New Exact Solitary Wave Solutions of a Coupled Nonlinear Wave Equation

Abstract

By using the theory of planar dynamical systems to a coupled nonlinear wave equation, the existence of bell-shaped solitary wave solutions, kink-shaped solitary wave solutions, and periodic wave solutions is obtained. Under the different parametric values, various sufficient conditions to guarantee the existence of the above solutions are given. With the help of three different undetermined coefficient methods, we investigated the new exact explicit expression of all three bell-shaped solitary wave solutions and one kink solitary wave solutions with nonzero asymptotic value for a coupled nonlinear wave equation. The solutions cannot be deduced from the former references.

1. Introduction

For the investigation of traveling wave solutions to nonlinear partial differential equations, which have been the subject of study in various branches of mathematical physical sciences such as physics, biology, and chemistry, many effective methods (see, e.g., [1–12]) have been presented, such as inverse scattering transform method [1], Hirota’s method [2], Backlund and Darboux transformation method [5], the Jacobi (Weierstrass) elliptic function method [7], undetermined coefficient method [9, 10],𝐺^′/𝐺-expansion method and others.

Following the work of Hirota and Satsuma in [13], recently, considerable attention has been focused on the study of coupled nonlinear partial differential equation (see, e.g., [14–19]) that can be solved exactly. However, less work on the coupled version of the higher KdV equation seems to have been reported.

In [20], Guha-Roy has been presented a system of coupled nonlinear wave equations as follows 𝑢_𝑡+𝛽𝑢²𝑢_𝑥+𝛼𝑣²𝑣_𝑥+𝜆𝑢𝑢_𝑥+𝛾𝑢_𝑥𝑥𝑥𝑣=0,_𝑡+𝛿(𝑢𝑣)_𝑥+𝑒𝑣𝑣_𝑥=0, (1) where the subscripts refer to partial differentiations with respect to the indicated variables, and 𝛼,𝛽,𝛾,𝛿,𝜆,𝑒 are arbitrary parameters. Equation (1) is the coupled version of combined form of the higher (modified) KdV equation and KdV equation. It is interesting to point out that, as is outlined in Wadati [21], (1) shares properties with the KdV and the modified KdV equation, under certain conditions.

Guha-Roy [22] supposed that|𝜉|=|𝑥−𝑐𝑡|→+∞ with 𝑢(𝜉),𝑢^′(𝜉),𝑢^′′(𝜉)→0 by transforming to𝑢(𝜉)=1/𝜙(𝜉) that Weierstrauss elliptic function, some exact solitary wave solutions in special conditions to (1) are obtained. Lu et al. [23] obtain the kink-antikink solitary wave solutions of (1) by using a truncated expansion. However, there has not been found any literature on the general analysis of the existences of the solitary wave solution and the exact expressions of the solitary wave solution with nonzero asymptotic value for (1).

In this work, we investigate generally the existence of bell-shaped, kink-shaped solitary wave solutions and periodic wave solutions by using the theory of planar dynamical system under different parameter conditions. It follows that we obtain the new exact expressions of all three bell-shaped solitary wave solutions, and one kink-shaped solitary wave solution with nonzero asymptotic value for (1) by undetermined coefficient methods. These solutions obtained here cannot be deduced from [20, 23].

2. Existence of the Bounded Traveling Wave Solutions to (1)

By introducing an analogue of the stream function, Guha-Roy et al. [18] have shown that if one of the solutions of some coupled nonlinear equations is of the traveling wave type, then the other must also exhibit the same form. Keeping this in mind, we choose a new variable 𝜉=𝑥−𝑐𝑡, where𝑐is the wave speed, such that 𝑢(𝑥,𝑡)=𝜑(𝜉) and 𝑣(𝑥,𝑡)=𝜓(𝜉), substituting them to (1) yields −𝑐𝜑^′+𝛽𝜑²𝜑^′+𝛼𝜓²𝜓^′+𝜆𝜑𝜑^′+𝛾𝜑^′′′=0,−𝑐𝜓^′+𝛿𝜙𝜑^′+𝑒𝜓𝜓^′=0, (2) where the prime denotes the derivative with respect to𝜉.

