Exp-Function Method for a Generalized MKdV Equation

Abstract

Under investigation in this paper is a generalized MKdV equation, which describes the propagation of shallow water in fluid mechanics. In this paper, we have derived the exact solutions for the generalized MKdV equation including the bright soliton, dark soliton, two-peak bright soliton, two-peak dark soliton, shock soliton and periodic wave solution via Exp-function method. By figures and symbolic computations, we have discussed the propagation characteristics of those solitons under different values of those coefficients in the generalized MKdV equation. The method constructing soliton solutions in this paper may be useful for the investigations on the other nonlinear mathematical physics model and the conclusions of this paper can give theory support for the study of dynamic features of models in the shallow water.

1. Introduction

Nonlinear science studies nonlinear phenomena of the world, which is cross-disciplinary [1–4]. In fact, any science, whether natural science or social science, has its own nonlinear phenomena and problems. To study these phenomena and problems, many branches of nonlinear science have been promoted to be built and developed [5–7]. Obviously, most of the phenomena are not linear in the natural sciences and engineering practice; thus many problems cannot be researched and solved by the linear methods, which makes the study of nonlinear science very significant. Actually, nonlinear science, which mainly consists of soliton, chaos, and fractal [3, 4, 7], is not the simple superposition and comprehensive of these nonlinear branches but a comprehensive subject to study the various communist rules in nonlinear phenomena. With the rapid development of nonlinear science, the study of exact solutions of nonlinear evolution equations has attracted much attention of many mathematicians and physicists.

In 1834, the solitary wave phenomenon was observed by Huang et al. [8] and Russell [9]. Later, people named the isolation water peak, which kept moving with constant shape and speed on the surface of the water, as a solitary wave [8]. The discovery of a solitary wave turned people into a new field in the study of the waves of the convection. In 1895, Holland’s Professor Kortewrg and his disciples Vries derived the famous KdV equation from the research of shallow water wave motion [10]. The analysis to the KdV equation has improved to recover inverse scattering method, on which people expanded the new research directions of algebra and geometry [8]: 𝑢_𝑡±6𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=0. (1) MKdV equation plays an important role in describing the plasmas and phonon in anharmonic lattice. In the nineteen fifties, physicist Fermi, Pasta, and Ulam made a famous FPU experiment, connecting the 64 particles by nonlinear spring, thereby forming a nonlinear vibrating string. Although the FPU experiment did not gain the solutions of solitary wave, it will expand the study of the solitary wave to the field outside the mechanics [11]. Later, Fermi et al. studied the nonlinear vibration problem of FPU model and obtained the solitary wave solutions [11], which is the right answer for the question of FPU. It is the first time to find solitary wave solutions in the field outside mechanics after the solution was found in the KdV equation, which give rise to the scientists’ interest in researching the solitary wave phenomenon. In 1965, Zabusky and Kruskal studied solitary waves in plasma [12]; they found that the waveform does not change nature before and after collision of nonlinear solitary waves, which is similar to particle collisions, so Zabusky and Kruskal named the solitary wave with the impact properties of collisions as soliton [12]. Soliton concept is an important milepost on the history of the development of the soliton theory. The next few decades, the soliton theory had a rapid development and penetrated into many areas, such as fluid mechanics, nonlinear optical fiber communication, plasma physics, fluid physics, chemistry, life science, and marine science [13–15].

Recently many new approaches to nonlinear wave equations have been proposed, such as Tanh-function method [16–18], F-expansion method [19], Jacobian elliptic function method [20], Darboux transformation method [21–26], adomian method [27–29], variational approach [30], and homotopy perturbation method [31]. All methods mentioned above have their limitations in their applications. We will apply Exp-function method to a generalized MKdV equation to gain exact solutions: 𝑢_𝑡+𝛼𝑢²𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥=0, (2) where 𝛼 and 𝛽 are real parameters. When 𝑢(𝑥,𝑡)=𝑐 (𝑐 is a constant) as 𝑡→±∞, solitons can be obtained.

