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

Abstract

The modified (𝐺^′/𝐺)-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.

1. 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], 𝐹-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.

Within recent years, a new method called (𝐺^′/𝐺)-expansion [13] has been proposed for finding the traveling wave solutions of the nonlinear evolution equations. Many equations have been investigated and many solutions have been found using the method, including KdV equation, Hirota-Satsuma equation [13], coupled Boussinesq equation [14], generalized Bretherton equation [15], the mKdV equation [16], the Burgers-KdV equation, the Benjamin-Bona-Mahony equation [17], the Whitham-Broer-Kaup-like equation [18], the Kolmogorov-Petrovskii-Piskunov equation [19], KdV-Burgers equation [20], and Drinfeld-Sokolov-Satsuma-Hirota equation [21].

The mKdV equation, a modified version of the Korteweg-de 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 [22–25]. 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 (𝐺^′/𝐺)-expansion method and use it to solve the generalized mKdV equation. In next section, we briefly introduce the modified (𝐺^′/𝐺)-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.

2. An Introduction to the Modified (𝐺^′/𝐺)-Expansion Method

Recently, a new approach called (𝐺^′/𝐺)-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 (𝐺^′/𝐺)-expansion method in the following.

Step 1. We consider a given 𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑡,𝑢_𝑡𝑡,𝑢_𝑥𝑥,…=0, (1) where 𝑃 is a polynomial for its arguments and 𝑢=𝑢(𝑥,𝑡) is the unknown function. Introducing the new variable 𝜉 and supposing that 𝑢(𝜉)=𝑢(𝑥,𝑡) and 𝜉=𝑥−𝑣𝑡, hence, the partial differential equation (PDE) (1) is reduced to an ordinary differential equation (ODE) for 𝑢=𝑢(𝜉) as 𝑃𝑢,−𝑣𝑢^′,𝑢^′,𝑣²𝑢^′′,−𝑣𝑢^′′,𝑢^′′,…=0. (2)

Step 2. For ODE (2) above, the solution could be expressed by a polynomial in 𝐺^′/𝐺 as 𝑢(𝜉)=^𝑚_𝑖=−𝑚𝑎_𝑖𝐺^′𝐺^𝑖, (3) where 𝐺=𝐺(𝜉) is the solution of a second order linear ODE 𝐺^′′(𝜉)+𝜆𝐺^′(𝜉)+𝜇𝐺(𝜉)=0 (4) with constants 𝜆, 𝜇 to be determined later. Positive integer 𝑚 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 𝐺^′/𝐺 in three different cases read as follows.
(1) When 𝜆²−4𝜇>0, 𝐺^′𝐺=𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹−𝜆2. (5)
(2) When 𝜆²−4𝜇<0, 𝐺^′𝐺=4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹−𝜆2. (6)
(3) When 𝜆²−4𝜇=0, 𝐺^′𝐺=𝐶₂𝐶₁+𝐶₂𝜉−𝜆2, (7) where 𝐶₁ and 𝐶₂ in above three solutions (5), (6), and (7) of (4) are integrate constants.

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 𝐺^′/𝐺, the left-hand side of ODE (2) becomes a long expression in the form of polynomial of 𝐺^′/𝐺. Making all coefficients of each order of 𝐺^′/𝐺 equal to zero, the solution sets of the parameters 𝜆, 𝜇, 𝑎_𝑖, 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 𝑎_𝑖, 𝑣, and so forth into solution (3), we could reach different types of travelling wave solutions of the PDE (1).

