Traveling Wave Solutions for the Generalized Zakharov Equations

Abstract

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended.

1. Introduction

The Zakharov equations 𝑖𝑢_𝑡+𝑢_𝑥𝑥𝑣−𝑢𝑣=0,_𝑡𝑡−𝑣_𝑥𝑥+|𝑢|²_𝑥𝑥=0, (1.1) which is one of the fundamental models governing dynamics of nonlinear waves in one-dimensional systems. It describes the interaction between high-frequency and low-frequency waves. The physically most important example involves the interaction between the Langmuir and ion-acoustic waves in plasmas [1]. The equations can be derived from a hydrodynamic description of the plasma [2, 3]. However, some important effects such as transit-time damping and ion nonlinearities, which are also implied by the fact that the values used for the ion damping have been anomalously large from the point of view of linear ion-acoustic wave dynamics, have been ignored in (1.1). This is equivalent to saying that (1.1) is a simplified model of strong Langmuir turbulence. Thus we have to generalize (1.1) by taking more elements into account. Starting from the dynamical plasma equations with the help of relaxed Zakharov simplification assumptions, and through making use of the time-averaged two-time-scale two-fluid plasma description, (1.1) are generalized to contain the self-generated magnetic field [4, 5], and the first related study on magnetized plasmas in [6, 7]. The generalized Zakharov equations are a set of coupled equations and may be written as [8] 𝑖𝑢_𝑡+𝑢_𝑥𝑥−2𝜆|𝑢|²𝑣𝑢+2𝑢𝑣=0,_𝑡𝑡−𝑣_𝑥𝑥+|𝑢|²_𝑥𝑥=0. (1.2)

Malomed et al. [8] analyzed internal vibrations of a solitary wave in (1.2) by means of a variational approach. Wang and Li [9] obtained a number of periodic wave solutions of (1.2) by using extended F-expansion method. Javidi and Golbabai [10] used the He's variational iteration method to obtain solitary wave solutions of (1.2). Zhang [11] obtained the exact traveling wave solutions of (1.2) by using the direct algebraic method. Zhang [12] used He’s semi-inverse method to search for solitary wave solutions of (1.2). Javidi and Golbabai [13] obtained the exact and numerical solutions of (1.2) by using the variational iteration method. Li et al. [14] used the Exp-function method to seek exact solutions of (1.2). Borhanifar et al. [15] obtained the generalized solitary solutions and periodic solutions of (1.2) by using the Exp-function method. Khan et al. [16] used He's variational approach to obtain new soliton solutions of (1.2).

The aim of this paper is to study the traveling wave solutions and their limits for (1.2) by using the bifurcation method and qualitative theory of dynamical systems [17–24]. Through some special phase orbits, we obtain many smooth periodic wave solutions and periodic blow-up solutions. Their limits contain kink-profile solitary wave solutions, unbounded wave solutions, periodic blow-up solutions, and solitary wave solutions.

The remainder of this paper is organized as follows. In Section 2, by using the bifurcation theory of planar dynamical systems, two-phase portraits for the corresponding traveling wave system of (1.2) are given under different parameter conditions. The relations between the traveling wave solutions and the Hamiltonian ℎ are presented. In Section 3, we obtain a number of traveling wave solutions of (1.2) and give the relations of the traveling wave solutions. A short conclusion will be given in Section 4.

2. Phase Portraits and Qualitative Analysis

We assume that the traveling wave solutions of (1.2) is of the form 𝑢(𝑥,𝑡)=𝑒^𝑖𝜂𝜑(𝜉),𝑣(𝑥,𝑡)=𝜓(𝜉),𝜂=𝑝𝑥+𝑞𝑡,𝜉=𝑘𝑥−2𝑝𝑡, (2.1) where 𝜑(𝜉) and 𝜓(𝜉) are real functions; 𝑝,𝑞, and 𝑘 are real constants.