This paper will be organized as follows. In Section 2, the basic idea of Exp-function method is introduced. In Section 3, carrying on calculating and illustrating by the mathematical software MATHEMATIC, we will solve the solitary wave solutions of (2) based on the Exp-function method. In Section 4, we will obtain shock soliton, bright soliton, two-peak bright soliton, dark soliton, two-peak dark soliton, and periodic wave solutions and analyze the dynamic features of soliton solutions by using some figures. Finally, our conclusions will be addressed in Section 5.

2. Basic Idea of Exp-Function Method

In order to illustrate the basic idea of the suggested method, we consider firstly the following general partial differential equation: 𝐹𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑡,𝑢_𝑥𝑥,…=0. (3) We aim at its exact solutions, so we introduce a complex variable, 𝜉, defined as 𝜉=𝑘𝑥+𝜔𝑡, (4) where 𝜔 and 𝑘 are constants unknown to be further determined. Therefore we can convert (3) into an ordinary differential equation with respect to 𝜉: 𝐹𝑢,𝑢_𝜉,𝑢_𝜉𝜉,…=0. (5) Very simple and straightforward, the Exp-function method is based on the assumption that traveling wave solutions can be expressed in the following form: 𝑢(𝜉)=∑^𝑐_𝑛=−𝑑𝑎_𝑛exp(𝑛𝜉)∑^𝑝_𝑚=−𝑞𝑏_𝑚(),exp𝑚𝜉 (6) where 𝑐, 𝑑, 𝑝, and 𝑞 are positive integers which are unknown to be further determined and 𝑎_𝑛 and 𝑏_𝑚 are unknown constants.

We suppose that 𝑢(𝜉) which is the solution of (5) can be expressed as 𝑢(𝜉)=𝑎_𝑐exp(𝑐𝜉)+⋯+𝑎_−𝑑exp(−𝑑𝜉)𝑏_𝑝exp𝑝𝜉+⋯+𝑏_−𝑞.exp−𝑞𝜉 (7)

To determine the values of 𝑐 and 𝑝, we balance the linear term of highest order in (5) with the highest order nonlinear term. Similarly we balance the lowest orders to confirm 𝑑 and 𝑞. For simplicity, we set some particular values for 𝑐, 𝑝, 𝑑, and 𝑞 and then change the left side of (5) into the polynomial of exp(𝑛𝜉). Equating the coefficients of exp(𝑛𝜉) to be zero results in a set of algebraic equations. Then solving the algebraic system with symbolic computation system, we can gain the solution. Substituting it into (3), we have the general form of the exact solution expressed as the form of exp(𝑛𝜉). Taking the form of the solutions into consideration, we can have more extensive solutions via Exp-function method.

3. Solving Generalized MKdV Equation via Exp-Function Method

Now we consider the follwing equation [32, 33]: 𝑢_𝑡+𝑢²𝑢_𝑥+𝑢_𝑥𝑥𝑥=0. (8) This equation is called modified KdV equation, which arises in the process of understanding the role of nonlinear dispersion and in the formation of structures like liquid drops. The KdV equation is one of the most familiar models for solitons and the foundation which studies other equations. Developing upon this foundation, the isolated theories are treated as the milestone of mathematics physical method. During researching the isolated theories, the remarkable application should be laser shooting practice and fiber-optic communication.

For researching the variety of the solutions for the modified KdV equation, we carry on the MKdV equation to expand. Thus, we have 𝑢_𝑡+𝛼𝑢²𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥=0, (9) where 𝛼 and 𝛽 are real parameters. In the following, we consider (9).

Introduce a complex variable, 𝜂, defined as 𝑢=𝑢𝜂,𝜂=𝑘𝑥+𝜔𝑡. (10) By (10), (9) becomes 𝜔𝑢^′+𝛼𝑘𝑢²𝑢^′+𝛽𝑘³𝑢^′′′=0, (11) which denotes the differential with respect to 𝜂.

The Exp-function method is based on the assumption that traveling wave solutions can be expressed in the following form: 𝑢𝜂=∑^𝑑_𝑛=−𝑐𝑎_𝑛exp𝑛𝜂∑^𝑞_𝑚=−𝑝𝑏_𝑚,exp𝑚𝜂 (12) where 𝑐, 𝑑, 𝑝, and 𝑞 are positive integers unknown to be further determined and 𝑎_𝑛 and 𝑏_𝑚 are unknown constants.