3. Application to the Generalized mKdV Equation

Now, we consider the generalized mKdV equation with the form [28] 𝑢_𝑡+𝛼𝑢^𝛾𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥=0, (8) while parameters 𝛼 and 𝛽 are real constants. Denoting the travelling wave solution as 𝑢(𝑥,𝑡)=𝑢(𝜉), (9) with 𝜉=𝑥−𝑣𝑡, then the PDE (8) becomes an ODE. After integrating with single variable 𝜉 and setting the integrate constant to zero, we have 𝛼−𝑣𝑢+𝑢1+𝛾^1+𝛾+𝛽𝑢^′′=0. (10) It can be easily seen that the standard (𝐺^′/𝐺)-expansion cannot be applied directly to this situation because of the arbitrary power index 𝛾, which results in the noninteger power index of 𝐺^′/𝐺. So it is necessary to introduce a transformation to deal with it via assuming that 𝑢(𝜉)=𝑤^1/𝛾(𝜉); (11) then (10) becomes 𝛽𝛾1+𝛾𝑤𝑤^′′+𝛽1−𝛾²𝑤^′2−𝑣𝛾²1+𝛾𝑤²+𝛼𝛾²𝑤³=0. (12) The balancing between the highest order nonlinear term and the highest order derivative term leads to the balance parameter 𝑚=2; hence the solution (3) could be expressed as 𝑤(𝜉)=𝑎₀+𝑎₁𝜙(𝜉)+𝑎₂𝜙²(𝜉)+𝑎₋₁𝜙(𝜉)+𝑎₋₂𝜙²(𝜉), (13) with 𝜙(𝜉)=𝐺^′(𝜉)/𝐺(𝜉) and 𝑎₀, 𝑎₁, 𝑎₂, 𝑎₋₁, and 𝑎₋₂ being constants to be determined later.

Substituting (13) into (12), making use of (4), a polynomial of 𝜙(𝜉) is obtained; a set of nonlinear algebra equations about undetermined constants 𝑎₀, 𝑎₁, 𝑎₂, 𝑎₋₁, 𝑎₋₂, 𝜆, 𝜇, and 𝑣 are reached through setting the coefficients of each order of 𝜙 to zero. These equations are expressed as follows.

(1)  𝜙⁰-order: −𝑣𝑎²₀−2𝑎₁𝑎₋₁−2𝑎₂𝑎₋₂𝛾³+4𝑎₁𝑎₋₁+4𝑎₂𝑎₋₂𝜆²+17𝑎₁𝑎₋₂+17𝜇𝑎₂𝑎₋₁+𝑎₀𝑎₋₁+𝜇𝑎₀𝑎₁𝜆+2𝑎₀𝑎₂−𝑎²₁𝜇²+8𝑎₁𝑎₋₁+4𝑎₂𝑎₋₂𝜇−𝑎²₋₁+2𝑎₀𝑎₋₂𝛽+𝛼𝑎³₀−𝑣𝑎²₀+6𝛼𝑎₁𝑎₋₁+𝑎₂𝑎₋₂𝑎₀+3𝛼𝑎²₁−2𝑣𝑎₂𝑎₋₂+3𝛼𝑎₂𝑎²₋₁−2𝑣𝑎₁𝑎₋₁𝛾²+2𝑎₁𝑎₋₁+4𝑎₂𝑎₋₂𝜆²+𝜇𝑎₀𝑎₁+𝑎₀𝑎₋₁+9𝜇𝑎₂𝑎₋₁+9𝑎₁𝑎₋₂𝜆+2𝑎₀𝑎₂𝜇²+4𝑎₁𝑎₋₁+4𝑎₂𝑎₋₂𝜇+2𝑎₀𝑎₋₂+𝑎𝛽𝛾−2₁𝑎₋₁+4𝑎₂𝑎₋₂𝜆²−8𝜇𝑎₂𝑎₋₁+𝑎₁𝑎₋₂𝜆+𝜇²𝑎²₁−4𝑎₁𝑎₋₁+4𝑎₂𝑎₋₂𝜇+𝑎²₋₁𝛽=0. (14)

