Further Results on the Traveling Wave Solutions for an Integrable Equation

Abstract

The objective of this paper is to extend some results of pioneers for the nonlinear equation 𝑚_𝑡=(1/2)(1/𝑚^𝑘)_𝑥𝑥𝑥−(1/2)(1/𝑚^𝑘)_𝑥 introduced by Qiao. The equivalent relationship of the traveling wave solutions between the integrable equation and the generalized KdV equation is revealed. Moreover, when 𝑘=−(𝑝/𝑞)(𝑝≠𝑞 and 𝑝,𝑞∈ℤ⁺), we obtain some explicit traveling wave solutions by the bifurcation method of dynamical systems.

1. Introduction

Qiao [1] introduced the following equation: 𝑚_𝑡=121𝑚²_𝑥𝑥𝑥−121𝑚²_𝑥, (1) which is the second positive member in a new completely integrable hierarchy. Equation (1) possesses a Lax representation and bi-Hamiltonian structure [1, 2]. In [1, 2] the traveling wave solutions of (1) were studied. Sakovich [3] found the transformation 𝑥=𝑣𝑦,𝑡,𝑚(𝑥,𝑡)=−1𝑣_𝑦,𝑦,𝑡 (2) which relates (1) with the well-known modified KdV equation and obtained three types of smooth solitons of (1). Moreover, the equivalence of (1) and the modified CBS equation is proved in [4]. Yang and Chen [5] obtained two potentials and two pseudopotentials of (1). Equation (1) is derived from the two-dimensional Euler equation and proven to have Lax pair and bi-Hamiltonian structures [6].

To study the bifurcations of traveling wave solutions, Li and Qiao [7] considered the following nonlinear equation: 𝑚_𝑡=121𝑚^𝑘_𝑥𝑥𝑥−121𝑚^𝑘_𝑥, (3) where 𝑘∈ℝ, 𝑘≠−1, 0. They used the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems [8–10] to find all possible bounded traveling wave solutions and their parametric representations for the cases of |𝑘|=1/2, 2, respectively. In fact, when 𝑘=1/2, (3) reads as a Harry Dym-type equation, which is actually the first member in the positive Camassa-Holm hierarchy [11–13]. The Harry Dym equation is an important integrable model in soliton theory [13]. It is also related to the classical string problem and has many applications in theoretical and experimental physics [14].

Sakovich [3] established the equivalent relationship between (1) and the modified KdV equation by the transformation for 𝑘=2. When 𝑘≥2, 𝑘∈ℤ, one objective of this paper is to find some transformations which relate the traveling wave solutions of (3) with that of the generalized KdV equation [15]: 𝑢_𝑡+𝑎𝑢^𝑛_𝑥+𝑢_𝑥𝑥𝑥=0, (4) where 𝑛 is a positive integer and 𝑎≠0. For the results of traveling wave solutions on (4), we refer the readers to [16–21]. The other objective is to extend the results of Li and Qiao [7]. We continue to consider the problems on explicit traveling wave solutions of (3) and their bifurcations, but we do not use the theory of the singular traveling wave systems. Instead, we apply the transformations which transform (3) into a traveling wave system without singular straight line [21, 22]. Then, by using the bifurcation method of dynamical systems [21–28], we obtain some explicit traveling wave solutions of (3) for the case of 𝑘=−(𝑝/𝑞)(𝑝≠𝑞and𝑝,𝑞∈ℤ⁺). Not only the existence of these solutions are proved, but also their concrete expressions are presented.

The rest of the paper is organized as follows. In Section 2, we reveal the equivalent relationship of the traveling wave solutions between (3) and (4). In Section 3, various planar systems and their bifurcation phase portraits of (3) are given. We state the explicit traveling wave solutions of (3) and present their theoretical derivation in Section 4. Some conclusions are given in Section 5.

2. Equivalent Relationship of (3) and (4)

In order to study the equivalent relationship of the traveling wave solutions between (3) and (4), we transform both (3) and (4) into traveling wave systems.