Integrating the second equation of (2), we have 𝑘𝜑=𝜓+𝑐𝛿−𝑒𝜓,2𝛿 (3) where 𝑘 is the integration constant treated as an arbitrary parameters. In order to have a regular 𝜑 everywhere, we have to impose 𝑘=0. It may be noted that 𝜑(𝜉) satisfies the following boundary conditions: lim_𝑥→+∞𝜑(𝜉)=𝐷₊,lim_{𝑥→−∞}𝜑(𝜉)=𝐷₋,𝜑^′(𝜉)⟶0,𝜑^′′(𝜉)⟶0, (4)(5) as |𝜉|→∞. Condition (4) is different from those considered by Guha-Roy [20], in Guha-Roy’s, 𝜑(𝜉)was found to be vanished in the infinity.

Thus, (3) reduces to 𝜓=2𝑐−2𝛿𝜑𝑒. (6) This shows that 𝜓(𝜉) is directly related to 𝜑(𝜉). By (6) and the first equation of (2), we get, after rearrangement, 𝛾𝜑^′′+𝑏₃3𝜑³+𝑏₂2𝜑²−𝑏₁𝜑=𝑔, (7) where 𝑏₁=𝑐+(8𝛼𝛿𝑐²/𝑒³), 𝑏₂=𝜆+(16𝛼𝛿²𝑐/𝑒³), 𝑏₃=𝛽−(8𝛼𝛿³/𝑒³), and 𝑔 is an arbitrary integration constant.

After the translation 𝜑=𝜑+(𝑏₂/2𝑏₃), we rewrite (7) to 𝛾𝜑^′′+𝑏₃3𝜑³+𝑝𝜑+𝑞=0, (8) where 𝑝=−(12𝑏₁𝑏₃+3𝑏²₂)/4𝑏²₃ and 𝑞=(𝑏³₂+6𝑏₁𝑏₂𝑏₃−12𝑏²₃𝑔)/4𝑏³₃.

It follows 𝑥=𝜑(𝜉) and𝑦=𝜑^′(𝜉) that (8) is equivalent to the two dimensional system 𝑑𝑥𝑑𝜉=𝑦,𝑑𝑦𝑏𝑑𝜉=−³𝑥3𝛾³+𝑝𝑥+𝑞, (9) which has the first integral 𝐻=𝑦𝑥,𝑦²2+𝑏₃𝑝3𝛾𝑞𝑥+2𝑥²+14𝑥⁴=ℎ. (10) System (9) is a four-parameter planar dynamical system depending on the parameter group (𝛾,𝑏₃,𝑝,𝑞). Because of the phase orbits defined by the vector fields of system (9) that determine all traveling wave solutions of (8), we will investigate the phase portraits of (9) in the phase plane (𝑥,𝑦) as the parameters 𝛾,𝑏₃,𝑝,𝑞 are changed.

We point out that here we are considering a physical model where only bounded traveling waves solutions are meaningful, so that we only pay attention to the bounded solutions of system (9).

To investigate the equilibrium points of system (9), we need to find all real zeros of the function 𝑓(𝑥)=𝑥³+𝑝𝑥+𝑞. Suppose that 𝑝<0 and Δ=(𝑞/2)²+(𝑝/3)³≤0, clearly, 𝑓(𝑥) has three real zeros at most, denoted by 𝑥₁,𝑥₂, and 𝑥₃. Therefore, system (9) has three equilibrium points at 𝑝_𝑖(𝑥_𝑖,0), 𝑖=1,2,3, at most.