(2)  𝜙¹-order: −2𝑣𝑎₀𝑎₁+𝑎₂𝑎₋₁𝛾³+𝛽𝑎₀𝑎₁+9𝑎₂𝑎₋₁𝜆²+𝛽𝜇6𝑎₀𝑎₁−𝑎²₁+8𝛽𝑎₁𝑎₋₁+4𝑎₂𝑎₋₂𝜆−2𝛽𝜇²𝑎₁𝑎₂+2𝛽𝑎₀𝑎₁+9𝑎₂𝑎₋₁𝜇+3𝑎²₁𝑎₋₁+𝑎²₀𝑎₁+2𝑎₁𝑎₂𝑎₋₁+2𝑎₀𝑎₂𝑎₋₁𝛼−2𝑣𝑎₀𝑎₁−2𝑣𝑎₂𝑎₋₁+8𝛽𝑎₁𝑎₋₂𝛾²+𝛽𝛾𝑎₀𝑎₁+5𝛾𝑎₂𝑎₋₁−4𝑎₂𝑎₋₁𝜆²+𝛽2+𝛾𝑎²₁+6𝛾𝑎₀𝑎₂𝜇+4𝛾−1𝑎₁𝑎₋₁+4𝑎₂𝑎₋₂𝜆+𝛽2𝛾+2𝑎₁𝑎₂𝜇²+2𝛾𝑎₀𝑎₁+5𝛾𝑎₂𝑎₋₁−4𝑎₂𝑎₋₁𝜇+4𝛾−1𝑎₁𝑎₋₁=0. (15)

(3)  𝜙²-order: 𝛽1+𝛾𝑎²₁+4𝑎₀𝑎₂𝜆²+𝛽−𝛾+1𝛾−8𝑎₁𝑎₂𝜇+3𝛾1+𝛾𝑎₀𝑎₁+𝛾+119𝛾−8𝑎₂𝑎₋₁𝜆−2𝛽𝛾+1𝛾−2𝑎²₂𝜇²+2𝛽𝛾+1𝑎²₁+4𝛾𝑎₀𝑎₂𝜇−𝑣𝛾³2𝑎₀𝑎₂+𝑎²₁+𝛾²4𝛽𝑎₁𝑎₋₁+4𝑎₂𝑎₋₂−𝑣𝑎²₁+2𝑎₀𝑎₂+3𝛼𝑎₀𝑎²₁+𝑎²₀𝑎₂+𝑎²₂𝑎₋₂+2𝑎₁𝑎₂𝑎₋₁=0. (16)

(4)  𝜙³-order: 𝛽𝑎₁𝑎₂𝛾+4𝛾+1𝜆²×𝑎+𝛽𝛾+1𝛾+2²₁+2𝛾+4𝜇𝑎²₂+10𝛾𝑎₀𝑎₂𝜆−2𝑣𝛾³𝑎₁𝑎₂+2𝛽𝑎₀𝑎₁+5𝑎₂𝑎₋₁+2𝑎₁𝑎₂𝑎𝛽𝜇−𝑣+𝛼³₁+3𝑎²₂𝑎₋₁+6𝑎₀𝑎₁𝑎₂𝛾²+2𝛽3𝛾−2𝑎₂𝑎₋₁+5𝛾+4𝑎₁𝑎₂𝜇+𝛾𝑎₀𝑎₁=0. (17)

(5)  𝜙⁴-order: 41+𝛾𝛽𝑎²₂𝜆²+𝛾+15𝛾+8𝛽𝑎₁𝑎₂𝜆−𝑣𝛾³𝑎²₂+3𝛼𝑎₂𝑎²₁+𝑎₀𝑎₂+𝛽𝑎²₁+6𝑎₀𝑎₂−𝑣𝑎²₂𝛾²+2𝛽𝛾𝑎²₁+3𝑎₀𝑎₂+4𝑎²₂𝜇𝛾+8𝛽𝑎²₂𝜇+𝑎²₁=0. (18)

(6)  𝜙⁵-order: 𝛾²4𝛽𝑎₁+2𝛽𝑎₂𝜆+3𝛼𝑎₁𝑎₂𝑎₂+2𝛾𝛽5𝜆𝑎₂+4𝑎₁+4𝛽2𝜆𝑎₂+𝑎₁𝑎₂=0. (19)

(7)  𝜙⁶-order: 𝛼𝛾²𝑎³₂+2𝛽𝛾𝛾+1𝛾+2𝑎²₂=0. (20) Equations for the coefficients of 𝜙^−𝑖 (𝑖∈[1,6]) are similar to the above equations and hence not shown here.

