Approximate Damped Oscillatory Solutions for Generalized KdV-Burgers Equation and Their Error Estimates

Abstract

We focus on studying approximate solutions of damped oscillatory solutions of generalized KdV-Burgers equation and their error estimates. The theory of planar dynamical systems is employed to make qualitative analysis to the dynamical systems which traveling wave solutions of this equation correspond to. We investigate the relations between the behaviors of bounded traveling wave solutions and dissipation coefficient, and give two critical values 𝜆₁ and 𝜆₂ which can characterize the scale of dissipation effect, for right and left-traveling wave solution, respectively. We obtain that for the right-traveling wave solution if dissipation coefficient 𝛼≥𝜆₁, it appears as a monotone kink profile solitary wave solution; that if 0<𝛼<𝜆₁, it appears as a damped oscillatory solution. This is similar for the left-traveling wave solution. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, we obtain their approximate damped oscillatory solutions by undetermined coefficients method. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between exact and approximate solutions. The errors are infinitesimal decreasing in the exponential form.

1. Introduction

Generalized KdV equation with dissipation term 𝑢_𝑡+𝑏𝑢^𝑝𝑢_𝑥−𝛼𝑢_𝑥𝑥+𝑢_𝑥𝑥𝑥=0(1.1) is the physical model describing the long-wave propagating in nonlinear media with dispersion-dissipation [1], where 𝛼≥0, 𝑏 is any real number and 𝑝 is any positive integer. It is very important in theory and application. If 𝑝=1, (1.1) becomes KdV-Burgers equation 𝑢_𝑡+𝑏𝑢𝑢_𝑥−𝛼𝑢_𝑥𝑥+𝑢_𝑥𝑥𝑥=0.(1.2) Equation (1.2) can be regarded as a control equation for many kinds of practical problems with some dissipation effect, such as shadow wave in viscous liquid, liquid flowing, and waving in elastic pipe, magnetosonic wave in plasma, and so on [2–8]. If 𝛼=0, (1.1) has no dissipation effect and becomes generalized KdV equation 𝑢_𝑡+𝑏𝑢^𝑝𝑢_𝑥+𝑢_𝑥𝑥𝑥=0.(1.3) Obviously, (1.1) is KdV equation [9–11] if 𝑝=1, while it is MKdV equation [12–15] if 𝑝=2. The stability of solitary wave solutions for (1.3) was studied in [16–22], and the results that the solitary wave solutions are stable if 𝑝<4 while that they are unstable if 𝑝>4 were obtained. Pego et al. summarized the discussion of [4, 20] in [1] and gave the following conclusion.

If 𝛼>0,𝑐>0, (1.1) has a unique traveling wave solution 𝑢(𝑥,𝑡)=𝜙(𝑥−𝑐𝑡) when 𝑏=1, where 𝜙(𝜉) satisfies −𝑐𝜙+1𝑝+1𝜙^𝑝+1𝜙(𝜉)⟶+𝑢𝜕^2𝜉_𝐿𝜙=𝛼𝜕_𝜉𝜙,𝜉∈𝑅,0𝜉𝜉⟶⟶+−∞∞,,(1.4) where 𝑢_𝐿=[𝑐(𝑝+1)]^1/𝑝 is the unique positive solution of −𝑐𝑢_𝐿+(1/(𝑝+1))𝑢^𝐿𝑝+1=0. If 𝛼>2√𝑝𝑐, 𝜙(𝜉) monotonically decreases. If 𝛼<2√𝑝𝑐, 𝜙(𝜉) approaches 𝑢_𝐿 in oscillatory form as 𝜉→−∞; while as 𝜉→+∞, 𝜙(𝜉) satisfies 𝜙(𝜉)∼𝑒^{𝜈(𝜉−𝜉0)},𝜕_𝜉𝜙(𝜉)∼𝜈𝑒^{𝜈(𝜉−𝜉0)},(1.5) where 𝜈=(1/2)(𝛼−√𝛼²+4𝑐)<0.

Studies in [1] focused on discussing the instability of oscillatory traveling wave solution of (1.1) when 𝑏=1 by the method of numerically simulating Evans' function 𝐷(𝜆) and obtained that linear instability takes place when, (1) for fix positive 𝑐 and𝑝>4, 𝛼 is made sufficiently small; (2) for fix positive 𝛼 and 𝑝>4, 𝑐 is made sufficiently large; (3) for fix positive 𝛼 and 𝑐, 𝑝 is made sufficiently large.