Substituting (2.1) into (1.2), we have 𝑘²𝜑^′′+2𝜑𝜓−𝑝²+𝑞𝜑−2𝜆𝜑³𝑘=0,₂4𝑝²−1𝜓^′′+𝑘²𝜑^2′′=0. (2.2)

Integrating the second equation of (2.2) twice, and letting the first integral constant be zero, we have 𝜑𝜓=²1−4𝑝²1+𝑔,𝑝≠2, (2.3) where 𝑔 is integral constant.

Substituting (2.3) into the first equation of (2.2), we have 𝑘²𝜑^′′+2𝑔−𝑝²1−𝑞𝜑+21−4𝑝²−𝜆𝜑³=0. (2.4)

Letting 𝜑^′=𝑦, 𝛼=(2/𝑘²)(𝜆−1/(1−4𝑝²)), and 𝛽=(2𝑔−𝑝²−𝑞)/𝑘², then we get the following planar system d𝜑d𝜉=𝑦,d𝑦d𝜉=𝛼𝜑³−𝛽𝜑. (2.5)

Obviously, the above system (2.5) is a Hamiltonian system with Hamiltonian function 𝐻𝜑,𝑦=𝑦²−12𝛼𝜑⁴+𝛽𝜑². (2.6)

In order to investigate the phase portrait of (2.5), set 𝑓𝜑=𝛼𝜑³−𝛽𝜑. (2.7)

Obviously, 𝑓(𝜑) has three zero points, 𝜑₋, 𝜑₀, and 𝜑₊, which are given as follows: 𝜑₋=−𝛽𝛼,𝜑₀=0,𝜑₊=𝛽𝛼. (2.8)

Letting (𝜑_𝑖,0) be one of the singular points of system (2.5), then the characteristic values of the linearized system of system (2.5) at the singular points (𝜑_𝑖,0) are 𝜆_±=±𝑓′𝜑_𝑖. (2.9)

From the qualitative theory of dynamical systems, we know that:(1)if 𝑓′(𝜑_𝑖)>0, (𝜑_𝑖,0) is a saddle point;(2)if 𝑓′(𝜑_𝑖)<0, (𝜑_𝑖,0) is a center point;(3)if 𝑓′(𝜑_𝑖)=0, (𝜑_𝑖,0) is a degenerate saddle point;

Therefore, we obtain the phase portraits of system (2.5) in Figure 1.

(a)

(b)

Figure 1 

The phase portraits of system (2.5). (a) 𝛼>0,𝛽>0, (b) 𝛼<0,𝛽<0.

Let 𝐻𝜑,𝑦=ℎ, (2.10) where ℎ is Hamiltonian.

Next, we consider the relations between the orbits of (2.5) and the Hamiltonian ℎ.

Set ℎ^∗=𝐻𝜑₊,0=𝐻𝜑₋=𝛽,0²2|𝛼|. (2.11) According to Figure 1, we get the following propositions.

Proposition 2.1. Suppose that 𝛼>0 and 𝛽>0, we have the following. (1)When ℎ<0 or ℎ>ℎ^∗, system (2.5) does not have any closed orbit.(2)When 0<ℎ<ℎ^∗, system (2.5) has three periodic orbits Γ₁, Γ₂, and Γ₃.(3)When ℎ=0, system (2.5) has two periodic orbits Γ₄ and Γ₅.(4)When ℎ=ℎ^∗, system (2.5) has two heteroclinic orbits Γ₆ and Γ₇.

Proposition 2.2. Suppose that 𝛼<0 and 𝛽<0, we have the following.(1)When ℎ⩽−ℎ^∗, system (2.5) does not have any closed orbit.(2)When −ℎ^∗<ℎ<0, system (2.5) has two periodic orbits Γ₈ and Γ₉.(3)When ℎ=0, system (2.5) has two homoclinic orbits Γ₁₀ and Γ₁₁.(4)When ℎ>0, system (2.5) has a periodic orbit Γ₁₂.

From the qualitative theory of dynamical systems, we know that a smooth solitary wave solution of a partial differential system corresponds to a smooth homoclinic orbit of a traveling wave equation. A smooth kink wave solution or an unbounded wave solution corresponds to a smooth heteroclinic orbit of a traveling wave equation. Similarly, a periodic orbit of a traveling wave equation corresponds to a periodic traveling wave solution of a partial differential system. According to the above analysis, we have the following propositions.