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 𝜆, 𝜇, and 𝑣 are arbitrary.

Case ii. All the coefficients in (13) are equal to 0 except for 𝑎₀=(1+𝛾)𝑣/𝛼, and 𝜆, 𝜇, 𝑣 are arbitrary.

Case iii. All the coefficients in (13) equal to 0 except for 𝑎₂=−2𝛽(𝛾²+3𝛾+2)/𝛼𝛾².

Case iv
Consider 𝑎₀𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²,𝑎₋₂𝛾=−2𝛽²+3𝛾+2𝜇²𝛼𝛾²,𝑎₁=𝑎₂=𝑎₋₁=0,𝜆=0,𝑣=−4𝛽𝜇𝛾². (21)

Case v
Consider 𝑎₁𝛾=−2𝛽²+3𝛾+2𝜆𝛼𝛾²,𝑎₂𝛾=−2𝛽²+3𝛾+2𝛼𝛾²,𝑎₀=𝑎₋₁=𝑎₋₂=0,𝜇=0,𝑣=𝛽𝜆²𝛾². (22)

Case vi
Consider 𝑎₀𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²,𝑎₁𝛾=−2𝛽²+3𝛾+2𝜆𝛼𝛾²,𝑎₂𝛾=−2𝛽²+3𝛾+2𝛼𝛾²,𝑎₋₁=𝑎₋₂𝛽𝜆=0,𝑣=²−4𝜇𝛾². (23)

Case vii
Consider 𝑎₀𝛽𝛾=−²+3𝛾+2𝜆²2𝛼𝛾²,𝑎₁𝛾=−2𝛽²+3𝛾+2𝜆𝛼𝛾²,𝑎₂𝛾=−2𝛽²+3𝛾+2𝛼𝛾²,𝑎₋₁=𝑎₋₂𝜆=0,𝜇=²4,𝑣=0. (24)

Case viii
Consider 𝑎₀𝛾=−4𝛽²+3𝛾+2𝜇𝛼𝛾²,𝑎₂𝛾=−2𝛽²+3𝛾+2𝛼𝛾²,𝑎₋₂𝛾=−2𝛽²+3𝛾+2𝜇²𝛼𝛾²,𝑎₁=𝑎₋₁=0,𝜆=0,𝑣=−16𝛽𝜇𝛾². (25)

Case ix
Consider 𝑎₀𝛽𝛾=−²+3𝛾+2𝜆²2𝛼𝛾²,𝑎₋₁𝛽𝛾=−²+3𝛾+2𝜆³2𝛼𝛾²,𝑎₋₂𝛽𝛾=−²+3𝛾+2𝜆⁴8𝛼𝛾²,𝑎₁=𝑎₂𝜆=0,𝜇=²4,𝑣=0. (26)

Case x
Consider 𝑎₀𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²,𝑎₋₁𝛾=−2𝛽²+3𝛾+2𝜆𝜇𝛼𝛾²,𝑎₋₂𝛾=−2𝛽²+3𝛾+2𝜇²𝛼𝛾²,𝑎₁=𝑎₂𝛽𝜆=0,𝑣=²−4𝜇𝛾². (27)

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 𝜉=𝑥+4𝛽𝜇𝑡/𝛾².