About the damped oscillatory traveling wave solutions of (1.1) when 𝑝=1, there have been many references. From these references, we know that Grad and Hu [3] and Bona and Schonbek [4] obtained the existence of a damped oscillatory solution for KdV-Burgers equation (1.2) by the method of planar dynamical systems, respectively. Besides the existence of a damped oscillatory solution for KdV-Burgers equation (1.2), Johnson also presented the asymptotic expansion for this solution in [5]. Canosa and Gazdag [23] gave the numerical solution for a damped oscillatory solution of KdV-Burgers equation (1.2). By using qualitative theory of ordinary differential equations, Gao and Guan [24] studied the bounded non trivial traveling wave solutions of KdV-Burgers equation (1.2). In the meantime, they proved that when the dissipation coefficient is greater than some value, that is, it satisfied (3.3) in [24], the traveling wave solution for (1.2) is monotone, and its properties are the same as the properties of the kink profile solitary wave solution of Burgers equation; when the dissipation coefficient is less than some value, that is, it satisfied (3.9) in [25], the traveling wave solution of (1.2) is a damped oscillatory solution which has a bell profile head. However, they did not present any analytic solution or approximate solution for (1.2). Xiong [25] obtained a kink profile solitary wave solution for KdV-Burgers equation. In [26], S. D. Liu and S. D. Liu obtained an approximate damped oscillatory solution to a saddle-focus kink profile solitary solution of KdV-Burgers equation, without giving its error estimate. However, it is very important to give its error estimate, or people will feel unreliable.

From the above references, we can see that there is not only theoretical but also practical significance to find damped oscillatory solutions. However, the above references did not give exact or approximate damped oscillatory solutions of (1.1) when 𝑝≥1, and we have not seen the references about it. In this paper, we focus on studying the relations between the behaviors of bounded traveling wave solutions and the dissipation coefficient, showing the reason why the damped oscillatory solutions take place and how to obtain the approximate damped oscillatory solutions and giving their error estimates. We will obtain all the results in [1, Theorem  1.1] when 𝑝 is natural number, as well as the existent number of bounded traveling wave solutions and relations between the behaviors of bounded traveling wave solutions and the dissipation coefficient 𝛼 in the case of 𝑏>0,𝑐<0,𝑏<0,𝑐>0, and 𝑏<0,𝑐<0, respectively. More importantly, we will give approximate damped oscillatory solution and its error estimate in the case of 𝑐>0,0<𝛼<2√𝑝𝑐 when 𝑝 is any natural number.

This paper is organized as follows. In Section 2, we carry out qualitative analysis for the planar dynamical system corresponding to (1.1). We present all global phase portraits of this planar dynamical system and give the existent conditions and number of bounded traveling wave solutions of (1.1). We obtain that, if 𝛼=0, (1.1) at most has two bell profile solitary wave solutions or two kink profile solitary wave solutions and that, if 𝛼≠0, (1.1) at most has two bounded traveling wave solutions (kink profile or oscillatory traveling wave solutions). In Section 3, we discuss the relations between the behaviors of bounded traveling wave solutions and the dissipation coefficient 𝛼. We find out two critical values 𝜆₁=2√𝑝𝑐 and 𝜆₂=2√−𝑐 and obtain that for the right-traveling wave of the equation, a bounded traveling wave solution appears as a monotones kink profile solitary wave solution if dissipation coefficient 𝛼≥𝜆₁, while it appears as a damped oscillatory wave if 0<𝛼<𝜆₁; for the left-traveling wave of the equation, a bounded traveling wave solution appears as a monotones kink profile solitary wave solution if dissipation coefficient 𝛼≥𝜆₂, while it appears as a damped oscillatory wave if 0<𝛼<𝜆₂. In Section 4, the exact bell profile and kink profile solitary wave solutions of (1.1) without dissipation effect are presented. Furthermore, according to the evolution relation of solution orbits in global phase portraits, by undetermined coefficients method, we obtain approximate damped oscillatory solutions of (1.1). In Section 5, we study the error estimate between approximate damped oscillatory solutions and their exact solutions. The difficulty of this problem is that we only know the approximate damped oscillatory solutions, but do not know their exact solutions. To overcome it, we use some transformations and the idea of homogenization principle and then establish the integral equations reflecting the relations between the exact solutions and approximate damped oscillatory solutions. Thus, we give error estimates for the approximate solutions obtained in Section 4. We can see that the errors between the exact solutions and approximate damped oscillatory solutions we obtained by this method are infinitesimal decreasing in exponential form.

