An Extended Auxiliary Function Method and Its Application in mKdV Equation

Abstract

An extended auxiliary function method is presented for constructing exact traveling wave solutions to nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions to the elliptic equation to construct exact traveling wave solutions for nonlinear partial differential equations. mKdV equation is chosen to illustrate the application of the extended auxiliary function method. Consequently, more new exact traveling wave solutions are derived that are not obtained by the previously known methods.

1. Introduction

The investigation of exact solutions, in particular solitons, for nonlinear mathematical physics equations is an important and interesting subject. Many authors have paid attention to this subject. And many effective methods have been presented, such as Bäcklund transformation, Darboux transformation, variable separation approach, various tanh methods, Painlevé method, generalized hyperbolic-function method, homogeneous balance method, and similarity reduction method [1–17].

With the rapid development of computer algebraic systems like Maple or Mathematical, many powerful algebraic methods have been presented to obtain a great many exact solutions (especially traveling wave solutions) for nonlinear partial differential equation (NLPDE), for example, sine-cosine method [18], Fu's method [19], the hyperbolic tangent function expansion method [8, 20], Riccati equation expansion method [9, 21], the generalized Riccati equation method [22], auxiliary ordinary differential equation method [23], modified extended tanh method [24], Fan's unified algebraic method [25], the extended Fan's subequation method [26], and so on [26–36]. Generally speaking, these methods have a common characteristic: the possible solutions are constructed from the known functions or the solutions for some simple and solvable nonlinear ordinary differential equations such as sine-cosine function, tanh function, Jacobi elliptic function or the Riccati equation, the coupled Riccati equation, and sinh-Gordon equation. Recently, Xu presents the elliptic equation expansion method to construct new and more general solutions of NLPDE [37]. The main purpose of this paper is to summarize and extend Xu’s method to unified basic frame so that it can be used to obtain more exact traveling wave solutions for NLPDE. For illustration, we apply the extended method to the modified Korteweg-de Vries (mKdV) equation and get successfully many new exact solutions.

This paper is organized as follows. In Section 2, we give the main steps of our extended algorithm for computing exact solutions of NLPDE. In Section 3, many families of traveling wave solutions of mKdV equation are obtained. Finally, some conclusions and discussions are provided in Section 4.

2. The Extended Elliptic Equation Method and Its Algorithm

In the following we would like to outline the main steps of our method.

Step 1 (reduce the nonlinear PDE to the nonlinear ODE). For a Given nonlinear partial differential equation with the physical field 𝑢(𝑥,𝑡) in two variables 𝑥, 𝑡Δ𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑥𝑡,𝑢_𝑥𝑥,…=0. (1)
We consider its traveling wave solution in the form 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑘(𝑥−𝑐𝑡), (2) where 𝑘 and 𝑐 are constants to be determined later. Substituting (2) into (1) gives rise to a nonlinear ordinary differential equation with constant coefficients Θ𝑈,𝑈^′,𝑈^′′,…=0. (3)

Step 2 (set the series of formal solutions). To seek the traveling wave solutions to (3), in [37] Xu assumes that (3) has the solutions in the form of 𝑈(𝜉)=𝑎₀+^𝑛_𝑖=1𝑎_𝑖𝜑^𝑖(𝜉), (4) with the new variable 𝜑(𝜉) satisfying the elliptic equation 𝜑^′2=𝐴𝜑𝜑−𝛼𝜑−𝛽𝜑−𝛾, (5) where 𝐴, 𝛼, 𝛽, and 𝛾 are constants and 𝑛 is the integer to be determined later.
In order to get more exact solutions for (3), here we assume that (3) has the following formal solutions: 𝑈(𝜉)=𝑎₀+^𝑛_𝑖=1𝑎_𝑖𝜑^𝑖(𝜉)+𝑏_𝑖𝜑^𝑖−1(𝜉)𝜑^′(𝜉)+𝑐_𝑖𝜑^−𝑖(𝜉)+𝑑_𝑖𝜑^−𝑖(𝜉)𝜑^′(𝜉). (6)

Step 3 (determine the truncation expansion terms in (6)). In order to obtain the value of 𝑛 in (6). We define the degree of 𝑈(𝜉) as 𝐷[𝑈(𝜉)]=𝑛 which gives rise to the degrees of other expressions as 𝐷𝑑^𝑞𝑈𝑑𝜉^𝑞=𝑛+𝑞,𝐷𝑈^𝑝𝑑^𝑞𝑢𝑑𝜉^𝑞𝑠=𝑛𝑝+𝑠𝑛+𝑞. (7) Therefore we can get the value of 𝑛 in (6) by balancing the highest-order contributions from such terms in (1) or (3). If 𝑛 is a nonnegative integer, we first make the transformation 𝑈=𝑤^𝑛.