Let 𝐽𝑥_𝑖=−,0𝛽01₂𝑓3𝛾^′𝑥_𝑖0, (11) where 𝑓^′(𝑥_𝑖)=3𝑥²_𝑖+𝑝, 𝑖=1,2,3, is the coefficient matrix of the linearized system of (9) at equilibrium point 𝑝_𝑖(𝑥_𝑖,0), 𝑖=1,2,3. At this equilibrium point, we obtain the determinant of matrix 𝐽(𝑥_𝑖,0) which is det𝐽𝑥_𝑖=𝑏,0³𝑓3𝛾^′𝑥_𝑖,𝑖=1,2,3. (12) By the theory of planar dynamical systems [24–26] for an equilibrium point of a planar dynamical (Hamiltonian) system, if det𝐽(𝑥_𝑖,0)<0, then the equilibrium point𝑝_𝑖is a saddle point; if det𝐽(𝑥_𝑖,0)>0, then the equilibrium point𝑝_𝑖is a center point; and if det𝐽(𝑥_𝑖,0)=0and the Poincare index of the equilibrium point is zero, then the equilibrium point𝑝_𝑖is a cusp point. So, we have(1)for 𝑏₃>0 and Δ<0, there exists three equilibrium points of system (9) at 𝑝_𝑖(𝑥_𝑖,0), 𝑖=1,2,3, with 𝑥₁<𝑥₂<𝑥₃. The points 𝑝₁ and𝑝₃are center points, 𝑝₂ is a saddle point. There is two homoclinic orbits to the saddle point 𝑝₂, in which there exists a family of periodic orbits surrounding the center 𝑝₁ and 𝑝₃. The phase portrait is shown in Figures 1(b), 1(c), and 1(e).(2)For 𝑏₃>0 and Δ=0, there exists two equilibrium points of system (9) at𝑝_𝑖(𝑥_𝑖,0). If 𝑞<0, the point 𝑝₁=𝑝₂ is a cusp point, 𝑝₃ is a center point; if 𝑞>0, the point 𝑝₁ is a center point, 𝑝₃=𝑝₂ is a cusp point. There is a homoclinic orbit to the cusp point 𝑝₂, in which there exists a family of periodic orbits surrounding the center 𝑝₁ (or 𝑝₃). The phase portrait is shown in Figures 1(a) and 1(d).(3)For 𝑏₃<0 and Δ<0, there exists three equilibrium points of system (9) at 𝑝_𝑖(𝑥_𝑖,0) with 𝑥₁<𝑥₂<𝑥₃. The points 𝑝₁ and 𝑝₃ are saddle points, 𝑝₂ is a center point. If 𝑞≠0, There is a homoclinic orbit to the saddle points 𝑝₁ and 𝑝₃, respectively, if 𝑞=0. There are two heteroclinic orbits connecting the saddle point 𝑝₁ and 𝑝₃, in which there exists a family of periodic orbits surrounding the center 𝑝₂. The phase portrait is shown in Figures 2(b), 2(c), and 2(e).(4)For𝑏₃<0andΔ=0, there exists two equilibrium points of system (9). If𝑞<0, the point 𝑝₁=𝑝₂is a cusp point,𝑝₃is a saddle point; if𝑞>0, the point𝑝₁is a saddle point,𝑝₃=𝑝₂is a cusp point. There does not exist bounded orbits. The phase portraits are shown in Figures 2(a) and 2(d).

(a)   𝑞<0,Δ=0

(b) 𝑞<0,Δ<0

(c) 𝑞=0,Δ<0

(d) 𝑞>0,Δ=0

(e) 𝑞>0,Δ<0

Figure 1 

The phase portraits of system (9) for𝑏3>0.

(a)  𝑞<0,Δ=0

(b) 𝑞<0,Δ<0

(c) 𝑞=0,Δ<0

(d) 𝑞>0,Δ=0

(e) 𝑞>0,Δ<0

Figure 2 

The phase portraits of system (9) for𝑏3<0.

Because orbits cannot be changed by the transformation𝜑=𝜑+(𝑏₂/2𝑏₃), it follows that (6), (7), (8), and the above discussion the following.

Theorem 1. Suppose that 𝑏₃>0, wave speed 𝑐, and integration constant𝑔satisfy 4𝑏₁𝑏₃+𝑏²₂>0; then(1)whenΔ<0, (1) has two bell-shaped solitary wave solutions and uncountable infinite many periodic traveling wave solutions in the case of𝑞>0, 𝑞=0, and𝑞<0, respectively, (see Figures 1(b), 1(c), and 1(e)).(2)WhenΔ=0, (1) has one bell-shaped solitary wave solution and uncountably infinite many periodic traveling wave solutions in the case of𝑞>0and𝑞<0, respectively, (see Figures 1(a), and 1(d)).