2. Qualitative Analysis to Bounded Traveling Wave Solutions of (1.1)

Assume that 𝑢(𝑥,𝑡)=𝑢(𝜉)=𝑢(𝑥−𝑐𝑡) is a traveling wave solution of (1.1), and 𝑢(𝜉) satisfies −𝑐𝑢^(𝜉)+𝑏𝑢^𝑝(𝜉)𝑢^(𝜉)−𝛼𝑢^(𝜉)+𝑢^(𝜉)=0,(2.1) where 𝑐 is the wave speed. Integrating the above equation once yields 𝑢^(𝜉)−𝛼𝑢^(𝜉)−𝑐𝑢(𝜉)+𝑏𝑝+1𝑢^𝑝+1(𝜉)=𝑔,(2.2) where 𝑔 is an integral constant. Owing that we focus on studying dissipation effect to the system, we assume that the traveling wave solutions we study satisfy 𝑢^(𝜉),𝑢^(𝜉)⟶0,||𝜉||⟶∞(2.3) and the asymptotic values C₊ and 𝐶₋(𝐶₊=lim_𝜉→+∞𝑢(𝜉), 𝐶₋=lim_{𝜉→−∞}(𝜉)) satisfy 𝑏𝑝+1𝑥^𝑝+1−𝑐𝑥=0,(2.4) so under the hypothesis (2.3) and (2.4), the traveling wave solutions of (1.1) satisfy 𝑢^(𝜉)−𝛼𝑢^(𝜉)−𝑐𝑢(𝜉)+𝑏𝑝+1𝑢^𝑝+1(𝜉)=0(2.5)

Remark 2.1. In the following discussion, we will always assume that the traveling wave solutions of (1.1) satisfy (2.3) and (2.4).
Letting 𝑥=𝑢(𝜉) and 𝑦=𝑢^(𝜉), then (2.5) can be reformulated as a planar dynamical system 𝑑𝑥𝑑𝑦𝑑𝜉=𝑦≡𝑃(𝑥,𝑦),𝑑𝜉=𝛼𝑦+𝑐𝑥−𝑝+𝑏1𝑥^𝑝+1≡𝑄(𝑥,𝑦).(2.6)
On (𝑥,𝑦) phase plane, the number of singular points of system (2.6) depends on the number of real roots of 𝑓(𝑥)=(𝑏/(𝑝+1))𝑥^𝑝+1−𝑐𝑥=0. Denote 𝑥₀=0, 𝑥₁=[(𝑐(𝑝+1))/𝑏]^1/𝑝, 𝑥₂=−𝑥₁. It is easy to know the following results on the real roots of 𝑓(𝑥)=0.
(1) If 𝑏𝑐>0, when 𝑝 is an even integer, 𝑓(𝑥) has three different real roots 𝑥₀, 𝑥₁ and 𝑥₂; when 𝑝 is an odd integer, 𝑓(𝑥) has two different real roots 𝑥₀ and 𝑥₁.
(2) If 𝑏𝑐<0, when 𝑝 is an even integer, 𝑓(𝑥) only has one real root 𝑥₀; when 𝑝 is an odd integer, 𝑓(𝑥) has two different real roots 𝑥₀ and 𝑥₁.
Since system (2.6) has and only has one singular point when 𝑏𝑐<0, and 𝑝 is an even integer, there does not exist traveling wave solutions in (1.1). We will not consider this case. We use 𝐽𝑥_𝑖,0=⎛⎜⎜⎝−𝑓_01𝑥_𝑖𝛼⎞⎟⎟⎠,𝑖=0,1,2(2.7) to denote the Jacobian matrix of system (2.6) at singular points 𝑃_𝑖(𝑥_𝑖,0). Therefore, determinant of 𝐽(𝑥_𝑖,0), that is, det(𝐽(𝑥_𝑖,0)), is 𝑓^(𝑥_𝑖),𝑖=0,1,2. Use Δ_𝑖 to denote the discriminant of characteristic equation at 𝑃_𝑖(𝑥_𝑖,0), Δ_𝑖=𝛼²−4𝑓′(𝑥_𝑖),𝑖=0,1,2. It is easy to know Δ₀=𝛼²+4𝑐, Δ₁=𝛼²−4𝑝𝑐; if 𝑝 is an even integer, Δ₁=Δ₂.
In the following, we employ the theory and method of planar dynamical systems [27–29] to discuss the type of singular points of system (2.6) and give the global phase portraits.