First of all, we substitute 𝑚=𝑈(𝜉) with 𝜉=𝑥−𝑐𝑡  (𝑐≠0) into (3). Then, we get −𝑐𝑈^′=121𝑈^{𝑘′′′}−121𝑈^𝑘′, (5) where 𝑐 is a constant wave speed. Letting 𝑘=−𝑠 and integrating (5) once, we have 1−𝑐𝑈=2𝑈^𝑠′′−12𝑈^𝑠+𝑔, (6) where 𝑔 is an integral constant. Letting 𝑈=𝜑^1/𝑠, (7) then (6) can be rewritten as −𝑐𝜑^1/𝑠=12𝜑^′′−12𝜑+𝑔. (8)

Letting 𝑛=1/𝑠, we have 𝜑^′′+2𝑐𝜑^𝑛−𝜑+2𝑔=0. (9)

Next, we also transform (4) into traveling wave system. Substituting 𝑢(𝑥,𝑡)=𝜓(𝜂) with 𝜂=𝑐^1/3𝑥−𝑐𝑡 into (4), we have −𝑐𝜓^′+𝑎𝑐^1/3(𝜓^𝑛)^′+𝑐𝜓^′′′=0. (10) Integrating (10) once and letting 𝑎=2𝑐^5/3 leads to 𝜓^′′+2𝑐𝜓^𝑛−𝜓+𝑔₁=0, (11) where 𝑔₁ is an integral constant.

Finally, according to (9) and (11), we know that, from the traveling wave solution (3), we can drive the solutions of (4) for 𝑘∈ℤ. When 𝑘∈ℤ, one can notice that the traveling wave system of (4) is more general than (9) because of the arbitrary coefficient 𝑎. However, the traveling wave solutions of (3) cannot be derived from the solutions of (4) for 𝑘∉ℤ. Next, we study the traveling wave solutions of (3) and their bifurcations for 𝑘∉ℤ.

3. Planar Systems and Their Bifurcation Phase Portraits

In this section, we derive the traveling wave systems of (3) for the different cases of 𝑘 and draw their bifurcation phase portraits which are the basis for constructing nonlinear wave solutions.

When 𝑘=−𝑝/𝑞, that is, 𝑛=𝑞/𝑝, putting 𝑦=𝜑^′ and 𝑔=0, from (9), we obtain the following planar system 𝑑𝜑𝑑𝜉=𝑦,𝑑𝑦𝑑𝜉=𝜑−2𝑐𝜑^𝑞/𝑝. (12) Letting 𝜑=𝜙^𝑝, (13) we have 𝑑𝜑=𝑝𝜙^𝑝−1𝑑𝜙. (14)

Then, system (12) can be written as 𝑝𝜙^𝑝−1𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦𝑑𝜉=𝜙^𝑝−2𝑐𝜙^𝑞. (15) Under the transformation 𝑑𝜏=𝑑𝜉/𝑝𝜙^𝑝−1, we have 𝑑𝜙𝑑𝜏=𝑦,𝑑𝑦𝑑𝜏=𝑝𝜙^2𝑝−1−2𝑐𝑝𝜙^{𝑝+𝑞−1}. (16)

System (16) has the first integral 𝐻𝜙,𝑦=ℎ, (17) where 𝐻𝜙,𝑦=𝑦²−𝜙^2𝑝+4𝑐𝑝𝜙𝑝+𝑞^𝑝+𝑞. (18)

Next, we discuss the phase portraits of system. (16) for two different cases of 𝑝−𝑞.

(a) 𝑝−𝑞 is even and 𝑐>0.

Letting 𝑓𝜙=𝑝𝜙^2𝑝−1−2𝑐𝑝𝜙^{𝑝+𝑞−1}=𝑝𝜙^𝑝−1𝜙^𝑝−2𝑐𝜙^𝑞, (19)

we have 𝑓^′𝜙=𝑝2𝑝−1𝜙^2𝑝−2−2𝑐𝑝𝑝+𝑞−1𝜙^{𝑝+𝑞−2}. (20)

Solving 𝑓(𝜙)=0, we get 𝜙₁=0,𝜙_2±1=±2𝑐^{1/(𝑝−𝑞)}. (21)

At the singular point (𝜙,0), it is easy to obtain that the linearized system of system (16) has the eigenvalues 𝜆_±𝜙,0=±𝑓^′𝜙. (22)

From (12) and (18), we get the properties of the singular points (𝜙_𝑖,0)(𝑖=1,2) as follow.(i) If 𝑓^′(𝜙_𝑖)>0, then (𝜙_𝑖,0) is a saddle point of system (16).(ii) If 𝑓^′(𝜙_𝑖)<0, then (𝜙_𝑖,0) is a center point of system (16).(iii) If 𝑓^′(𝜙_𝑖)=0, then (𝜙_𝑖,0) is a degenerate singular point of system (16).