Step 4 (lead to the set of algebraic equations). We substitute (6) along with (5) into (3) and then set to zero the coefficients of like powers of 𝜑^𝑖𝜑^′𝑗  (𝑖=0,1,2,…; 𝑗=0,1) to obtain a set of nonlinear algebraic equations with respect to the unknowns 𝑎₀, 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖, and 𝑑_𝑖(𝑖=1,2,…).

Step 5 (solve the algebraic equations). By solving the overdetermined system of nonlinear algebraic equations by use of symbolic computation system Maple, we can get these unknowns 𝑎₀, 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖 and 𝑑_𝑖(𝑖=1,2,…).

Step 6 (solve (5)). Motivated by [37], we obtain some special solutions for (5) which are now listed in the following

Case 1. If 𝐴>0 and 𝛼<0<𝛽<𝛾, (5) has the solution 𝜑(𝜉)=𝛼𝛽𝑠𝑛²𝛼₁𝜉𝛼−𝛽𝑐𝑛²𝛼₁𝜉,𝛼₁=12𝐴𝛾𝛽−𝛼,𝑚=𝛽𝛾−𝛼𝛾.𝛽−𝛼 (8)

Case 2. If 𝐴>0 and 𝛼<𝛽<0<𝛾, (5) has the solution 𝜑(𝜉)=𝛼𝛾𝑠𝑛²𝛼₁𝜉𝛾−𝛼𝑐𝑛²𝛼₁𝜉,𝛼₁=12𝐴𝛽𝛼−𝛾,𝑚=𝛼𝛾−𝛽𝛽.𝛾−𝛼 (9)

Case 3. If 𝐴<0 and 0<𝛼<𝛽<𝛾, (5) has the solution 𝜑(𝜉)=𝛼𝛾−𝛽𝑠𝑛²𝛼₁𝜉−𝛽𝛾−𝛼𝛾−𝛽𝑠𝑛²𝛼₁𝜉−,𝛼𝛾−𝛼₁=12𝐴𝛽𝛼−𝛾,𝑚=𝛼𝛾−𝛽𝛽.𝛾−𝛼 (10)

Case 4. If 𝐴<0 and 𝛼<0<𝛽<𝛾, (5) has the solution 𝜑(𝜉)=𝛽𝛾𝛾+𝛽−𝛾𝑠𝑛²𝛼₁𝜉,𝛼₁=12−𝐴𝛾𝛽−𝛼,𝑚=𝛼𝛽−𝛾𝛾.𝛽−𝛼 (11)

Case 5. If 𝐴<0 and 𝛼<𝛽<0<𝛾, (5) has the solution 𝜑(𝜉)=𝛽𝛾𝑠𝑛²𝛼₁𝜉𝛽−𝛾𝑐𝑛²𝛼₁𝜉,𝛼₁=12𝐴𝛼𝛾−𝛽,𝑚=𝛾𝛼−𝛽𝛼.𝛾−𝛽 (12)

Step 7 (obtain the exact solutions to (3)). By inserting each solution to the above set of algebraic equations into (6) and making use of the solutions (8)–(12), some new types of elliptic function solutions for (3) can be obtained.

Remark 1. The method proposed here is more general than Xu's method for finding exact solutions for NLPDE. When 𝑏_𝑖=0, 𝑐_𝑖=0 and 𝑑_𝑖=0  (𝑖=1,2,…), our method can degenerate to Xu’s method.

3. The Application of the Elliptic Equation Expansion Method

In the following, we will apply the extended elliptic equation method to construct new explicit solutions of mKdV equation [1]: 𝑢_𝑡+𝛼𝑢²𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥=0, (13) where 𝛼 and 𝛽 are constants.