2.1. In the Case of 𝛼=0

In this case, system (2.6) has the first integral 𝐻(𝑥,𝑦)≡𝑦²+𝑥²(2𝑏𝑥^𝑝−𝑐(𝑝+1)(𝑝+2))(𝑝+1)(𝑝+2)=ℎ,ℎ∈𝑅.(2.8)

It is easy to see that 𝑃₀ is a saddle point if 𝑐>0 and that 𝑃₀ is a center if 𝑐<0. The types of singular points 𝑃_𝑖 are shown as follows:(1)𝑏>0,𝑐>0. (i) If 𝑝 is an even integer, system (2.6) has three singular points 𝑃₀(0,0) and 𝑃_𝑖(𝑥_𝑖,0), 𝑖=1,2. Since (2.8) holds and det(𝐽(𝑥_𝑖,0))=𝑓^(𝑥_𝑖)=𝑝𝑐>0, 𝑃_𝑖(𝑥_𝑖,0), 𝑖=1,2 are centers. (ii) If 𝑝 is an odd integer, system (2.6) has two singular points 𝑃₀(0,0) and 𝑃₁(𝑥₁,0). Since (2.8) holds and det(𝐽(𝑥₁,0))=𝑓^(𝑥₁)=𝑝𝑐>0, 𝑃₁ is a center;(2)𝑏<0,𝑐<0. (i) If 𝑝 is an even integer, system (2.6) has three singular points 𝑃₀(0,0) and 𝑃_𝑖(𝑥_𝑖,0), 𝑖=1,2. Since 𝑓^(𝑥_𝑖)<0, 𝑃_𝑖(𝑥_𝑖,0),𝑖=1,2 are saddle points. (ii) If 𝑝 is an odd integer, system (2.6) has two singular points 𝑃₀(0,0) and 𝑃₁(𝑥₁,0). Since 𝑓^(𝑥₁)<0, 𝑃₁ is a saddle point;(3)𝑏>0,𝑐<0. If 𝑝 is an odd integer, system (2.6) has two singular points 𝑃₀(0,0) and 𝑃₁(𝑥₁,0). Since 𝑓^(𝑥₁)<0, 𝑃₁ is a saddle point;(4)𝑏<0,𝑐>0. If 𝑝 is an odd integer, system (2.6) has two singular points 𝑃₀(0,0) and 𝑃₁(𝑥₁,0). Since (2.8) holds and 𝑓^(𝑥₁)>0, 𝑃₁ is a center.

2.2. In the Case of 𝛼>0

It is easy to see that 𝑃₀ is a saddle point if 𝑐>0, while that 𝑃₀ is an unstable singular point if 𝑐<0, where 𝑃₀ is an unstable node point if Δ₀>0 and 𝑃₀ is an unstable focus point if Δ₀<0. The types of singular points 𝑃_𝑖 are shown as follows:(1)𝑏>0,𝑐>0. (i) If 𝑝 is an even integer, since 𝑓^(𝑥_𝑖)>0, 𝑃_𝑖, 𝑖=1,2 are unstable node points if Δ₁>0, and 𝑃_𝑖, 𝑖=1,2 are unstable focus points if Δ₁<0. (ii) If 𝑝 is an odd integer, since 𝑓^(𝑥_𝑖)>0, 𝑃₁ is unstable node point if Δ₁>0 and is unstable focus point if Δ₁<0;(2)𝑏<0,𝑐<0. (i) If 𝑝 is an even integer, since 𝑓^(𝑥_𝑖)<0, 𝑃_𝑖(𝑥_𝑖,0), 𝑖=1,2 are saddle points. (ii) If 𝑝 is an odd integer, since 𝑓^(𝑥₁)<0, 𝑃₁ is a saddle point;(3)𝑏>0,𝑐<0. If 𝑝 is an odd integer, system (2.6) has two singular points 𝑃₀(0,0) and 𝑃₁(𝑥₁,0). Since 𝑓^(𝑥₁)<0, 𝑃₁ is a saddle point;(4)𝑏<0,𝑐>0. If 𝑝 is an odd integer, system (2.6) has two singular points 𝑃₀(0,0) and 𝑃₁(𝑥₁,0). Since 𝑓^(𝑥₁)>0, 𝑃₁ is unstable node point if Δ₁>0 and is unstable focus point if Δ₁<0.