We suppose that the solution of (11) can be expressed as 𝑢𝜂=𝑎_𝑐exp𝑐𝜂+⋯+𝑎_−𝑑exp−𝑑𝜂𝑏_𝑝exp𝑝𝜂+⋯+𝑏_−𝑞.exp−𝑞𝜂 (13) To determine values of 𝑐 and 𝑝, we balance the linear term of highest order in (11) with the highest order nonlinear term. By simple calculation, we have 𝑢^′′′=𝑐₁exp7𝑝+𝑐𝜂+⋯𝑐₂,𝑢exp8𝑝𝜂+⋯₂𝑢^′=𝑐₃exp𝑝+3𝑐𝜂+⋯𝑐₄=𝑐exp4𝑝𝜂+⋯³exp5𝑝+3𝑐𝜂+⋯𝑐₄,exp8𝑝𝜂+⋯ (14) where 𝑐_𝑖(𝑖=1,2,3,4) are coefficients only for simplicity.

Balancing highest order of exponential function in (14), we have 5𝑝+𝑐=7𝑝+𝑐, (15) which leads to the result 𝑝=𝑐. (16) Similarly to make sure of the values of 𝑑 and 𝑞, we balance the linear term of lowest order in (11): 𝑢^′′′=⋯+𝑑₁exp−7𝑞+𝑑𝜂⋯+𝑑₂,𝑢exp−8𝑞𝜂₂𝑢^′=⋯+𝑑₃exp−𝑞+3𝑑𝜂⋯+𝑑₄=exp−8𝑞𝜂⋯+𝑑₃exp−5𝑞+3𝑑𝜂⋯+𝑑₄,exp−8𝑞𝜂 (17) where 𝑑_𝑖(𝑖=1,2,3,4) are determined to be coefficients only for simplicity.

Balancing lowest order of exponential function in (17), we have −7𝑞+𝑑=−5𝑞+3𝑑, (18) which leads to the result 𝑞=𝑑. (19) For simplicity, we set 𝑝=𝑐=1 and 𝑞=𝑑=1, so (13) reduces to 𝑢𝜂=𝑎₁exp𝜂+𝑎₀+𝑎₋₁exp−𝜂exp𝜂+𝑏₀+𝑏₋₁.exp−𝜂 (20) Substituting (20) into (11), and by the help of Mathematica, we have 1𝐴𝐶₃exp3𝜂+𝐶₂exp2𝜂+𝐶₁exp𝜂+𝐶₀+𝐶₋₁exp−𝜂+𝐶₋₂exp−2𝜂+𝐶₋₃exp−3𝜂=0, (21) where 𝐴=𝑏₋₁+𝑒^2𝜂+𝑏₀𝑒^𝜂4,𝐶₃=ℎ+𝑘𝛼𝑎²₁−𝑎₀+𝑎₁𝑏₀,𝐶₂=−2𝑘𝛼𝑎²₀𝑎₁−2𝑎₋₁3𝑘³𝛽+ℎ+𝑘𝛼𝑎²₁+2𝑎₀3𝑘³𝛽−ℎ+𝑘𝛼𝑎²₁𝑏₀+2𝑎₁3𝑘³𝛽+ℎ+𝑘𝛼𝑎²₁𝑏₋₁+−3𝑘³𝛽+ℎ𝑏²₀,𝐶₁=𝑎₀−𝑘𝛼𝑎²₀+6𝑎₋₁𝑎₁+24𝑘³𝛽−ℎ+5𝑘𝛼𝑎²₁𝑏₋₁−ℎ𝑏²₀𝑎₀−𝑎₁𝑏₀+𝑎₋₁−5ℎ+𝑘𝛼𝑎²₁+𝑎₁𝑘𝛼𝑎²₀+6−4𝑘³𝛽+ℎ𝑏₋₁𝑏₀,𝐶₀=−4𝑎₋₁−𝑎₁𝑏₋₁×𝑘𝛼𝑎²₀+𝑎₋₁𝑎₁−9𝑘³𝛽−ℎ𝑏₋₁+ℎ𝑏²₀,𝐶₋₁=𝑎₋₁6𝑘𝛼𝑎₀𝑎₁𝑏₋₁+−𝑘𝛼𝑎²₀+64𝑘³𝛽−ℎ𝑏₋₁−ℎ𝑏²₀𝑏₀−𝑘𝛼𝑎²₋₁5𝑎₀+𝑎₁𝑏₀+𝑏₋₁𝑘𝛼𝑎³₀+5ℎ𝑎₁𝑏₋₁𝑏₀+𝑎₀−24𝑘³𝛽+ℎ𝑏₋₁+ℎ𝑏²₀,𝐶₋₂=2𝑏²₋₁𝑎₋₁−𝑎₁𝑏₋₁3𝑘³𝛽+ℎ+2𝑏₀𝑏₋₁𝑎₋₁𝑏₀−𝑎₀𝑏₋₁−3𝑘³𝛽+ℎ−2𝑘𝛼𝑎₋₁×𝑎₋₁𝑎₋₁+𝑎₀𝑏₀−𝑏₋₁𝑎²₀+𝑎₁𝑎₋₁,𝐶₋₃=−𝑘𝛼𝑎²₋₁+ℎ𝑏²₋₁−𝑎₀𝑏₋₁+𝑎₋₁𝑏₀, (22) with ℎ=𝑘³𝛽+𝜔.