Therefore, we have the following results.(1) When 𝑝>𝑞, (𝜙₁,0) is a center point, and (𝜙₂₊,0) and (𝜙₂₋,0) are saddle points. Due to 𝐻(𝜙₂₊,0)=𝐻(𝜙₂₋,0), the orbits connecting with (𝜙₂₊,0) and (𝜙₂₋,0) are heteroclinic orbits.(2) When 𝑝<𝑞, (𝜙₁,0) is a saddle point, and (𝜙₂₊,0) and (𝜙₂₋,0) are center points.

From the previous discussion, we get the phase portraits of system (13) as Figures 1(a) and 1(b).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

The phase portraits of Sy. (16). When 𝑝−𝑞 is even and 𝑐>0, (a) and (b) are the phase portraits of system (16) for 𝑝>𝑞 and 𝑝<𝑞, respectively. When 𝑝−𝑞 is odd, (c)–(f) are the phase portraits of system (16) for 𝑝>𝑞 and 𝑐>0, 𝑝>𝑞 and 𝑐<0, 𝑝<𝑞 and 𝑐>0, 𝑝<𝑞 and 𝑐<0, respectively.

(b) 𝑝−𝑞 is odd.

In this case, 𝑓(𝜙) has two zero points 𝜙₃ and 𝜙₄, where 𝜙₃=0,𝜙₄=12𝑐^{1/(𝑝−𝑞)}. (23)

Similar to the previous discussion, we obtain the following results.(1)When 𝑝>𝑞 and 𝑐>0, (𝜙₃,0) is a center point and (𝜙₄,0) is a saddle point.(2)When 𝑝>𝑞 and 𝑐<0, (𝜙₃,0) is a saddle point and (𝜙₄,0) is a center point.(3)When 𝑝<𝑞 and 𝑐>0, (𝜙₃,0) is a saddle point and (𝜙₄,0) is a center point.(4)When 𝑝<𝑞 and 𝑐<0, (𝜙₃,0) is a saddle point and (𝜙₄,0) is a center point.

Through the discussion mentioned above, we obtain the phase portraits of Sy. (16) as Figures 1(c)–1(f).

4. New Exact Solutions and Theoretical Derivation

When 𝑘=−𝑝/𝑞, 𝑝≠𝑞, 𝑝, 𝑞∈ℤ⁺, we have the following results.(1)When 𝑝>𝑞 and 𝑝−𝑞 is odd, (3) has the following exact solution: 𝑚₁()=𝑒𝑥,𝑡_{−(𝑥−𝑐𝑡)−𝑋}16(𝑝+𝑞)²𝑒^{𝑥−𝑐𝑡}+16𝑐𝑝²𝑒^𝑋+16𝑐𝑝𝑞𝑒^{𝑋2𝑞/(𝑝−𝑞)}, (24) where 𝑋=𝑞(𝑥−𝑐𝑡)/𝑝.(2)When 𝑝>𝑞, 𝑝−𝑞 is even and 𝑐>0, (3) also has the solution of the same expression as 𝑚₁(𝑥,𝑡).(3)When 𝑝<𝑞 and 𝑝−𝑞 is odd, (3) has the solitary wave solution 𝑚₂(𝑥,𝑡)=𝑝+𝑞4𝑐𝑝sech²(𝑞−𝑝)(𝑥−𝑐𝑡)2𝑝^{𝑞/(𝑞−𝑝)}, (25) and the blow-up solution 𝑚₃(𝑥,𝑡)=−𝑝+𝑞4𝑐𝑝csch²(𝑞−𝑝)(𝑥−𝑐𝑡)2𝑝^{𝑞/(𝑞−𝑝)}. (26)(4)When 𝑝<𝑞, 𝑝−𝑞 is even and 𝑐>0, (3) also has the solitary wave solution of the same expression as 𝑚₂(𝑥,𝑡).

Next, we give the demonstration for the previous results of (3) by two cases.