Applying Poincaré transformation to analyze singular points at infinity of system (2.6), it is clear that there only exists a couple of singular points 𝐴_𝑖, 𝑖=1,2 at infinity on 𝑦 axis, where 𝐴₁ lies on the positive semiaxis of 𝑦 and 𝐴₂ lies on the negative semiaxis of 𝑦. There, respectively, exists a hyperbolic type region around 𝐴_𝑖 if 𝑝 is an even integer and 𝑏>0. There, respectively, exists a elliptic type region around 𝐴_𝑖 if 𝑝 is an even integer and 𝑏<0. There, respectively, exists a parabolic type region around 𝐴_𝑖 if 𝑝 is an odd integer. Moreover, the circumference of Poincaré disk is orbits.

For system (2.6), owing to 𝜕𝑃/𝜕𝑥+𝜕𝑄/𝜕𝑦=𝛼, by Bendixson-Dulac's criterion [27–29], the following proposition holds.

Proposition 2.2. If 𝛼>0, then system (2.6) does not have any closed orbit or singular closed orbit with finite number of singular points on (𝑥,𝑦) phase plane. Further, as 𝛼>0, there exists no periodic traveling wave solution or bell profile solitary wave solution in (1.1).

According to the above analysis, we present the global phase portraits of system (2.6) under different parameter conditions (see Figures 1 and 12):(1) global phase portraits in case of 𝛼=0,(i)𝑝 is an even integer,(ii)𝑝 is an odd integer;(2) Global phase portraits in case of 𝛼>0,(i)𝑝 is an even integer,(ii)𝑝 is an odd integer.

Figure 1 

(𝑏>0, 𝑐<0).

From Figures 1–12, we can derive the following propositions.

Figure 2 

(𝑏<0, 𝑐<0).

Figure 3 

(𝑏>0,𝑐>0).

Figure 4 

(𝑏<0, 𝑐<0).

Figure 5 

(𝑏<0, 𝑐>0).

Figure 6 

(𝑏>0, 𝑐<0).

(a) (Δ1>0)

(b) (Δ1<0)

Figure 7 

(𝑏>0, 𝑐>0).

(a) (Δ0>0)

(b) (Δ0<0)

Figure 8 

(𝑏<0, 𝑐<0).

(a) (Δ1>0)

(b) (Δ1<0)

Figure 9 

(𝑏>0, 𝑐>0).

(a) (Δ0>0)

(b) (Δ0<0)

Figure 10 

(𝑏<0, 𝑐<0).

(a) (Δ1>0)

(b) (Δ1<0)

Figure 11 

(𝑏<0, 𝑐>0).

(a) (Δ0>0)

(b) (Δ0<0)

Figure 12 

(𝑏>0, 𝑐<0).

Proposition 2.3. (1) If 𝛼=0, then except 𝑃_𝑖(𝑖=0,1,2), 𝐿(𝑃₀,𝑃₂), 𝐿(𝑃₁,𝑃₁), 𝐿(𝑃₁,𝑃₂), and 𝐿(𝑃₂,𝑃₁), the nonperiodic orbits of system (2.6) are unbounded. Moreover, the coordinate values of points on these orbits tend to infinity.
(2) If 𝛼>0, then except 𝑃_𝑖(𝑖=0,1,2) and 𝐿(𝑃_𝑖,𝑃₀), 𝐿(𝑃₀,𝑃_𝑖), the nonperiodic orbits of system (2.6) are unbounded. Moreover, the coordinate values of points on these orbits tend to infinity.