The mKdV equation is of important significance in many branches of nonlinear science field. When 𝛼=6, (13) is called the positive mKdV equation, while 𝛼=−6, (13) is called the negative mKdV equation. The mKdV equation appears in many fields such as acoustic waves in certain anharmonic lattices, Alfvén waves in a collisionless plasma, transmission lines in Schottky barrier, models of traffic congestion, ion acoustic solitons, elastic media, and so forth. It possesses many remarkable properties such as miura transformation, conservation laws, inverse scattering transformation, bilinear transformation, N-solitons, breather solutions, Bäcklund transformation, Painlevé integrability, Darboux transformation, and doublyperiodic solutions.

According to Step 1 in Section 2, in order to look for the traveling wave solutions for (13), we make the following traveling wave transformation: 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑘(𝑥−𝑐𝑡), (14) and substitute (14) into (13). Then (13) changes into the form −𝑐𝑈^′+𝛼𝑈²𝑈^′+𝛽𝑘²𝑈^′′′=0. (15)

According to Steps 2 and 3 in Section 2, by balancing the highest-order partial derivative term 𝑈^′′′ with the highest nonlinear term 𝑈²𝑈^′ in (15), we can obtain 𝑛=1. So we have 𝑈=𝑎₀+𝑎₁𝜑+𝑏₁𝜑^′+𝑐₁𝜑+𝑑₁𝜑^′𝜑, (16) where 𝑎₀, 𝑎₁, 𝑏₁, 𝑐₁ and 𝑑₁ are constants to be determined later and 𝜑 satisfies (5).