Theorem 2. Suppose that 𝑏₃<0, wave speed 𝑐, and integration constant 𝑔 satisfy 4𝑏₁𝑏₃+𝑏²₂>0; then(1)whenΔ<0, (1) has one bell-shaped solitary wave solution and uncountably infinite many periodic traveling wave solutions in the case of 𝑞>0and 𝑞<0, respectively, and two kink-shaped solitary wave solutions and uncountably infinite many periodic traveling wave solutions in the case of𝑞=0 (see Figures 2(b), 2(c), and 2(e)).(2)WhenΔ=0, (1) does not exist bounded traveling wave solutions (see Figures 2(a), and 2(d)).

3. Exact Explicit Representations of Bell-Shaped and Kink-Shaped Solitary Wave Solutions

According to the discussion in Section 2, we assume that (7) has solution with the following form: 𝜑(𝜉)=𝐴𝑒^{𝑚(𝜉+𝜉0)}1+𝑒^{𝑚(𝜉+𝜉0)2}+𝐵𝑒^{𝑚(𝜉+𝜉0)}=+𝐷𝐴sech²𝑚𝜉+𝜉₀/24+𝐵𝐴sech²𝑚𝜉+𝜉₀/2+𝐷, (13) where 𝐴,𝐵,𝐷, and 𝑚 are undetermined real parameters and 𝜉₀ is arbitrary constant.

Substituting (13) and 𝜑^′(𝜉), 𝜑^′′(𝜉) into (7), by using the linear independence of 𝑒^{𝑘𝑚(𝜉+𝜉0)}, 𝑘=0,1,…,6, we obtain that the following algebraic equations with 𝐴,𝐵,𝐷, and 𝑚: 2𝑏₃𝐷³+3𝑏₂𝐷²−6𝑏₁𝐷−6𝑔=0,𝛾𝑚²+𝑏₂𝐷+𝑏₃𝐷²−𝑏₁𝑏=0,₂+2𝑏₃𝐷𝐴−6𝛾𝑚²(2+𝐵)=0,2𝑏₃𝐴²+(2+𝐵)3𝑏₂+6𝑏₃𝐷𝐴−6𝑚²(2+𝐵)²−48𝑚²=0. (14) Suppose that𝐷satisfy 2𝑏₃𝐷³+3𝑏₂𝐷²−6𝑏₁𝐷−6𝑔=0. (15) By solving the above equations; we have the following.

Case 1. For 𝑏₂+2𝑏₃𝐷≠0, we have 𝑚=±−𝑏₃𝐷²+𝑏₂𝐷−𝑏₁𝛾,𝐴=6𝛾𝑚²(2+𝐵)𝑏₂+2𝑏₃𝐷,𝑏2+𝐵=±²+2𝑏₃𝐷𝑏²₂+6𝑏₃𝑏₁−2𝑏²₃𝐷²−2𝑏₂𝑏₃𝐷. (16)

Case 2. For𝑏₂+2𝑏₃𝐷=0,𝑏₃>0,𝑏₁>𝑏₂𝐷/2, we have 𝑚=±𝑏₁𝛾−𝑏₂𝐷2𝛾,𝐴=±12𝑏₃2𝑏₁−𝑏₂𝐷,𝐵=−2. (17)
By (6), (13), and the above conclusions, we can write the exact solutions of (1).

Theorem 3. For 𝛾>0, 𝜉=𝑥−𝑐𝑡, 𝜉₀ is an arbitrary constant; suppose that 𝐷 is real and satisfies 2𝑏₃𝐷³+3𝑏₂𝐷²−6𝑏₁𝐷−6𝑔=0; then(1) when 𝐴₁=𝑏₂+2𝑏₃𝐷≠0,𝐴₂=𝑏₃𝐷²+𝑏₂𝐷−𝑏₁<0, and 𝐴₃=𝑏²₂+6𝑏₃𝑏₁−2𝑏²₃𝐷²−2𝑏₂𝑏₃𝐷>0, (1) has the following bell-shaped solitary wave solutions: 𝜑^±₁(𝜉)=±6𝐴₂𝐴₁/𝐴₃𝐴₁sech²𝜂2+−1±𝐴₁/𝐴₃sech²𝜂𝜓+𝐷,_±1(𝜉)=2𝑐−2𝛿𝜑^±₁𝑒, (18)where 𝜂=(1/2)−(𝑏₃𝐷²+𝑏₂𝐷−𝑏₁)/𝛾(𝜉+𝜉₀).(2) When 𝐴₁=𝑏₂+2𝑏₃𝐷=0,𝑏₃>0 and 𝑏₁>(𝑏₂/2)𝐷, (1) has the following bell-shaped solitary wave solutions 𝜑^±₂(𝜉)=±3𝑏₃2𝑏₁−𝑏₂𝐷sech2𝑏₁−𝑏₂𝐷2𝜉+𝜉₀𝜓+𝐷,_±2(𝜉)=2𝑐−2𝛿𝜑^±₂𝑒. (19)