Equating the coefficients of exp(𝑛𝜂) to be zero, we have 𝐶₃=0,𝐶₂=0,𝐶₁=0,𝐶₀𝐶=0,₋₃=0,𝐶₋₂=0,𝐶₋₁=0. (23) Solving (23), simultaneously, we obtain 𝜔=−𝑘³𝛽−𝑘𝛼𝑎²₁,𝑎₋₁=3𝑘²𝛽𝑏²₀+2𝛼𝑎²₁𝑏²₀8𝛼𝑎₁,𝑎₀=3𝑘²𝛽𝑏₀+𝛼𝑎²₁𝑏₀𝛼𝑎₁,𝑏₋₁=3𝑘²𝛽𝑏²₀+2𝛼𝑎²₁𝑏²₀8𝛼𝑎²₁. (24) Substituting (24) into (20) results in a compact-like solution, which reads 𝑢(𝑥,𝑡)=𝑎₁1+24𝑦𝑘²𝛽𝑏₀3𝑘²𝛽𝑏²₀+2𝛼𝑎²₁2𝑦+𝑏₀², (25) where 𝑎₁, 𝑏₀, 𝛼, 𝛽, and 𝑘 are real parameters and 𝑦=𝑒^{𝑘𝑥+𝑡(−𝑘3𝛽−𝑘𝛼𝑎21)}. Besides, the obtained solution equation (25) is a generalized soliton solution of (9).

In case, 𝑘 is an imaginary parameter, the obtained soliton solution can be converted into the periodic solution or compact-like solution. We write 𝑘=𝑖𝐾, (26) where 𝐾 is a real parameter.

Use the transformations exp𝑘𝑥+𝑘²+6𝑎₁+1𝑘𝑡=cos𝐾𝑥+−𝐾²+6𝑎₁+1𝐾𝑡+𝑖sin𝐾𝑥+−𝐾²+6𝑎₁+1𝐾𝑡,exp−𝑘𝑥−𝑘²+6𝑎₁+1𝑘𝑡=cos𝐾𝑥+−𝐾²+6𝑎₁+1𝐾𝑡−𝑖sin𝐾𝑥+−𝐾²+6𝑎₁+1𝐾𝑡. (27) Equation (25) becomes 𝑢(𝑥,𝑡)=𝑎₁1+−24𝐾²𝛽cos𝑦₁+𝑖sin𝑦₁𝑏₀×−3𝐾²𝛽𝑏²₀+2𝛼𝑎²₁×2cos𝑦₁+𝑖sin𝑦₁+𝑏₀²⁻¹, (28) where 𝑎₁, 𝑏₀, 𝛼, 𝛽, and 𝐾 are real parameters and 𝑦₁=𝐾𝑥−(𝛼𝐾𝑎²₁−𝛽𝐾³)𝑡. Simplifying the above formula, we have 𝑢(𝑥,𝑡)=𝑎₁+−3𝐾²𝑏₀𝛽/𝛼𝑎₁𝑏₀+1+𝜌cos𝑦₁+𝑖1−𝜌sin𝑦₁, (29) where 𝑎₁, 𝑏₀, 𝛼, 𝛽, and 𝐾 are real parameters and 𝜌=𝑏²₀(−3𝛽𝐾²+2𝛼𝑎²₁)/8𝛼𝑎²₁.