Case 1 (𝑝<𝑞). When 𝑝<𝑞, 𝑝−𝑞 is odd and 𝑐>0, system (16) has two singular points (0,0) and (𝜙₂,0), where 𝜙₂=(1/2𝑐)^{1/(𝑝−𝑞)}. The singular point (𝜙₂,0) is a center point, and (0,0) is a saddle point.
When 𝑝<𝑞, 𝑝−𝑞 is even and 𝑐>0, system (16) has three singular points (𝜙₄,0), (0,0) and (−𝜙₄,0), where 𝜙₄=(1/2𝑐)^{1/(𝑝−𝑞)}. The singular points (𝜙₄,0) and (−𝜙₄,0) are center points, and the singular point (0,0) is a saddle point.
Assume that (𝜙₀,0) is an initial point of system (13), then we have the following results.(1)When 0<𝜙₀≤𝜙₁=(4𝑐𝑝/(𝑝+𝑞))^{1/(𝑝−𝑞)}, the boundary of the closed orbit denoted by Γ₁ is a homoclinic orbit which passes (𝜙₁,0) and connects with (0,0) (see Figure 2(a)).(2)When 𝜙₀<0, there are two special orbits denoted by Γ⁺₂ and Γ⁻₂ which pass through (0,0) (see Figure 2(a)). (3)When 0<𝜙₀≤𝜙₃, we use Γ₃ to sign the orbit which passes through (𝜙₃,0) and connects with (0,0), where 𝜙₃=(4𝑐𝑝/(𝑝+𝑞))^{1/(𝑝−𝑞)} (see Figure 2(b)).

(a)

(b)

Figure 2 

The phase portraits of Sy. (16). (a) For 𝑝<𝑞, 𝑝−𝑞 is odd and 𝑐>0. (b) For 𝑝<𝑞, 𝑝−𝑞 is even and 𝑐>0.

On the 𝜙−𝑦 plane, the orbits Γ₁, Γ^±₂, and Γ₃ have the same expression which can be written as 𝑦=±𝜙^2𝑝−4𝑐𝑝𝜙𝑝+𝑞^𝑝+𝑞. (27)

Substituting the previous expression into 𝑝𝜙^𝑝−1𝑑𝜙/𝑑𝜉=𝑦 and integrating it along different orbits, we have ^𝜙_𝜙1𝑝𝑑𝑠𝑠1−4𝑐𝑝/𝑝+𝑞𝑠^𝑞−𝑝=𝜉alongΓ₁,_𝜙−∞𝑝𝑑𝑠𝑠1−4𝑐𝑝/𝑝+𝑞𝑠^𝑞−𝑝=𝜉alongΓ^±₂,_𝜙𝜙3𝑝𝑑𝑠𝑠1−4𝑐𝑝/𝑝+𝑞𝑠^𝑞−𝑝=𝜉alongΓ₃. (28)(29)(30)

In (28), completing the integration and solving the equation for 𝜙, it follows that 𝜙₂(𝜉)=𝑝+𝑞4𝑐𝑝sech²(𝑞−𝑝)𝜉2𝑝^{1/(𝑞−𝑝)}. (31)

Similarly, via (29) we have 𝜙₃(𝜉)=−𝑝+𝑞4𝑐𝑝csch²(𝑞−𝑝)𝜉2𝑝^{1/(𝑞−𝑝)}. (32)

Via (30), we get the solution of the same expression as 𝜙₂(𝜉).

When 𝑝<𝑞 and 𝑐<0, we can obtain the solutions of the same expressions as 𝜙₂(𝜉) and 𝜙₃(𝜉).

Note that the transformation 𝑚=𝜙^𝑞(𝜉), we get the smooth solitary wave solution 𝑚₂(𝑥,𝑡) and the blow-up solution 𝑚₃(𝑥,𝑡) of (3).