Remark 4. (1)  (𝜑⁺_𝑖,𝜓⁺_𝑖), 𝑖=1,2, denotes the solitary wave solutions taking “+” in expression of (16) and (17), (𝜑⁻_𝑖,𝜓⁻_𝑖), 𝑖=1,2, is similar.
(2) Substituting 𝐷=0,𝛽=8𝛼𝛿³/𝑒³ into solutions (𝜑⁺₁,𝜓⁺₁) of expressions (18) yields 𝜑⁺₁(𝜉)=3𝑏₁𝑏₂sech²12𝑏₁𝛾𝜉+𝜉₀,𝜓⁺₁(𝜉)=2𝑐−2𝛿𝜑⁺₁𝑒, (20) where 𝑏₁>0 and 𝑏₂>0 are denoted by (7). Solution (20) is the solitary wave solution (17) and (7) of Guha-Roy [20].
(3) Substituting 𝐷=0 and 𝜆=−16𝛼𝛿²𝑐/𝑒³into solutions (𝜑^±₂,𝜓^±₂) of expressions (19) yields 𝜑^±₂(𝜉)=±6𝑏₁𝑏₃sech𝑏₁𝛾𝜉+𝜉₀,𝜓^±₂(𝜉)=2𝑐−2𝛿𝜑^±₂𝑒, (21) where 𝑏₁>0 and 𝑏₃>0 are denoted by (7). Solution (21) is the solitary wave solution (20) and (7) of Guha-Roy [20].
(4) Solutions (18) and (19) cannot be obtained by the method used in Guha-Roy [20].
(5) For 𝑏₃>0, 𝑞<0, and Δ<0, the roots 𝐷₁,𝐷₂, and 𝐷₃ of 2𝑏₃𝐷³+3𝑏₂𝐷²−6𝑏₁𝐷−6𝑔=0 satisfy 𝑥_𝑖=𝐷_𝑖+(𝑏₂/2𝑏₃), 𝑖=1,2,3, where 𝑥₁,𝑥₂, and 𝑥₃ are the roots of 𝑓(𝑥)=0. After computation, we know that 𝐷₁<𝐷₂<−(𝑏₂/2𝑏₃)<𝐷₃, 𝑏₂+2𝑏₃𝐷₂<0, and 𝑏₃𝐷²₂+𝑏₂𝐷₂−𝑏₁<0, so, solution 𝜑⁺₁ in expression (18) is less than 𝐷₂ and 𝜑⁻₁ in expression (18) is over 𝐷₂. Then, two homoclinic orbits at a saddle point (𝐷₂,0) can be denoted by solutions 𝜑^±₁ in (18), in here, 𝜑⁺₁ denotes the left hand homoclinic orbit and 𝜑⁻₁ denotes the right hand homoclinic orbit in Figure 1(b). Similarly, we can explain that solutions 𝜑^±₁ denote the two homoclinic orbits in Figure 1(e).