According to Step 4 in Section 2, with the aid of Maple, substituting (16) into (15) yields a set of algebraic equations for 𝜑^𝑖𝜑^′𝑗(𝑖=−1,0,1; 𝑗=0,1). Setting the coefficients of these terms 𝜑^𝑖𝜑^′𝑗 of the resulting system's numerator to zero yields a set of overdetermined algebraic equations with respect to the unknowns 𝑎₀, 𝑎₁, 𝑏₁, 𝑐₁, and 𝑑₁: 𝛼3𝛽𝑘²𝑏³₁𝛼𝐴=0,𝛽𝑘²𝑐²₁𝑑₁𝛼=0,𝛽𝑘²𝑎₁𝑏²₁𝛼𝐴=0,3𝛽𝑘²𝑐³₁−𝛼=0,3𝛽𝑘²−3𝑏²₁𝐴𝑑₁+𝑏³₁𝐴𝛽+𝑏³₁𝐴𝛼+𝑏³₁𝐴𝛾=0,6𝑏₁𝛼𝐴−3𝛽𝑘²−3𝑎²₁𝑏₁−𝑏³₁𝐴𝛼𝛾−𝑏³₁𝐴𝛼𝛽+3𝑏²₁𝐴𝑑₁𝛽+3𝑏²₁𝐴𝑑₁𝛼−𝑏³₁𝐴𝛽𝛾−3𝑏₁𝐴𝑑²₁+3𝑏²₁𝐴𝑑₁𝛾=0,2𝑑₁𝐴−3ℎ₃𝑎²₁𝑑₁−3ℎ₃𝑏²₁𝐴𝑑₁𝛼𝛾−3ℎ₃𝑏²₁𝐴𝑑₁𝛼𝛽−3ℎ₃𝑏²₁𝐴𝑑₁𝛽𝛾−3𝑏₁𝐴𝛾−3𝑏₁𝐴𝛽+ℎ₃𝑏³₁𝐴𝛼𝛽𝛾+3ℎ₃𝑏₁𝐴𝑑²₁𝛾−3𝑏₁𝐴𝛼+3ℎ₃𝑏₁𝐴𝑑²₁𝛼−ℎ₃𝑑³₁𝐴−6ℎ₃𝑎₀𝑎₁𝑏₁+3ℎ₃𝑏₁𝐴𝑑²₁−𝛼𝛽=0,𝛽𝑘²−𝑎₀𝑏²₁𝐴−2𝑎₁𝑏₁𝐴𝑑₁+𝑎₁𝑏²₁𝐴𝛼+𝑎₁𝑏²₁𝐴𝛾+𝑎₁𝑏²₁𝐴𝛽=0,2𝑎₁𝛼𝐴−3𝛽𝑘²6𝑎₁𝑏₁𝐴𝑑₁𝛽−3𝑎₁𝑏²₁𝐴𝛽𝛾−3𝑎₁𝑏²₁𝐴𝛼𝛾−3𝑎₁𝑏²₁𝐴𝛼𝛽−3𝑏²₁𝐴𝑐₁+3𝑎₀𝑏²₁𝐴𝛼+6𝑎₁𝑏₁𝐴𝑑₁𝛼−𝑎³₁−3𝑎₁𝑑²₁𝐴+3𝑎₀𝑏²₁𝐴𝛾−6𝑎₀𝑏₁𝐴𝑑₁+6𝑎₁𝑏₁𝐴𝑑₁𝛾+3𝑎₀𝑏²₁−3𝐴𝛽=0,2𝑎₁3𝐴𝛼−2𝑎₁3𝐴𝛽−2𝑎₁−𝛼𝐴𝛾𝛽𝑘²2𝑎₀𝑏₁𝐴𝑑₁𝛼−2𝑎₁𝑏₁𝐴𝑑₁𝛼𝛽−2𝑎₁𝑏₁𝐴𝑑₁𝛼𝛾−𝑎₀𝑏²₁𝐴𝛼𝛽−2𝑏₁𝐴𝑐₁𝑑₁+𝑏²₁𝐴𝑐₁𝛼+2𝑎₀𝑏₁𝐴𝑑₁𝛽−𝑎₀𝑎²₁+𝑏²₁𝐴𝑐₁𝛽−𝑎₀𝑏²₁𝐴𝛼𝛾−𝑎₀𝑏²₁𝐴𝛽𝛾+2𝑎₀𝑏₁𝐴𝑑₁−𝑎₀𝑑²₁𝐴+𝑏²₁𝐴𝑐₁𝛾+𝑎₁𝑑²₁𝐴𝛽+𝑎₁𝑏²₁𝐴𝛼𝛽𝛾+𝑎₁𝑑²₁𝐴𝛾−2𝑎₁𝑏₁𝐴𝑑₁𝛽𝛾+𝑎₁𝑑²₁𝑎𝐴𝛼=0,₁𝐴−𝑐𝛼𝛽+𝛽𝛾+𝛼𝛾𝛽𝑘²𝑎₁−𝛼𝛽𝑘²2𝑏₁𝐴𝑐₁𝑑₁𝛾+2𝑏₁𝐴𝑐₁𝑑₁𝛽−𝑎²₀𝑎₁+2𝑏₁𝐴𝑐₁𝑑₁𝛼−𝑎₁𝑑²₁𝐴𝛽𝛾−𝑎₁𝑑²₁𝐴𝛼𝛾−𝑎₁𝑑²₁𝐴𝛼𝛽−𝑏²₁𝐴𝑐₁𝛽𝛾−𝑎²₁𝑐₁−𝑏²₁𝐴𝑐₁𝛼𝛾−𝑏²₁𝐴𝑐₁𝛼𝛽+𝑎₀𝑑²₁𝐴𝛾+𝑎₀𝑑²₁𝐴𝛽−𝑐₁𝑑²₁𝐴+𝑎₀𝑑²₁𝐴𝛼+2𝑎₁𝑏₁𝐴𝑑₁𝛼𝛽𝛾+𝑎₀𝑏²₁𝐴𝛼𝛽𝛾−2𝑎₀𝑏₁𝐴𝑑₁𝛼𝛾−2𝑎₀𝑏₁𝐴𝑑₁𝛽𝛾−2𝑎₀𝑏₁𝐴𝑑₁𝛼𝛽=0. (17)

According to Step 5 in Section 2, by use of the Maple software package “Charsets” by Dongming Wang, which is based on the Wu-elimination method [38], solving the previous overdetermined algebraic equations, we can get the following three sets of explicit expressions for 𝑎₀, 𝑎₁, 𝑏₁, 𝑐₁, and 𝑑₁.