Proposition 2.3. If 𝛼>0 and 𝛽>0, we have the following.(1)When 0<ℎ<ℎ^∗, (1.2) has two periodic wave solutions (corresponding to the periodic orbit Γ₂ in Figure 1) and two periodic blow-up wave solutions (corresponding to the periodic orbits Γ₁ and Γ₃ in Figure 1).(2)When ℎ=0, (1.2) has two periodic blow-up wave solutions (corresponding to the periodic orbits Γ₄ and Γ₅ in Figure 1).(3)When ℎ=ℎ^∗, (1.2) has two kink-profile solitary wave solutions and two unbounded wave solutions (corresponding to the heteroclinic orbits Γ₆ and Γ₇ in Figure 1).

Proposition 2.4. If 𝛼<0 and 𝛽<0, we have the following.(1)When −ℎ^∗<ℎ<0, (1.2) has two periodic wave solutions (corresponding to the periodic orbits Γ₈ and Γ₉ in Figure 1).(2)When ℎ=0, (1.2) has two solitary wave solutions (corresponding to the homoclinic orbits Γ₁₀ and Γ₁₁ in Figure 1).(3)When ℎ>0, (1.2) has two periodic wave solutions (corresponding to the periodic orbit Γ₁₂ in Figure 1).

3. Traveling Wave Solutions and Their Relations

Firstly, we will obtain the explicit expressions of traveling wave solutions for the (1.2) when 𝛼>0 and 𝛽>0.

(1) From the phase portrait, we note that there are three periodic orbits Γ₁, Γ₂, and Γ₃ passing the points (𝜑₁,0),(𝜑₂,0),(𝜑₃,0), and (𝜑₄,0). In (𝜑,𝑦) plane the expressions of the orbits are given as 𝑦=±𝛼2𝜑−𝜑₁𝜑−𝜑₂𝜑−𝜑₃𝜑−𝜑₄, (3.1) where 𝜑₁=−(𝛽+𝛽²−2𝛼ℎ)/𝛼, 𝜑₂=−(𝛽−𝛽²−2𝛼ℎ)/𝛼, 𝜑₃=(𝛽−𝛽²−2𝛼ℎ)/𝛼, 𝜑₄=(𝛽+𝛽²−2𝛼ℎ)/𝛼, and 0<ℎ<ℎ^∗.

Substituting (3.1) into d𝜑/d𝜉=𝑦 and integrating them along Γ₁,Γ₂, and Γ₃, we have ±^∞_𝜑1𝑠−𝜑₁𝑠−𝜑₂𝑠−𝜑₃𝑠−𝜑₄d𝑠=𝛼2^𝜉₀±d𝑠,_𝜑01𝑠−𝜑₁𝑠−𝜑₂𝑠−𝜑₃𝑠−𝜑₄d𝑠=𝛼2^𝜉₀d𝑠. (3.2)

Completing above integrals we obtain 𝜑𝜑=±⁴sn𝜑₄𝛼/2𝜉,𝜑₃/𝜑₄,𝜑=±𝜑₃sn𝜑₄𝛼2𝜑𝜉,³𝜑₄. (3.3)

Noting that (2.1) and (2.3), we get the following periodic wave solutions: 𝑢₁()𝑒𝑥,𝑡=±^𝑖𝜂𝜑₄sn𝜑₄𝛼/2𝜉,𝜑₃/𝜑₄,𝑣₁()=𝜑𝑥,𝑡²⁴1−4𝑝²sn𝜑₄𝛼/2𝜉,𝜑₃/𝜑₄²𝑢+𝑔,₂(𝑥,𝑡)=±𝑒^𝑖𝜂𝜑₃sn𝜑₄𝛼2𝜑𝜉,³𝜑₄,𝑣₂()=𝜑𝑥,𝑡³sn𝜑₄𝛼/2𝜉,𝜑₃/𝜑₄²1−4𝑝²+𝑔, (3.4) where 𝜂=𝑝𝑥+𝑞𝑡 and 𝜉=𝑘(𝑥−2𝑝𝑡).