Proof. (1) When 𝛼=0, then except 𝑃_𝑖(𝑖=0,1,2), 𝐿(𝑃₀,𝑃₂), 𝐿(𝑃₁,𝑃₁), 𝐿(𝑃₁,𝑃₂), and 𝐿(𝑃₂,𝑃₁), the nonperiodic orbits of system (2.6) are unbounded. Moreover, the coordinate values of points on these orbits tend to infinity. In fact, since these nonperiodic orbits either tend to 𝐴₁ or 𝐴₂ as |𝜉|→+∞, the ordinate values on 𝑦 axis of these nonperiodic orbits must be unbounded.
Now, we prove that the abscissas of these orbits are unbounded by reduction to absurdity. Assume that the abscissas of these orbits are bounded. Since the tangent slope of the orbits at arbitrary point satisfies 𝑑𝑦𝑑𝑥=1𝑦𝑐𝑥−𝑝+𝑏1𝑥^𝑝+1(2.9)𝑑𝑦/𝑑𝑥→0 as |𝑦|→+∞. However, 𝑑𝑦/𝑑𝑥 cannot keep bounded as |𝑦|→+∞ according to the Differential Mean Value Theorem. Therefore, the abscissas of the orbits are unbounded.
(2) It can be proved similarly.

Proposition 2.4. (1) 𝛼=0. If 𝑝 is an even integer, there exists two homoclinic orbits 𝐿(𝑃₀,𝑃₀) or two heteroclinic orbits 𝐿(𝑃₁,𝑃₂) and 𝐿(𝑃₂,𝑃₁) in system (2.6) (see Figures 1 and 2); if 𝑝 is an odd integer, there exists a unique homoclinic orbit 𝐿(𝑃₀,𝑃₀) or 𝐿(𝑃₁,𝑃₁) in system (2.6) (see Figures 3 and 6).
(2) 𝛼>0. If 𝑝 is an even integer, there exists two heteroclinic orbits 𝐿(𝑃_𝑖,𝑃₀)𝑖=1,2 in system (2.6) (see Figures 7 and 8); if 𝑝 is an odd integer, there exists a unique heteroclinic orbit 𝐿(𝑃₀,𝑃₁) in system (2.6) (see Figures 9 and 12).

Considering that a homoclinic orbit or close orbit of a planar dynamical system corresponds to a bell profile solitary wave solution or periodic traveling wave solution of its corresponding nonlinear evolution equation and a heteroclinic orbit corresponds to a kink profile solitary wave solution or an oscillatory traveling wave solution, therefore, from Propositions 2.3 and 2.4 and Figures 1–12, we derive the following theorem.

Theorem 2.5. (1) 𝛼=0. If 𝑝 is an even integer, (1.1) has two bell profile solitary wave solutions (corresponding to the homoclinic orbits 𝐿(𝑃₀,𝑃₀) in Figure 1), or two kink profile solitary wave solutions (corresponding to the heteroclinic orbits 𝐿(𝑃₂,𝑃₁) and 𝐿(𝑃₁,𝑃₂) in Figure 2); if 𝑝 is an odd integer, (1.1) has a unique bell profile solitary wave solution (corresponding to the homoclinic orbit in Figures 3–6).
(2) 𝛼>0. If 𝑝 is an even integer, (1.1) has two bounded traveling wave solutions (corresponding to the heteroclinic orbits 𝐿(𝑃_𝑖,𝑃₀) or 𝐿(𝑃₀,𝑃_𝑖),𝑖=1,2 in Figures 7 and 8); if 𝑝 is an odd integer, (1.1) has a unique bounded traveling wave solution (corresponding to the heteroclinic orbit in Figures 9–12).

3. Relations between the Behaviors of Bounded Traveling Wave Solutions and Dissipation Coefficient 𝛼 of (1.1)

We firstly consider the case of wave speed 𝑐>0.