Family 1. Consider 𝑎₀1=∓4−6𝛽𝑘²𝐴𝛼,𝑎𝛼+𝛽+𝛾₁=±−6𝛽𝑘²𝐴𝛼,𝑏₁=0,𝑐₁=0,𝑑₁𝑐=0,𝛽𝑘²𝐴=−83𝛽²−2𝛼𝛽−2𝛽𝛾+3𝛼²−2𝛼𝛾+3𝛾². (18)

Family 2. Consider 𝑎₀=0,𝑎₁𝑏=0,₁=0,𝑐₁𝑑=0,₁=±−6𝛽𝑘²𝛼 (19)

Family 3. Consider 𝑎₀=0,𝑎₁=±−3𝛽𝑘²𝐴,𝑏2𝛼₁=0,𝑐₁=0,𝑑₁=±−3𝛽𝑘²,𝑐2𝛼𝛽𝑘²1=−2𝐴𝛼𝛽+𝛽𝛾+𝛼𝛾. (20)

According to Step 7 in Section 2, substituting (18)–(20) into (16), respectively, and combining Step 6 in Section 2, we get the following traveling wave solutions to (15).

Family 1. We obtained the following exact solutions: 𝑢₁₁1=∓4−6𝛽𝑘²𝐴𝛼±𝛼+𝛽+𝛾−6𝛽𝑘²𝐴𝛼𝛼𝛽𝑠𝑛²𝛼₁𝜉𝛼−𝛽𝑐𝑛²𝛼₁𝜉,𝑚=𝛽𝛾−𝛼𝛾,𝛼𝛽−𝛼₁=12,𝑐𝐴𝛾𝛽−𝛼𝛽𝑘²𝐴=−83𝛽²−2𝛼𝛽−2𝛽𝛾+3𝛼²−2𝛼𝛾+3𝛾²,𝑢₁₂=±−6𝛽𝑘²𝛼2𝛼₁𝛼−𝛽𝑐𝑛𝛼₁𝜉𝑑𝑛𝛼₁𝜉𝑠𝑛𝛼₁𝜉𝛼−𝛽𝑐𝑛²𝛼₁𝜉,𝛼₁=12𝐴𝛾𝛽−𝛼,𝑚=𝛽𝛾−𝛼𝛾,𝑢𝛽−𝛼₁₃=±−3𝛽𝑘²𝐴2𝛼𝛼𝛽𝑠𝑛²𝛼₁𝜉𝛼−𝛽𝑐𝑛²𝛼₁𝜉±−3𝛽𝑘²𝛼2𝛼¹𝛼−𝛽𝑐𝑛𝛼₁𝜉𝑑𝑛𝛼₁𝜉𝑠𝑛𝛼₁𝜉𝛼−𝛽𝑐𝑛²𝛼₁𝜉,𝛼₁=12𝐴𝛾𝛽−𝛼,𝑚=𝛽𝛾−𝛼𝛾,𝑐𝛽−𝛼𝛽𝑘²1=−2𝐴𝛼𝛽+𝛽𝛾+𝛼𝛾, (21) where 𝜉=𝑘(𝑥−𝑐𝑡), 𝐴>0, and 𝛼<0<𝛽<𝛾.

Family 2. We obtained the following exact solutions: 𝑢₂₁1=∓4−6𝛽𝑘²𝐴𝛼±𝛼+𝛽+𝛾−6𝛽𝑘²𝐴𝛼𝛼𝛾𝑠𝑛²𝛼₁𝜉𝛾−𝛼𝑐𝑛²𝛼₁𝜉,𝑚=𝛼𝛾−𝛽𝛽,𝛼𝛾−𝛼₁=12,𝑐𝐴𝛽𝛼−𝛾𝛽𝑘²𝐴=−83𝛽²−2𝛼𝛽−2𝛽𝛾+3𝛼²−2𝛼𝛾+3𝛾²,𝑢₂₂=±−6𝛽𝑘²𝛼2𝛼₁𝛾−𝛼𝑐𝑛𝛼₁𝜉𝑑𝑛𝛼₁𝜉𝑠𝑛𝛼₁𝜉𝛾−𝛼𝑐𝑛²𝛼₁𝜉,𝛼₁=12𝐴𝛽𝛼−𝛾,𝑚=𝛼𝛾−𝛽𝛽,𝑢𝛾−𝛼₂₃=±−3𝛽𝑘²𝐴2𝛼𝛼𝛾𝑠𝑛²𝛼₁𝜉𝛾−𝛼𝑐𝑛²𝛼₁𝜉±−3𝛽𝑘²2𝛼2𝛼₁𝛾−𝛼𝑐𝑛𝛼₁𝜉𝑑𝑛𝛼₁𝜉𝑠𝑛𝛼₁𝜉𝛾−𝛼𝑐𝑛²𝛼₁𝜉,𝛼₁=12𝐴𝛽𝛼−𝛾,𝑚=𝛼𝛾−𝛽𝛽,𝑐𝛾−𝛼𝛽𝑘²1=−2𝐴𝛼𝛽+𝛽𝛾+𝛼𝛾, (22) where 𝜉=𝑘(𝑥−𝑐𝑡), 𝐴>0, and 𝛼<𝛽<0<𝛾.