(2) From the phase portrait, we note that there are two special orbits Γ₄ and Γ₅, which have the same hamiltonian as that of the center point (0,0). In (𝜑,𝑦) plane the expressions of the orbits are given as 𝑦=±𝛼2𝜑𝜑−𝜑₅𝜑−𝜑₆, (3.5) where 𝜑₅=−2𝛽/𝛼 and 𝜑₆=2𝛽/𝛼.

Substituting (3.5) into d𝜑/d𝜉=𝑦, and integrating them along the two orbits Γ₄ and Γ₅, it follows that ±_𝜑^+∞1𝑠𝑠−𝜑₅𝑠−𝜑₆d𝑠=𝛼2^𝜉₀d𝑠. (3.6)

Completing above integrals we obtain𝜑=±2𝛽𝛼csc𝛽𝜉. (3.7)

Noting (2.1) and (2.3), we get the following periodic blow-up wave solutions: 𝑢₃(𝑥,𝑡)=±𝑒^𝑖𝜂2𝛽𝛼csc𝑣𝛽𝜉,₃(𝑥,𝑡)=2𝛽csc𝛽𝜉²𝛼1−4𝑝²+𝑔, (3.8) where 𝜂=𝑝𝑥+𝑞𝑡 and 𝜉=𝑘(𝑥−2𝑝𝑡).

(3) From the phase portrait, we see that there are two heteroclinic orbits Γ₆ and Γ₇ connected at saddle points (𝜑₋,0) and (𝜑₊,0). In (𝜑,𝑦) plane the expressions of the heteroclinic orbits are given as 𝑦=±𝛼2𝜑−𝜑₋²𝜑−𝜑₊². (3.9)

Substituting (3.9) into d𝜑/d𝜉=𝑦, and integrating them along the heteroclinic orbits Γ₆ and Γ₇, it follows that ±^𝜑₀1𝑠−𝜑₋𝜑₊−𝑠d𝑠=𝛼2^𝜉₀±d𝑠,_𝜑+∞1𝑠−𝜑₋𝑠−𝜑₊d𝑠=𝛼2^𝜉₀d𝑠. (3.10)

Completing above integrals we obtain 𝜑=±𝛽𝛼tanh𝛽2𝜉,𝜑=±𝛽𝛼coth𝛽2𝜉. (3.11)

Noting (2.1) and (2.3), we get the following kink profile solitary wave solutions: 𝑢₄(𝑥,𝑡)=±𝑒^𝑖𝜂𝛽𝛼tanh𝛽2𝑣𝜉,₄()=𝛽𝑥,𝑡tanh𝛽𝜉²𝛼1−4𝑝²+𝑔, (3.12) and unbounded wave solutions 𝑢₅(𝑥,𝑡)=±𝑒^𝑖𝜂𝛽𝛼coth𝛽2𝑣𝜉,₅()=𝛽𝑥,𝑡coth𝛽𝜉²𝛼1−4𝑝²+𝑔, (3.13) where 𝜂=𝑝𝑥+𝑞𝑡 and 𝜉=𝑘(𝑥−2𝑝𝑡).

Secondly, we will obtain the explicit expressions of traveling wave solutions for (1.2) when 𝛼<0 and 𝛽<0.

(1) From the phase portrait, we see that there are two closed orbits Γ₈ and Γ₉ passing the points (𝜑₇,0),(𝜑₈,0),(𝜑₉,0), and (𝜑₁₀,0). In (𝜑,𝑦) plane the expressions of the closed orbits are given as 𝑦=±−𝛼2𝜑−𝜑₇𝜑−𝜑₈𝜑−𝜑₉𝜑₁₀−𝜑, (3.14) where 𝜑₇=−(𝛽−𝛽²−2𝛼ℎ)/𝛼, 𝜑₈=−(𝛽+𝛽²−2𝛼ℎ)/𝛼, 𝜑₉=(𝛽+𝛽²−2𝛼ℎ)/𝛼, 𝜑₁₀=(𝛽−𝛽²−2𝛼ℎ)/𝛼, and −ℎ^∗<ℎ<0.