Case 2 (𝑝>𝑞). When 𝑘=−𝑝/𝑞, 𝑝>𝑞, 𝑝−𝑞 is odd and 𝑐>0, there is an interesting phenomenon concerning the traveling wave solutions of (3). Generally speaking, homoclinic orbit is corresponding to solitary wave solution. However, after applying the transformation to (3), we do not get solitary wave solution from homoclinic orbit. Therefore, we have the reason to believe that the transformation influences the corresponding relations between bifurcation orbits and traveling wave solutions.(1) When 𝑝>𝑞, 𝑝−𝑞 is odd and 𝑐>0, system (16) has two singular points (0,0) and (𝜙₅,0), where 𝜙₅=(2𝑐)^{1/(𝑝−𝑞)}. The singular point (0,0) is a center point and the singular point (𝜙₅,0) is a saddle point. The boundary of the closed orbit denoted by Γ₄ is a homoclinic orbit which passes through (𝜙₆,0) and connects with (0,0), where 𝐻(0,0)=𝐻(𝜙₆,0) and 𝜙₆=(4𝑐𝑝/(𝑝+𝑞))^{1/(𝑝−𝑞)} (see Figure 3(a)).(2) When 𝑝>𝑞, 𝑝−𝑞 is even and 𝑐>0, Sy. (16) has three singular points (0,0), (𝜙₈,0), and (−𝜙₈,0), where 𝜙₈=(2𝑐)^{1/(𝑝−𝑞)}. The singular points (𝜙₈,0) and (−𝜙₈,0) are saddle points. The singular point (0,0) is a center point. Due to 𝐻(𝜙₈,0)=𝐻(−𝜙₈,0), the orbits connecting with (𝜙₈,0) and (−𝜙₈,0) are two heteroclinic orbits. Here we only consider two special orbits which pass through (𝜙₇,0), where 𝜙₇=(4𝑐𝑝/(𝑝+𝑞))^{1/(𝑝−𝑞)}. We denote the orbits by Γ^±₅ (see Figure 3(b)).

(a)

(b)

Figure 3 

The phase portraits of Sy. (16). (a) For 𝑝>𝑞, 𝑝−𝑞 is odd and 𝑐>0. (b) For 𝑝>𝑞, 𝑝−𝑞 is even and 𝑐>0.

On 𝜙−𝑦 plane, the orbit Γ₄ and the orbits Γ^±₅ have the same expression 𝑦=±𝜙^2𝑝−4𝑐𝑝𝜙𝑝+𝑞^𝑝+𝑞. (33)

We take ((4𝑐𝑝/(𝑝+𝑞))^{1/(𝑝−𝑞)},0), that is, (𝜙₆,0) or (𝜙₇,0) as an initial point for system (16). Substituting the previous expression into 𝑝𝜙^𝑝−1𝑑𝜙/𝑑𝜉=𝑦 and integrating it along Γ^±₄ and Γ^±₅, respectively, it follows that ^𝜙_{(4𝑐𝑝/(𝑝+𝑞))1/(𝑝−𝑞)}𝑝𝑠^𝑝−1𝑑𝑠𝑠^2𝑝−4𝑐𝑝/𝑝+𝑞𝑠^𝑝+𝑞=^𝜙_{(4𝑐𝑝/(𝑝+𝑞))1/(𝑝−𝑞)}𝑝𝑠^{(𝑝−𝑞)/2}𝑑𝑠𝑠𝑠^𝑝−𝑞=𝜉.−4𝑐𝑝/𝑝+𝑞 (34)

In (34), completing the integration and solving the equation for 𝜙, it follows that 𝜙₁(𝜉)=𝑒^{−𝜉−𝜂}16(𝑝+𝑞)²𝑒^𝜉+16𝑐𝑝²𝑒^𝜂+16𝑐𝑝𝑞𝑒^{𝜂21/(𝑝−𝑞)}, (35) where 𝜂=𝑞𝜉/𝑝.

From the transformation 𝑚=𝜙^𝑞(𝜉), we obtain the traveling wave solution 𝑚₁(𝑥,𝑡) of (3).

The theoretical derivation of the other cases can be finished similarly. We omit it for convenience. Hereto, we have completed the demonstrations to the previous results of (3).

5. Conclusions

In this paper, we find some transformations which relate (3) with (4). Applying these transformations, we reveal the relationship of traveling wave solutions between (3) and (4). By using the bifurcation method of dynamical systems, we consider the further results on the explicit traveling wave solutions of (3) for the special case of 𝑘=−𝑝/𝑞, where 𝑝≠𝑞, 𝑝, 𝑞∈ℤ⁺. The correctness of these solutions is tested as well by using the software Mathematica.

Note that in this paper, there are two problems waiting to solve. The first one is that we only discuss the equivalent relationship of the traveling wave solutions between (3) and (4). We do not know whether the relationship of the other solutions of (3) and (4) is equivalent. The second one is that we have investigated the traveling wave solutions of (3) for the special cases of 𝑘=−𝑝/𝑞. But the traveling wave solutions of (3) for the other cases of 𝑘 await further study.

Conflicts of Interests

The authors have declared that there is no conflict of interests.

Acknowledgment

This research is supported by the National Natural Science Foundation of China (no. 11171115), Natural Science Foundation of Guangdong Province (no. S2012040007959), and the Fundamental Research Funds for the Central Universities (no. 2012ZM0057).