Family 3. We obtained the following exact solutions: 𝑢₃₁1=∓4−6𝛽𝑘²𝐴𝛼±𝛼+𝛽+𝛾−6𝛽𝑘²𝐴𝛼𝛼𝛾−𝛽𝑠𝑛²𝛼₁𝜉−𝛽𝛾−𝛼𝛾−𝛽𝑠𝑛²𝛼₁𝜉−,𝛼𝛾−𝛼₁=12𝐴𝛽𝛼−𝛾,𝑚=𝛼𝛾−𝛽𝛽,𝑐𝛾−𝛼𝛽𝑘²𝐴=−83𝛽²−2𝛼𝛽−2𝛽𝛾+3𝛼²−2𝛼𝛾+3𝛾²,𝑢₃₂=±−6𝛽𝑘²𝛼×2𝛼₁𝛼−𝛽𝛽−𝛾𝛾−𝛼𝑠𝑛𝛼₁𝜉𝑐𝑛𝛼₁𝜉𝑑𝑛𝛼₁𝜉𝛼𝛾−𝛽𝑠𝑛²𝛼₁𝜉−𝛽𝛾−𝛼𝛾−𝛽𝑠𝑛²𝛼₁𝜉−,𝛼𝛾−𝛼₁=12𝐴𝛽𝛼−𝛾,𝑚=𝛼𝛾−𝛽𝛽,𝑢𝛾−𝛼₃₃=±−3𝛽𝑘²×2𝛼2𝛼₁𝛼−𝛽𝛽−𝛾𝛾−𝛼𝑠𝑛𝛼₁𝜉𝑐𝑛𝛼₁𝜉𝑑𝑛𝛼₁𝜉𝛼𝛾−𝛽𝑠𝑛²𝛼₁𝜉−𝛽𝛾−𝛼𝛾−𝛽𝑠𝑛²𝛼₁𝜉−±𝛾−𝛼−3𝛽𝑘²𝐴𝛼2𝛼𝛾−𝛽𝑠𝑛²𝛼₁𝜉−𝛽𝛾−𝛼𝛾−𝛽𝑠𝑛²𝛼₁𝜉−,𝑐𝛾−𝛼𝛽𝑘²1=−2𝐴𝛼𝛽+𝛽𝛾+𝛼𝛾,𝛼₁=12𝐴𝛽𝛼−𝛾,𝑚=𝛼𝛾−𝛽𝛽,𝛾−𝛼 (23)where 𝜉=𝑘(𝑥−𝑐𝑡), 𝐴<0 and 0<𝛼<𝛽<𝛾.