Substituting (3.14) into d𝜑/d𝜉=𝑦, and integrating them along Γ₈ and Γ₉, we have ±^𝜑_𝜑71𝜑₁₀−𝑠𝜑₉−𝑠𝜑₈−𝑠𝑠−𝜑₇d𝑠=−𝛼2^𝜉₀±d𝑠,_𝜑𝜑101𝑠−𝜑₇𝑠−𝜑₈𝑠−𝜑₉𝜑₁₀−𝑠d𝑠=−𝛼2^𝜉₀d𝑠. (3.15)

Completing above integrals we obtain 𝜑𝜑=¹⁰−𝜑₈𝜑₇+𝜑₈−𝜑₇𝜑₁₀sn𝜔−(𝛼/2)𝜉,𝜅²𝜑₁₀−𝜑₈+𝜑₈−𝜑₇sn𝜔−(𝛼/2)𝜉,𝜅²,𝜑=𝜑²₁₀−𝜑²₁₀−𝜑²₉sn𝜑₁₀−𝛼2𝜉,𝜑²₁₀−𝜑²₉𝜑₁₀², (3.16) where 𝜔=(𝜑₁₀−𝜑₈)(𝜑₉−𝜑₇)/2 and 𝜅=(𝜑₁₀−𝜑₉)(𝜑₈−𝜑₇)/(𝜑₁₀−𝜑₈)(𝜑₉−𝜑₇).

Noting (2.1) and (2.3), we get the following periodic wave solutions: 𝑢₆()=𝑒𝑥,𝑡^𝑖𝜂𝜑₁₀−𝜑₈𝜑₇+𝜑₈−𝜑₇𝜑₁₀sn𝜔−(𝛼/2)𝜉,𝜅²𝜑₁₀−𝜑₈+𝜑₈−𝜑₇sn𝜔−(𝛼/2)𝜉,𝜅²,𝑣₆(𝑥,𝑡)=(𝜑₁₀−𝜑₈)𝜑₇+(𝜑₈−𝜑₇)𝜑₁₀sn𝜔−(𝛼/2)𝜉,𝜅²²1−4𝑝²𝜑₁₀−𝜑₈+(𝜑₈−𝜑₇)sn𝜔−(𝛼/2)𝜉,𝜅²²𝑢+𝑔,₇(𝑥,𝑡)=𝑒^𝑖𝜂𝜑²₁₀−𝜑²₁₀−𝜑²₉sn𝜑₁₀−(𝛼/2)𝜉,𝜑²₁₀−𝜑²₉𝜑₁₀²,𝑣₇()=𝜑𝑥,𝑡²¹⁰−𝜑²₁₀−𝜑²₉sn𝜑₁₀−(𝛼/2)𝜉,𝜑²₁₀−𝜑²₉/𝜑₁₀²1−4𝑝²+𝑔, (3.17) where 𝜂=𝑝𝑥+𝑞𝑡 and 𝜉=𝑘(𝑥−2𝑝𝑡).

(2) From the phase portrait, we see that there are two symmetric homoclinic orbits Γ₁₀ and Γ₁₁ connected at the saddle point (0,0). In (𝜑,𝑦) plane the expressions of the homoclinic orbits are given as 𝑦=±−𝛼2𝜑𝜑−𝜑₁₁𝜑₁₂−𝜑, (3.18) where 𝜑₁₁=−2𝛽/𝛼 and 𝜑₁₂=2𝛽/𝛼.