Theorem 3.1. Suppose that 𝑝 is an even integer, 𝑏>0, and wave speed 𝑐>0. (1)If 𝛼>2√𝑝𝑐, (1.1) has a monotonically decreasing kink profile solitary wave solution 𝑢(𝜉), satisfying 𝑢(−∞)=𝑥₁,𝑢(+∞)=0. Meanwhile, (1.1) has a monotonically increasing kink profile solitary wave solution 𝑢(𝜉), satisfying 𝑢(−∞)=𝑥₂,𝑢(+∞)=0 (𝑢(𝜉), resp., corresponds to the orbit 𝐿(𝑃_𝑖,𝑃₀),𝑖=1,2 in Figure 7(a)).(2)If 0<𝛼<2√𝑝𝑐, (1.1) has a damped oscillatory traveling wave solution 𝑢(𝜉) satisfying 𝑢(−∞)=𝑥₁ and 𝑢(+∞)=0. This solution has maximum at 𝜉̂₁. Moreover, it has monotonically decreasing property if 𝜉>𝜉̂₁, while it has damped property if 𝜉<𝜉̂₁. That is, there exist numerably infinite maximum points 𝜉̂_𝑖(𝑖=1,2,…,+∞) and minimum points 𝜉̌_𝑖(𝑖=1,2,…,+∞) on 𝜉 axis, such that −∞<⋯l<im𝜉_𝑛→∞̌_𝑛<𝜉̂_𝑛<⋯<𝜉̌₁<𝜉̂₁<+∞,𝑢(+∞)<𝑢𝜉̌₁𝜉̌_𝑛=lim_𝑛→∞𝜉̂_𝑛=−∞,(3.1)<⋯<lim𝑢_𝑛→∞𝜉̌_𝑛<⋯<𝑢(−∞)<⋯<𝑢𝜉̂_𝑛<⋯<𝑢𝜉̂₁,lim_𝑛→∞𝑢𝜉̌_𝑛=lim_𝑛→∞𝑢𝜉̌_𝑛=−∞,(3.2)𝜉̌_𝑛−𝜉̌_𝑛+1=lim_𝑛→∞𝜉̌_𝑛−𝜉̌_𝑛+1=4𝜋√4𝑝𝑐−𝛼²(3.3) hold. 𝑢(𝜉) corresponds to the orbit 𝐿(𝑃₁,𝑃₀) in Figure 7(b).
Meanwhile, (1.1) has a damped oscillatory traveling wave solution 𝑢(𝜉) satisfying 𝑢(−∞)=𝑥₂ and 𝑢(+∞)=0. This solution has minimum at 𝜉̌₁. Moreover, it has damped property if 𝜉<𝜉̌₁, while it has monotonically increasing property if 𝜉>𝜉̌₁. That is, there exists numerably infinite maximum points 𝜉̂_𝑖(𝑖=1,2,…,+∞) and minimum points 𝜉̌_𝑖(𝑖=1,2,…,+∞) on 𝜉 axis, such that −∞<𝜉lǐ₁m<_𝑛→∞𝜉̂₁<⋯<𝜉̌_𝑛<𝜉̂_𝑛<+∞,𝑢(−∞)<𝑢𝜉̂₁𝜉̌_𝑛=lim_𝑛→∞𝜉̂_𝑛=−∞,<lim⋯_𝑛→∞<𝑢𝜉̂_𝑛<𝑢(+∞)<𝑢𝜉̌_𝑛<⋯<𝑢𝜉̌₁,𝑢𝜉̂_𝑛=lim_𝑛→∞𝑢𝜉̌_𝑛=+∞,(3.4) and (3.3) hold. 𝑢(𝜉) corresponds to the orbit 𝐿(𝑃₂,𝑃₀) in Figure 7(b).