(6) For 𝑏₃<0, 𝑞<0, and Δ<0, the roots 𝐷₁,𝐷₂, and 𝐷₃ of 2𝑏₃𝐷³+3𝑏₂𝐷²−6𝑏₁𝐷−6𝑔=0 satisfy 𝑥_𝑖=𝐷_𝑖+(𝑏₂/2𝑏₃), 𝑖=1,2,3, where 𝑥₁,𝑥₂, and 𝑥₃ are the roots of 𝑓(𝑥)=0. (𝐷₁,0) and (𝐷₃,0) are saddle points and (𝐷₂,0) is a center point. After computation, we know that 𝐷₁<𝐷₂<−(𝑏₂/2𝑏₃)<𝐷₃, 𝑏₂+2𝑏₃𝐷₁>0 and 𝑏₃𝐷²₁+𝑏₂𝐷₁−𝑏₁<0, by 𝑞=𝐴₁𝐴₃/4𝑏³₃ yields 𝐴₃>0, 𝜑⁺₁(𝜉)>𝐷₁. It follows 𝑏₂+2𝑏₃𝐷₃<0 and 𝑏₃𝐷²₃+𝑏₂𝐷₃−𝑏₁<0, 𝑞=𝐴₁𝐴₃/4𝑏³₃ that 𝐴₃<0, so 𝜑⁻₁(𝜉) is unbounded. Therefore, the homoclinic orbit at a saddle point (𝐷₁,0) in Figure 2(b) is denoted by 𝜑⁺₁ in expression (18), 𝜑⁻₁ is unbounded. Similarly, we can explain that solutions 𝜑⁺₁ denote the homoclinic orbits in Figure 2(e).
(7) For 𝑏₃>0, 𝑞=0 and Δ<0, point (𝐷₁,0) and (𝐷₃,0) are corresponding to the centers 𝑝₁(−√−𝑝,0) and 𝑝₃(√−𝑝,0) of system (9), respectively, points (𝐷₂,0) is corresponding to a saddle point 𝑝₂(0,0) of system (9), where 𝐷₂=−𝑏₂/2𝑏₃, 𝐷₂<2𝑏₁/𝑏₂; then 𝜑^±₂ denotes the two homoclinic orbits in Figure 1(c).
If we suppose that (7) has solutions with the following form: 𝜑(𝑥−𝑐𝑡)=𝜑(𝜉)=𝐴𝑒^{𝑚(𝜉+𝜉0)}1+𝑒^{𝑚(𝜉+𝜉0)}+𝐷, (22) where 𝐴,𝐷, and 𝑚 are undetermined real parameters and 𝜉₀ is arbitrary constant.
Substituting (22) into (7) and using the linear independence of 𝑒^{𝑘(𝜉+𝜉0)}, 𝑘=0,1,2,3 yields 2𝑏₃𝐷³+3𝑏₂𝐷²−6𝑏₁𝐷−6𝑔=0,𝛾𝑚²+𝑏₂𝐷+𝑏₃𝐷²−𝑏₁=0,2𝑏₃𝐴²+3𝑏₂+6𝑏₃𝐷𝐴−6𝛾𝑚²=0,3𝑏₂+6𝑏₃𝐷𝐴−18𝛾𝑚²=0. (23) By solving the above equations, we obtain 𝑚=±−𝑏²₂+4𝑏₁𝑏₃2𝛾𝑏₃,𝐴=±3𝑏²₂+12𝑏₁𝑏₃𝑏²₃,𝑏𝐷=−²2𝑏₃±3𝑏²₂+12𝑏₁𝑏₃2𝑏₃. (24) It follows (22), (24) and (6) that