Substituting (3.18) into d𝜑/d𝜉=𝑦, and integrating them along the orbits Γ₁₀ and Γ₁₁, we have ±^𝜑_𝜑111𝑠𝑠−𝜑₁₁𝜑₁₂−𝑠d𝑠=−𝛼2^𝜉₀±d𝑠,_𝜑𝜑121𝑠𝑠−𝜑₁₁𝜑₁₂−𝑠d𝑠=−𝛼2^𝜉₀d𝑠. (3.19)

Completing above integrals we obtain 𝜑=2𝛽𝛼sech−𝛽𝜉,𝜑=−2𝛽𝛼sech−𝛽𝜉. (3.20) Noting (2.1) and (2.3), we get the following solitary wave solutions: 𝑢₈(𝑥,𝑡)=𝑒^𝑖𝜂2𝛽𝛼sech𝑣−𝛽𝜉,₈(𝑥,𝑡)=2𝛽sech−𝛽𝜉²𝛼1−4𝑝²𝑢+𝑔,₉(𝑥,𝑡)=−𝑒^𝑖𝜂2𝛽𝛼sech𝑣−𝛽𝜉,₉(𝑥,𝑡)=2𝛽sech−𝛽𝜉²𝛼1−4𝑝²+𝑔, (3.21) where 𝜂=𝑝𝑥+𝑞𝑡 and 𝜉=𝑘(𝑥−2𝑝𝑡).

(3) From the phase portrait, we see that there is a closed orbit Γ₁₂ passing the points (𝜑₁₃,0) and (𝜑₁₄,0). In (𝜑,𝑦) plane the expressions of the closed orbits are given as 𝑦=±−𝛼2𝜑₁₄−𝜑𝜑−𝜑₁₃𝜑−𝑐₁𝜑−𝑐₁, (3.22) where 𝜑₁₄=(𝛽−𝛽²−2𝛼ℎ)/𝛼, 𝜑₁₃=−(𝛽−𝛽²−2𝛼ℎ)/𝛼, 𝑐₁=𝑖(−𝛽−𝛽²−2𝛼ℎ)/𝛼, 𝑐₁=−𝑖(−𝛽−𝛽²−2𝛼ℎ)/𝛼, and ℎ>0.

Substituting (3.22) into d𝜑/d𝜉=𝑦, and integrating them along the orbit Γ₁₂, we have ±^𝜑_𝜑131𝜑₁₄−𝑠𝑠−𝜑₁₃𝑠−𝑐₁𝑠−𝑐₁d𝑠=−𝛼2^𝜉₀±d𝑠,_𝜑14𝜑1𝜑₁₄−𝑠𝑠−𝜑₁₃𝑠−𝑐₁𝑠−𝑐₁d𝑠=−𝛼2^𝜉₀d𝑠. (3.23)

Completing above integrals we obtain 𝜑=𝜑₁₃cn−𝛽𝜉,−𝜑₁₃𝛼,2𝛽𝜑=𝜑₁₄cn−𝛽𝜉,𝜑₁₄𝛼.2𝛽 (3.24)

Noting (2.1) and (2.3), we get the following periodic wave solutions: 𝑢₁₀(𝑥,𝑡)=𝑒^𝑖𝜂𝜑₁₃cn−𝛽𝜉,−𝜑₁₃𝛼,𝑣2𝛽₁₀()=𝜑𝑥,𝑡¹³cn−𝛽𝜉,−𝜑₁₃𝛼/2𝛽²1−4𝑝²𝑢+𝑔,₁₁(𝑥,𝑡)=𝑒^𝑖𝜂𝜑₁₄cn−𝛽𝜉,𝜑₁₄,𝑣𝛼/2𝛽₁₁()=𝜑𝑥,𝑡¹⁴cn−𝛽𝜉,𝜑₁₄𝛼/2𝛽²1−4𝑝²+𝑔, (3.25) where 𝜂=𝑝𝑥+𝑞𝑡 and 𝜉=𝑘(𝑥−2𝑝𝑡).