(iv-1) When 𝜇<0, we obtain the hyperbolic function solution: 𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²+2𝛽𝛾²+3𝛾+2𝜇𝛼𝛾²⋅𝐶²₁+𝐶²₂cosh²√−𝜇𝜉+2𝐶₁𝐶₂sinh√−𝜇𝜉cosh√−𝜇𝜉−𝐶²₂×𝐶²₁+𝐶²₂cosh²√−𝜇𝜉+2𝐶₁𝐶₂sinh√−𝜇𝜉cosh√−𝜇𝜉−𝐶²₁⁻¹, (28) where 𝐶₁ and 𝐶₂ are integration constants. Recalling (11) we get 𝑢(𝜉)=−2𝛽𝛾²+3𝛾+2𝜇𝛼𝛾²+2𝛽𝛾²+3𝛾+2𝜇𝛼𝛾²⋅𝐶²₁+𝐶²₂cosh²√−𝜇𝜉+2𝐶₁𝐶₂sinh√−𝜇𝜉cosh√−𝜇𝜉−𝐶²₂×𝐶²₁+𝐶²₂cosh²√−𝜇𝜉+2𝐶₁𝐶₂sinh√−𝜇𝜉cosh√−𝜇𝜉−𝐶²₁^−11/𝛾. (29) For simplicity, we only show expression for 𝑤 rather than 𝑢 in the following cases.

(iv-2) When 𝜇>0, we have the trigonometric function solution: 𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²+2𝛽𝛾²+3𝛾+2𝜇𝛼𝛾²⋅𝐶²₁−𝐶²₂cos²√𝜇𝜉+2𝐶₁𝐶₂sin√𝜇𝜉cos√𝜇𝜉+𝐶²₂×𝐶²₁−𝐶²₂cos²√𝜇𝜉+2𝐶₁𝐶₂sin√𝜇𝜉cos√𝜇𝜉−𝐶²₁⁻¹. (30)

For Case v, there are also two types of solution with 𝜉=𝑥−𝛽𝜆²𝑡/𝛾².

(v-1) When 𝜆≠0: 𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝜆𝛼𝛾²×𝜆2𝐶₁sinh(𝜆/2)𝜉+𝐶₂cosh(𝜆/2)𝜉𝐶₁cosh(𝜆/2)𝜉+𝐶₂()𝜉−𝜆sinh𝜆/22−2𝛽𝛾²+3𝛾+2𝛼𝛾²×𝜆2𝐶₁sinh(𝜆/2)𝜉+𝐶₂cosh(𝜆/2)𝜉𝐶₁cosh(𝜆/2)𝜉+𝐶₂()𝜉−𝜆sinh𝜆/22². (31)

(v-2) When 𝜆=0: 𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝛼𝛾²𝐶²₂𝐶₁+𝐶₂𝜉². (32)

For Case vi, there are three types of solution with 𝜉=𝑥−𝛽(𝜆²−4𝜇)𝑡/𝛾².

(vi-1) When 𝜆²−4𝜇>0: 𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²−2𝛽𝛾²+3𝛾+2𝜆𝛼𝛾²×𝜆²−4𝜇2⋅𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹−𝜆2−2𝛽𝛾²+3𝛾+2𝛼𝛾²⋅𝜆²−4𝜇2×𝐶₁sinh𝜆²−4𝜇2𝜉+𝐶₂cosh𝜆²−4𝜇2𝜉×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉⁻¹−𝜆2². (33)

(vi-2) When 𝜆²−4𝜇<0: 𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²−2𝛽𝛾²+3𝛾+2𝜆𝛼𝛾²×4𝜇−𝜆²2⋅−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹−𝜆2−2𝛽𝛾²+3𝛾+2𝛼𝛾²⋅4𝜇−𝜆²2×−𝐶₁sin4𝜇−𝜆²2𝜉+𝐶₂cos4𝜇−𝜆²2𝜉×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉⁻¹−𝜆2². (34)

(vi-3) When 𝜆²−4𝜇=0: 𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²−2𝛽𝛾²+3𝛾+2𝜆𝛼𝛾²𝐶₂𝐶₁+𝐶₂𝜉−𝜆2−2𝛽𝛾²+3𝛾+2𝛼𝛾²𝐶₂𝐶₁+𝐶₂𝜉−𝜆2². (35)

For Case vii, we only have one type of solution: 𝑤(𝜉)𝛽𝛾=−²+3𝛾+2𝜆²2𝛼𝛾²−2𝛽𝛾²+3𝛾+2𝜆𝛼𝛾²𝐶₂𝐶₁+𝐶₂𝜉−𝜆2−2𝛽𝛾²+3𝛾+2𝛼𝛾²𝐶₂𝐶₁+𝐶₂𝜉−𝜆2², (36) where 𝜉=𝑥.