If we search for a periodic solution or compact-like solution, the imaginary part in the denominator of (29) must be zero, which requires that 𝑏1−𝜌=1−²⁰−3𝛽𝐾²+2𝛼𝑎²₁8𝛼𝑎²₁=0. (30) Solving 𝑏₀ from (30), we obtain 𝑏₀=±8𝛼𝑎²₁−3𝛽𝐾²+2𝛼𝑎²₁. (31) Substituting (31) into (29) results in a compact-like solution, which reads 𝑢(𝑥,𝑡)=𝑎₁−6√2𝐾²𝛽𝑔2√2𝛼𝑎₁𝑔±cos𝑦₁−𝛼𝑎₁+𝑔²3𝐾²𝛽−2𝛼𝑎²₁, (32) where 𝑎₁ and 𝐾 are free parameters, and 𝑦₁=𝐾𝑥−(𝛼𝐾𝑎²₁−𝛽𝐾³)𝑡, 𝑔=𝛼𝑎²₁/(−3𝐾²𝛽+2𝛼𝑎²₁), and it requires that 𝛼𝑎²₁/(−3𝐾²𝛽+2𝛼𝑎²₁)>0.

4. Discussing the Forms of the Solutions

In the above, we gain two cases of solution equations (25) and (32). In order to get the richness of solutions of (9), we have these two cases discussed in the following.

4.1. The Solutions from (25)

Through discussions, we obtain the following results.

4.1.1. When 𝛼=0 and 𝛽=1

when 𝛼=0 and 𝛽=1, (9) becomes 𝑢_𝑡+𝑢_𝑥𝑥𝑥=0. Then we can obtain shock solitons. The solution formula is shown as follows: 𝑢(𝑥,𝑡)=𝑎₁1+8𝑒^{−𝑘3𝑡+𝑘𝑥}𝑏₀=𝑎₁+8𝑎₁cosh𝑘³𝑡−𝑘𝑥𝑏₀−8𝑎₁sinh𝑘³𝑡−𝑘𝑥𝑏₀, (33) where 𝑎₁, 𝑏₀, and 𝑘 are real parameters.

Case 1 (when 𝑎₁=1, 𝑏₀=2, 𝑘=3). Consider𝑢₁(𝑥,𝑡)=1+4𝑒^{−27𝑡+3𝑥}=1+4cosh(27𝑡−3𝑥)−4sinh(27𝑡−3𝑥). (34) Description is as shown in Figure 1(a).

(a)

(b)

Figure 1 

Evolution of the solutions of (25). (a) Shock soliton: parameters are 𝛼=0, 𝛽=1, 𝑎1=1, 𝑏0=2, and 𝑘=3. (b) A shock soliton in collision with a dark soliton: parameters are 𝛼=0, 𝛽=1, 𝑎1=3, 𝑏0=−2, and 𝑘=4.

Case 2 (when 𝑎₁=3, 𝑏₀=−2, and 𝑘=4). Consider𝑢₂(𝑥,𝑡)=31−4𝑒^{−64𝑡+4𝑥}=3−12cosh(64𝑡−4𝑥)+12sinh(64𝑡−4𝑥). (35) Description is as shown in Figure 1(b).