Thirdly, we will give the relations of the traveling wave solutions.(1)Letting ℎ→ℎ^∗−, it follows that 𝜑₄→𝛽/𝛼, 𝜑₃→𝛽/𝛼, 𝜑₃/𝜑₄→1 and sn(𝛽𝜉,1)=tanh𝛽𝜉. Therefore, we obtain 𝑢₁(𝑥,𝑡)→𝑢₅(𝑥,𝑡), 𝑣₁(𝑥,𝑡)→𝑣₅(𝑥,𝑡), 𝑢₂(𝑥,𝑡)→𝑢₄(𝑥,𝑡) and 𝑣₂(𝑥,𝑡)→𝑣₄(𝑥,𝑡).(2)Letting ℎ→0+, it follows that 𝜑₄→2𝛽/𝛼, 𝜑₃→0, 𝜑₃/𝜑₄→0 and sn(𝛽𝜉,0)=sin𝛽𝜉. Therefore, we obtain 𝑢₁(𝑥,𝑡)→𝑢₃(𝑥,𝑡) and 𝑣₁(𝑥,𝑡)→𝑣₃(𝑥,𝑡).(3)Letting ℎ→0−, it follows that 𝜑₁₀→2𝛽/𝛼, 𝜑₉→0, 𝜑₈→0, 𝜑₇→−2𝛽/𝛼, 𝜔→𝛽/2𝛼, 𝑘→1 and sn((−𝛽/2)𝜉,1)=tanh(−𝛽/2)𝜉. Therefore, we obtain 𝑢₆(𝑥,𝑡)→𝑢₈(𝑥,𝑡) and 𝑣₆(𝑥,𝑡)→𝑣₈(𝑥,𝑡).(4)Letting ℎ→0−, it follows that 𝜑₁₀→2𝛽/𝛼, 𝜑₉→0, 𝜑₈→0, 𝜑₇→−2𝛽/𝛼, 𝜑²₁₀−𝜑²₉/𝜑₁₀→1 and sn(−𝛽𝜉,1)=tanh−𝛽𝜉. Therefore, we obtain 𝑢₇(𝑥,𝑡)→𝑢₉(𝑥,𝑡) and 𝑣₇(𝑥,𝑡)→𝑣₉(𝑥,𝑡).(5)Letting ℎ→0+, it follows that 𝜑₁₄→2𝛽/𝛼, 𝜑₁₃→−2𝛽/𝛼, −𝜑₁₃𝛼/2𝛽→1, 𝜑₁₄𝛼/2𝛽→1 and cn(−𝛽𝜉,1)=sech−𝛽𝜉. Therefore, we obtain 𝑢₁₀(𝑥,𝑡)→𝑢₉(𝑥,𝑡), 𝑣₁₀(𝑥,𝑡)→𝑣₉(𝑥,𝑡), 𝑢₁₁(𝑥,𝑡)→𝑢₈(𝑥,𝑡) and 𝑣₁₁(𝑥,𝑡)→𝑣₈(𝑥,𝑡).

Finally, we will show that the periodic wave solutions 𝑢₂(𝑥,𝑡) evolute into the kink-profile solitary wave solutions 𝑢₄(𝑥,𝑡) when the Hamiltonian ℎ→ℎ^∗− (corresponding to the changes of phase orbits of Figure 1 as ℎ varies). We take some suitable choices of the parameters, such as 𝜆=1,𝑘=1,𝑝=1,𝑞=1,𝑔=2, (3.26) as an illustrative sample and draw their plots (see Figures 2 and 3).

(a)

(b)

(c)

Figure 2 

The real part of the periodic wave solution 𝑢2(𝑥,𝑡) evolutes into the kink-profile solitary wave solutions 𝑢4(𝑥,𝑡) with the conditions (3.26). (a) ℎ=0.5; (b) ℎ=0.749; (c) ℎ=0.75.

(a)

(b)

(c)

Figure 3 

The imaginary part of the periodic wave solution 𝑢2(𝑥,𝑡) evolutes into the kink-profile solitary wave solutions 𝑢4(𝑥,𝑡) with the conditions (3.26). (a) ℎ=0.5; (b) ℎ=0.749; (c) ℎ=0.75.

4. Conclusion

In this paper, we obtain phase portraits for the corresponding traveling wave system of (1.2) by using the bifurcation theory of planar dynamical systems. Furthermore, a number of exact traveling wave solutions are also obtained, and their relations are given. The method can be applied to many other nonlinear evolution equations, and we believe that many new results wait for further discovery by this method.

Acknowledgment

Research is supported by the National Natural Science Foundation of China (Grant no. 11171115) and the Natural Science Foundation of Yunnan Province (Grant no. 2010ZC154).