For Case viii, there are three types of solution with 𝜉=𝑥+16𝛽𝜇𝑡/𝛾².

(viii-1) When 𝜇<0: 𝑤(𝜉)𝛾=−4𝛽²+3𝛾+2𝜇𝛼𝛾²+2𝛽𝛾²+3𝛾+2𝜇𝛼𝛾²⋅𝐶²₁+𝐶²₂cosh²√−𝜇𝜉+2𝐶₁𝐶₂sinh√−𝜇𝜉cosh√−𝜇𝜉−𝐶²₁×𝐶²₁+𝐶²₂cosh²√−𝜇𝜉+2𝐶₁𝐶₂sinh√−𝜇𝜉cosh√−𝜇𝜉−𝐶²₂⁻¹+2𝛽𝛾²+3𝛾+2𝜇𝛼𝛾²⋅𝐶²₁+𝐶²₂cosh²√−𝜇𝜉+2𝐶₁𝐶₂sinh√−𝜇𝜉cosh√−𝜇𝜉−𝐶²₂×𝐶²₁+𝐶²₂cosh²√−𝜇𝜉+2𝐶₁𝐶₂sinh√−𝜇𝜉cosh√−𝜇𝜉−𝐶²₁⁻¹. (37)

(viii-2) When 𝜇>0: 𝑤(𝜉)𝛾=−4𝛽²+3𝛾+2𝜇𝛼𝛾²+2𝛽𝛾²+3𝛾+2𝜇𝛼𝛾²⋅𝐶²₁−𝐶²₂cos²√𝜇𝜉+2𝐶₁𝐶₂sin√𝜇𝜉cos√𝜇𝜉−𝐶²₁×𝐶²₁−𝐶²₂cos²√𝜇𝜉+2𝐶₁𝐶₂sin√𝜇𝜉cos√𝜇𝜉+𝐶²₂⁻¹+2𝛽𝛾²+3𝛾+2𝜇𝛼𝛾²⋅𝐶²₁−𝐶²₂cos²√𝜇𝜉+2𝐶₁𝐶₂sin√𝜇𝜉cos√𝜇𝜉+𝐶²₂×𝐶²₁−𝐶²₂cos²√𝜇𝜉+2𝐶₁𝐶₂sin√𝜇𝜉cos√𝜇𝜉−𝐶²₁⁻¹. (38)

(viii-3) When 𝜇=0: 𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝛼𝛾²𝐶₂𝐶₁+𝐶₂𝜉−𝜆2². (39)

For Case ix, only one type of solution exits; it is 𝑤(𝜉)𝛽𝛾=−²+3𝛾+2𝜆²2𝛼𝛾²+𝛽𝛾²+3𝛾+2𝜆³𝛼𝛾²𝐶₁+𝐶₂𝜉𝜆𝐶₁+𝐶₂𝜉−2𝐶₂−𝛽𝛾²+3𝛾+2𝜆⁴2𝛼𝛾²𝐶₁+𝐶₂𝜉²𝜆𝐶₁+𝐶₂𝜉−2𝐶₂², (40) where 𝜉=𝑥.

For Case x, there are three types of solution with 𝜉=𝑥−𝛽(𝜆²−4𝜇)𝑡/𝛾².