4.1.2. When 𝛼=1 and 𝛽=1

when 𝛼=1 and 𝛽=1, (9) becomes 𝑢_𝑡+𝑢²𝑢_𝑥+𝑢_𝑥𝑥𝑥=0. Then we can obtain one-peak bright solitons. The solution formula is shown as follows: 𝑢(𝑥,𝑡)=𝑎₁1+24𝑒^𝑦𝑘²𝑏₀3𝑘²𝑏²₀+2𝑎²₁2𝑒^𝑦+𝑏₀²=𝑎₁1+24𝑘²cosh𝑦+sinh𝑦𝑏₀3𝑘²𝑏²₀+2𝑎²₁2cosh𝑦+2sinh𝑦+𝑏₀², (36) where 𝑎₁, 𝑏₀, and 𝑘 are real parameters and 𝑦=𝑘𝑥+𝑡(−𝑘³−𝑘𝑎²₁).

Case 3 (when 𝑎₁=1, 𝑏₀=2, and 𝑘=2). Consider𝑢₃(𝑥,𝑡)=1+192𝑒^{−10𝑡+2𝑥}48+22+2𝑒^{−10𝑡+2𝑥2}(()(=1+192cosh10𝑡−2𝑥−sinh10𝑡−2𝑥))48+2(2+2cosh(10𝑡−2𝑥)−2sinh(10𝑡−2𝑥))². (37) Description is as shown in Figure 2(a).

(a)

(b)

Figure 2 

Evolution of the solutions of (25). (a) Bright soliton: parameters are 𝛼=1, 𝛽=1, 𝑎1=1, 𝑏0=2, and 𝑘=2. (b) A bright soliton in collision with two dark solitons: parameters are 𝛼=1, 𝛽=1, 𝑎1=1, 𝑏0=−1, and 𝑘=−3.

Case 4 (when 𝑎₁=1, 𝑏₀=−1, and 𝑘=−3). Consider𝑢₄(𝑥,𝑡)=1−216𝑒^{30𝑡−3𝑥}27+2−1+2𝑒^{30𝑡−3𝑥2}(()(=1−216cosh30𝑡−3𝑥+sinh30𝑡−3𝑥))27+2(−1+2cosh(30𝑡−3𝑥)+2sinh(30𝑡−3𝑥))². (38) Description is as shown in Figure 2(b).

4.1.3. When 𝛼=1 and 𝛽=−1

when 𝛼=1 and 𝛽=−1, (9) becomes 𝑢_𝑡+𝑢²𝑢_𝑥−𝑢_𝑥𝑥𝑥=0. Then we can obtain bright solitons and dark solitons. The solution formula is shown as follows: 𝑢(𝑥,𝑡)=𝑎₁1−24𝑒^𝑦𝑘²𝑏₀−3𝑘²𝑏²₀+2𝑎²₁2𝑒^𝑦+𝑏₀²=𝑎₁1−24𝑘²cosh𝑦+sinh𝑦𝑏₀−3𝑘²𝑏²₀+2𝑎²₁2cosh𝑦+2sinh𝑦+𝑏₀², (39) where 𝑎₁, 𝑏₀, and 𝑘 are real parameters and 𝑦=𝑘𝑥+𝑡(𝑘³−𝑘𝑎²₁).

Case 5 (when 𝑎₁=−5, 𝑏₀=−2, and 𝑘=3). Consider𝑢₅(𝑥,𝑡)=−51+432𝑒^{−48𝑡+3𝑥}−108+50−2+2𝑒^{−48𝑡+3𝑥2}(()(×(()(=−51+432cosh48𝑡−3𝑥+sinh48𝑡−3𝑥))−108+50−2+2cosh48𝑡−3𝑥−2sinh48𝑡−3𝑥))²⁻¹. (40) Description is as shown in Figure 3(a).

(a)

(b)

Figure 3 

Evolution of the solutions of (25). (a) Two-peak bright soliton: parameters are 𝛼=1, 𝛽=−1, 𝑎1=−5, 𝑏0=−2, and 𝑘=3. (b) A bright soliton in collision with a dark soliton: parameters are 𝛼=1, 𝛽=−1, 𝑎1=3, 𝑏0=−1, and 𝑘=4.

