadened when increasing ; (iii) in addition, we find that the velocity of the wave slowed down when  is bigger.