Family 4. We obtained the following exact solutions: 𝑢₄₁1=∓4−6𝛽𝑘²𝐴𝛼±𝛼+𝛽+𝛾−6𝛽𝑘²𝐴𝛼𝛽𝛾𝛾+𝛽−𝛾𝑠𝑛²𝛼₁𝜉,𝑚=𝛼𝛽−𝛾𝛾,𝛼𝛽−𝛼₁=12,𝑐−𝐴𝛾𝛽−𝛼𝛽𝑘²𝐴=−83𝛽²−2𝛼𝛽−2𝛽𝛾+3𝛼²−2𝛼𝛾+3𝛾²,𝑢₄₂=±−6𝛽𝑘²𝛼2𝛼₁𝛾−𝛽𝑠𝑛𝛼₁𝜉𝑐𝑛𝛼₁𝜉𝑑𝑛𝛼₁𝜉𝛾+𝛽−𝛾𝑠𝑛²𝛼₁𝜉,𝛼₁=12−𝐴𝛾𝛽−𝛼,𝑚=𝛼𝛽−𝛾𝛾,𝑢𝛽−𝛼₄₃=±−3𝛽𝑘²𝐴2𝛼𝛽𝛾𝛾+𝛽−𝛾𝑠𝑛²𝛼₁±𝜉,𝑚−3𝛽𝑘²2𝛼2𝛼₁𝛾−𝛽𝑠𝑛𝛼₁𝜉𝑐𝑛𝛼₁𝜉𝑑𝑛𝛼₁𝜉𝛾+𝛽−𝛾𝑠𝑛²𝛼₁𝜉,𝛼₁=12−𝐴𝛾𝛽−𝛼,𝑚=𝛼𝛽−𝛾𝛾,𝑐𝛽−𝛼𝛽𝑘²1=−2𝐴𝛼𝛽+𝛽𝛾+𝛼𝛾, (24) where 𝜉=𝑘(𝑥−𝑐𝑡), 𝐴<0, and 𝛼<0<𝛽<𝛾.

Family 5. We obtained the following exact solutions: 𝑢₅₁1=∓4−6𝛽𝑘²𝐴𝛼±𝛼+𝛽+𝛾−6𝛽𝑘²𝐴𝛼𝛽𝛾𝑠𝑛²𝛼₁𝜉𝛽−𝛾𝑐𝑛²𝛼₁𝜉,𝑚=𝛾𝛼−𝛽𝛼,𝛼𝛾−𝛽₁=12,𝑐𝐴𝛼𝛾−𝛽𝛽𝑘²𝐴=−83𝛽²−2𝛼𝛽−2𝛽𝛾+3𝛼²−2𝛼𝛾+3𝛾²,𝑢₅₂=±−6𝛽𝑘²𝛼2𝛼₁𝛽−𝛾𝑐𝑛𝛼₁𝜉𝑑𝑛𝛼₁𝜉𝑠𝑛𝛼₁𝜉𝛽−𝛾𝑐𝑛²𝛼₁𝜉,𝛼₁=12𝐴𝛼𝛾−𝛽,𝑚=𝛾𝛼−𝛽𝛼,𝑢𝛾−𝛽₅₃=±−3𝛽𝑘²𝐴2𝛼𝛽𝛾𝑠𝑛²𝛼₁𝜉𝛽−𝛾𝑐𝑛²𝛼₁𝜉±−3𝛽𝑘²2𝛼2𝛼₁𝛽−𝛾𝑐𝑛𝛼₁𝜉𝑑𝑛𝛼₁𝜉𝑠𝑛𝛼₁𝜉𝛽−𝛾𝑐𝑛²𝛼₁𝜉,𝛼₁=12𝐴𝛼𝛾−𝛽,𝑚=𝛾𝛼−𝛽𝛼,𝑐𝛾−𝛽𝛽𝑘²1=−2𝐴𝛼𝛽+𝛽𝛾+𝛼𝛾, (25) where 𝜉=𝑘(𝑥−𝑐𝑡), 𝐴<0, and 𝛼<𝛽<0<𝛾.

Remark 2. The previous solutions obtained here, to our knowledge, are all new families of rational formal solutions to mKdV equation.

4. Conclusion and Discussions

In summary, we have presented the extended elliptic equation method and its algorithm. As a result, some new exact traveling wave solutions for the mKdV equation are obtained which may be useful for describing certain nonlinear physical phenomena. The method which we have proposed in this paper is a standard, direct, and computerized method, which allow us to do complicated and tedious algebraic calculation. It is shown that the algorithm can be also applied to other NLPDE in mathematical physics. In addition, a natural problem is that whether the extended elliptic equation method can be further extended to generate more types of solutions of NLPDE in nonlinear science. Hence, further study of the extended elliptic equation method is needed.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

Many thanks are due to suggestions from the referees and support from the National Natural Science Foundation of China (Grant no. 61171179) and the Young Scientists Fund of the National Natural Science Foundation of China (Grant no. 10901145).