Theorem 5. Suppose that𝑏²₂+4𝑏₁𝑏₃>0,𝑏₃<0,𝛾>0,𝜉=𝑥−𝑐𝑡,𝜉₀is an arbitrary constant, and𝐷=−(𝑏₂/2𝑏₃)±(3𝑏²₂+12𝑏₁𝑏₃/2𝑏₃), (1) has the following kink-shaped solitary wave solutions: 𝜑^±₃(𝜉)=𝑏𝐷+²2𝑏₃1tanh2−𝑏²₂+4𝑏₁𝑏₃2𝛾𝑏₃𝜉+𝜉₀−𝑏₂2𝑏₃,𝜓^±₃(𝜉)=2𝑐−2𝛿𝜑^±₃𝑒. (25)

Remark 6. (1) In [20], expression (13) can be rewritten as follows: 𝑢(𝜉)=−3𝑏₁𝑏₂11±2𝑏₁6𝜉+𝜉₀, (26) where𝑏²₂=−6𝑏₁𝑏₃. Comparing expression (26) with𝜑^±₃ in expression (25), we know that the solution (13) solved by Guha-Roy [20] is in accordance with solutions𝜑^±₃of expression (25) in the case of𝑏²₂=−6𝑏₁𝑏₃. But the general solutions (25) cannot be solved by Guha-Roy [20].
(2) For 𝑏₃<0, 𝑞=0, and Δ<0, points (𝐷₁,0) and (𝐷₃,0) are two saddle points of system (9), and (𝐷₂,0)is a center point, where 𝐷_1,3=−(𝑏₂/2𝑏₃)±(3𝑏²₂+12𝑏₁𝑏₃/2𝑏₃), 𝐷₂=−𝑏₂/2𝑏₃. According to 𝑞=0 yields 𝑏²₂+6𝑏₁𝑏₃−2𝑏²₃𝐷²_𝑖−2𝑏₂𝑏₃𝐷_𝑖=0, 𝑖=1,3; thus,𝜑⁺₃(𝜉)and 𝜑⁻₃(𝜉) of expression (25) are corresponding to the two heteroclinic orbits in Figure 2(c).
If we suppose that (7) has solutions with the following form: 𝜑(𝜉)=𝐴𝐵+𝑚𝜉²+𝐷, (27) where 𝐴,𝐵,𝐷, and𝑚are undetermined real parameters and𝜉₀is arbitrary constant.
By using (27) and its derivations, it follows (7) that 𝑏₃𝐷²+𝑏₂𝐷−𝑏₁𝑏=0,𝐴=−2𝐵₂+2𝑏₃𝐷𝑏₃,𝐵𝑏𝑚=²²+4𝑏₁𝑏₃6𝛾𝑏₃𝑏,𝑔=¹𝑏₂−4𝑏₁𝑏₃𝐷−𝑏²₂𝐷6𝑏₃. (28)
Further, we obtain the following results.

Theorem 7. Suppose that 𝑏₃>0,𝛾>0, and 𝜉=𝑥−𝑐𝑡,𝜉₀ is an arbitrary constant, and 𝐷 satisfies 𝑏₃𝐷²+𝑏₂𝐷−b₁=0 integration constant 𝑔=(𝑏₁𝑏₂−4𝑏₁𝑏₃𝐷−𝑏²₂𝐷)/6𝑏₃; then (1) has the following bell-shaped solitary wave solutions: 𝜑₄(𝜉)𝑏=−12₂+2𝑏₃𝐷𝛾6𝛾𝑏₃+𝑏²₂+4𝑏₁𝑏₃𝜉+𝜉₀²𝜓+𝐷,₄(𝜉)=2𝑐−2𝛿𝜑₄𝑒. (29)

Remark 8. (1) By the hypothesis𝑝<0and 𝑏₃𝐷²+𝑏₂𝐷−𝑏₁=0, we know that 𝑏²₂+4𝑏₁𝑏₃>0, and that 𝐷₁=−𝑏₂/2𝑏₃+3𝑏²₂+12𝑏₁𝑏₃/2𝑏₃, 𝐷₂=−𝑏₂/2𝑏₃−3𝑏²₂+12𝑏₁𝑏₃/2𝑏₃. If we take𝐷=𝐷₁in solution (29), solution𝜑₄(𝜉)of expression (29) is corresponding to the homoclinic orbit in Figure 1(d). If we take 𝐷=𝐷₂in solution (29), solution𝜑₄(𝜉)of expression (29) is corresponding to the homoclinic orbit in Figure 1(a).

4. Discussion and Conclusion

In this paper, we obtain all the three bell-shaped and one kink-shaped solitary wave solutions of (1) by using three different undetermined coefficient methods. The conclusions have not been deduced from the method reported by Guha-Roy. The method is simple and can be applied to solve many couple nonlinear equations such as Ito equation, Ito-type equation, and coupled KdV equations.

Conflict of Interests

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

Acknowledgments

This project is supported by Fund of Guizhou Science and Technology Department ([2013]2138) and technical innovation talents support plan of Guizhou Education Department (KY[2012]092).