(x-1) When 𝜆²−4𝜇>0: 𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²−4𝛽𝛾²+3𝛾+2𝜆𝜇𝛼𝛾²×𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉×𝜆²−4𝜇𝐶₁−𝜆𝐶₂×sinh𝜆²−4𝜇2𝜉+𝜆²−4𝜇𝐶₂−𝜆𝐶₁×cosh𝜆²−4𝜇2𝜉⁻¹−8𝛽𝛾²+3𝛾+2𝜇²𝛼𝛾²⋅𝐶₁cosh𝜆²−4𝜇2𝜉+𝐶₂sinh𝜆²−4𝜇2𝜉²×𝜆²−4𝜇𝐶₁−𝜆𝐶₂sinh𝜆²−4𝜇2𝜉+𝜆²−4𝜇𝐶₂−𝜆𝐶₁×cosh𝜆²−4𝜇2𝜉²⁻¹. (41)

(x-2) When 𝜆²−4𝜇<0:𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²−4𝛽𝛾²+3𝛾+2𝜆𝜇𝛼𝛾²×𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉×−4𝜇−𝜆²𝐶₁+𝜆𝐶₂sin4𝜇−𝜆²2𝜉+4𝜇−𝜆²𝐶₂−𝜆𝐶₁×cos4𝜇−𝜆²2𝜉⁻¹−8𝛽𝛾²+3𝛾+2𝜇²𝛼𝛾²⋅𝐶₁cos4𝜇−𝜆²2𝜉+𝐶₂sin4𝜇−𝜆²2𝜉²×−4𝜇−𝜆²𝐶₁−𝜆𝐶₂sin4𝜇−𝜆²2𝜉+4𝜇−𝜆²𝐶₂−𝜆𝐶₁×cos4𝜇−𝜆²2𝜉²⁻¹. (42)

(x-3) When 𝜆²−4𝜇=0: 𝑤(𝜉)𝛾=−2𝛽²+3𝛾+2𝜇𝛼𝛾²+4𝛽𝛾²+3𝛾+2𝜆𝜇𝛼𝛾²𝐶₁+𝐶₂𝜉𝜆𝐶₁+𝐶₂𝜉−2𝐶₂−8𝛽𝛾²+3𝛾+2𝜇²𝛼𝛾²𝐶₁+𝐶₂𝜉²𝜆𝐶₁+𝐶₂𝜉−2𝐶₂². (43)

As an example, we consider solutions of Case (iv-1) when 𝐶₁=0 and 𝜇=−1; then the solution reduces to the soliton form as 𝑢(𝜉)=12𝛽𝛼sech²𝑥−4𝛽𝑡 (44) for 𝛾=1 as the standard KdV case and 𝑢(𝜉)=6𝛽𝛼sech𝑥−𝛽𝑡 (45) for the standard mKdV case when 𝛾=2. We show the diagrams of Case (iv-1) in Figure 1 to illustrate the behaviors of the solutions for different power index 𝛾. We choose coefficients of the generalized mKdV equation as 𝛼=6 and 𝛽=1; the constants of the solutions 𝐶₁ and 𝐶₂ equal 0.5 and 2, respectively, while the coefficient 𝜇 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 𝛾; (ii) it gives that the width of the wave packet broadened when increasing 𝛾; (iii) in addition, we find that the velocity of the wave slowed down when 𝛾 is bigger.

(a)

(b)

(c)

(d)

Figure 1 

Figures of solutions of Case (iv-1). The solution 𝑢(𝜉) evolutes with spatial coordinate 𝑥 and time 𝑡. The subscript of 𝑢 indicates the four kinds of different power index 𝛾 of the generalized mKdV equation.

4. Summary and Conclusion

In this paper, we use the modified (𝐺^′/𝐺)-expansion method to construct some types of solutions of the mKdV equation through the introduction of a proper transformation. Some new solutions are given, including the hyperbolic, trigonometric, and rational function solutions. It is shown that using the modified (𝐺^′/𝐺)-expansion method we can deal with the nonlinear evolution equations effectively and directly and abundant solutions could be obtained.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors acknowledge Qinghua Bu, Qingchun Zhou, and Mingxing Zhu for useful discussion. This work was supported by NSF-China under Grant nos. 11047101, 11105039, 11205071, and 11391240183.