Case 6 (when 𝑎₁=3, 𝑏₀=−1, and 𝑘=4). Consider𝑢₇(𝑥,𝑡)=31+384𝑒^28𝑡+4𝑥−48+18−1+2𝑒^{28𝑡+4𝑥2}(()(×(()(=31+384cosh28𝑡+4𝑥+sinh28𝑡+4𝑥))−48+18−1+2cosh28𝑡+4𝑥+2sinh28𝑡+4𝑥))²⁻¹. (41) Description is as shown in Figure 3(b).

Case 7 (when 𝑎₁=−5, 𝑏₀=2, and 𝑘=−3). Consider𝑢₆(𝑥,𝑡)=−51−432𝑒^{48𝑡−3𝑥}−108+502+2𝑒^{48𝑡−3𝑥2}(()(×(()(=−51−432cosh48𝑡−3𝑥+sinh48𝑡−3𝑥))−108−502+2cosh48𝑡−3𝑥+2sinh48𝑡−3𝑥))²⁻¹. (42) Description is as shown in Figure 4(a).

(a)

(b)

Figure 4 

Evolution of the solutions of (25). (a) Dark soliton: parameters are 𝛼=1, 𝛽=−1, 𝑎1=−5, 𝑏0=2, and 𝑘=−3. (b) Two-peak dark soliton: parameters are 𝛼=1, 𝛽=−1, 𝑎1=−5, 𝑏0=3, and 𝑘=4.

Case 8 (when 𝑎₁=−5, 𝑏₀=3, and 𝑘=4). Consider𝑢₈(𝑥,𝑡)=−51−1152𝑒^{−36𝑡+4𝑥}−432+503+2𝑒^{−36𝑡+4𝑥2}(()(×(()(=−51−1152cosh36𝑡−4𝑥−sinh36𝑡−4𝑥))−432+503+2cosh36𝑡−4𝑥−2sinh36𝑡−4𝑥))²⁻¹. (43) Description is as shown in Figure 4(b).

4.2. The Solutions from (32)

Based on the form of (32), if we set 𝛼=0 and 𝛽=0 at the same time, the solution should be zero. Therefore, it is not significative to study and analyse. So, in the following, we consider 𝛼 and 𝛽 under the circumstance that 𝛼=0 and 𝛽=0 are not zero in the meantime.

When 𝛼=1 and 𝛽=1, (9) becomes 𝑢_𝑡+𝑢²𝑢_𝑥+𝑢_𝑥𝑥𝑥=0. Then we can obtain periodic wave solutions. The solution formula is shown as follows: 𝑢(𝑥,𝑡)=𝑎₁−6√2𝐾²𝑔2√2𝑎₁𝑔±cos𝑦−𝑎₁+𝑔²3𝐾²−2𝑎²₁, (44) where 𝑦=𝐾𝑥−(𝐾𝑎²₁−𝐾³)𝑡 and 𝑔=𝑎²₁/(−3𝐾²+2𝑎²₁).

To illustrate the analysis, we obtain the following results.

Case 9 (when 𝑎₁=5, 𝐾=−2). Consider𝑢₉(𝑥,𝑡)=5−24√1910/√(2(.19+6cos21𝑡−𝑥)) (45) Description is as shown in Figure 5.

Figure 5 

Evolution of the solutions of (32). Periodic wave solution: parameters are 𝛼=1, 𝛽=1, 𝑎1=5, and 𝐾=−2.

5. Conclusions

In this paper, our main attention has been focused on seeking the soliton solutions of (9) via Exp-function method. By applying Exp-function method, we have obtained shock soliton, bright soliton, two-peak bright soliton, dark soliton, two-peak dark soliton, and periodic wave solution. In addition, with figures and symbolic computations, we have described the propagation characteristics of those solitons under different values of those coefficients in the generalized MKdV equation. Furthermore, the problem solving process and the algorithm, by the help of Mathematica, can be easily extended to all kinds of nonlinear equations.

Conflict of Interests

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

Acknowledgment

This work is supported by the National Natural Science Foundation of China (grant no. 11172194), by the Special Funds of the National Natural Science Foundation of China under Grant No. 11347165, by Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi under Grant No. 2013110.