Proof. We make use of the transformation 𝑉(𝜉)=(𝑢(𝜉)−𝑥₂)/(𝑥₁−𝑥₂) to (2.5) and note that 𝑥₂=−𝑥₁ and 𝑥₁=(𝑐(𝑝+1)/𝑏)^1/𝑝. Then we have 𝑉^(𝜉)−𝛼𝑉^(𝜉)+2^𝑝𝑐𝑉(𝜉)−12𝑉(𝜉)−21^𝑝−12^𝑝=0.(3.5) If 𝑝 is an even integer, (3.5) is equivalent to 𝑉^(𝜉)−𝛼𝑉^(𝜉)+2^𝑝𝑐𝑉(𝜉)𝑉(𝜉)−1×𝑉(𝜉)−12(𝑉(𝜉)−1)2^𝑝−2+𝑉(𝜉)−12^𝑝−414+⋯+12^𝑝−2=0.(3.6) Obviously, if 𝑝 is an even integer, 𝑃^1(0,0), 𝑃^2(1/2,0), and 𝑃^3(1,0) are the singular points of the system which (3.5) corresponds to. They correspond to singular points 𝑃₁(𝑥₂,0), 𝑃₂(0,0), 𝑃₃(𝑥₁,0) of system (2.6), respectively. Since the linear transformation keeps the properties of singular points, the results on 𝑃_𝑖,𝑖=1,2,3 given in Section 2 also hold for 𝑃^𝑖,𝑖=1,2,3 under corresponding conditions.
In the case that 𝑝 is an even integer and 𝑐>0, from the qualitative analysis carried out in Section 2, we have the fact that (2.5) does not have the bounded solution satisfying 𝑢(−∞)=𝑢(+∞). It only has the bounded solution satisfying the following alternative:(A)𝑢(−∞)=𝑥₁,𝑢(+∞)=0, (B)𝑢(−∞)=𝑥₂,𝑢(+∞)=0. Namely, a bounded solution of (3.5) satisfies one of the following two cases:(A₁)𝑉(−∞)=1,𝑉(+∞)=1/2, (B₁)𝑉(−∞)=0,𝑉(+∞)=1/2.
We will use the following lemma in the proof.
Lemma 3.2. Assume that 𝑓∈𝐶¹[0,1],𝑓(0)=𝑓(1)=0,𝑓′(0)>0,𝑓′(1)<0, and forall𝑢∈(0,1), 𝑓(𝑢)>0 holds. Then there exists 𝑟^∗ satisfying −2sup𝑓(𝑢)𝑢≤𝑟^∗≤−2𝑓^(0),(3.7) such that the necessary and sufficient condition under which problem 𝑢(−𝑢∞^+)=𝑟𝑢0^,+𝑓𝑢((+𝑢)∞=)0=1,(3.8) has a monotone solution is 𝑟≤𝑟^∗.
This lemma is quoted from [30–32]. Now we consider the solution of (3.5) which satisfies (𝐴₁).
Let 𝜔=2(1−𝑉), namely, 𝑉=1−(1/2)𝜔. And then the solution of (3.5) which satisfies (𝐴₁) is equivalent to the solution of 𝜔^−𝜔𝛼(𝜔−^∞+)𝑐=(𝜔0,−1)(𝜔1(−+∞𝜔))^𝑝=−11,=0(3.9) Assume 𝑓(𝜔)=𝑐(𝜔−1)[(1−𝜔)^𝑝−1], forall𝜔∈[0,1]. It is easy to verify that 𝑓(0)=0,𝑓(1)=0. Since 𝑓^=𝑐[(𝑝+1)(1−𝜔)^𝑝−1], 𝑓^(0)=𝑝𝑐,𝑓^(1)=−𝑐<0 and forall𝜔∈(0,1)𝑓(𝜔)>0 hold.
According to Lemma 3.2, there exists 𝑟^∗ satisfying (3.7), such that, when −𝛼≤𝑟^∗, (3.9) has a monotone solution. Since 𝑓𝜔(𝜔)^=𝜔𝑓^(𝜔)−𝑓(𝜔)𝜔²=𝑐(1−𝜔)^𝑝(𝑝𝜔+1)−1𝜔²=𝑐𝑔(𝜔)𝜔²,∀𝜔∈(0,1),(3.10) where 𝑔(𝜔)=(1−𝜔)^𝑝(𝑝𝜔+1)−1,forall𝜔∈[0,1].Forall𝜔∈(0,1),𝑔^(𝜔)=−𝑝(𝑝+1)𝜔(1−𝜔)^𝑝−1<0(𝑝 is an even integer). From 𝑔(0)=0, we have forall𝜔∈(0,1),𝑔(𝜔)<0.
Therefore, forall𝜔∈(0,1),(𝑓(𝜔)/𝜔)^=𝑐𝑔(𝜔)/𝜔²<0, namely, 𝑓(𝜔)/𝜔 monotonically decreases in (0,1). So sup_(0,1)𝑓(𝜔)𝜔=lim^𝜔→0𝑓(𝜔)𝜔=lim^𝜔→0𝑓^(𝜔)=𝑝𝑐.(3.11) According to (3.7) in Lemma 3.2, we know 𝑟^∗=−2√𝑝𝑐. So when −𝛼≤𝑟^∗=−2√𝑝𝑐, namely, 𝛼≥2√𝑝𝑐, there exists monotonically increasing solution in (3.9).
Since 𝑉(𝜉)=1−(1/2)𝜔(𝜉),𝑢(𝜉)=𝑥₂+(𝑥₁−𝑥₂)𝑉(𝜉), if 𝛼≥2√𝑝𝑐, the traveling wave solution of (1.1) 𝑢(𝜉) satisfying condition (𝐴) monotonically decreases and appears as a monotone decreasing kink profile solitary wave solution.
Consider the solution of (3.5) satisfying condition (𝐵₁). Let 𝜔=2𝑉, namely, 𝑉=(1/2)𝜔. And then the solution of (3.5) which satisfies (𝐵₁) is equivalent to the solution of 𝜔^−𝜔𝛼(𝜔−^∞+)𝑐=(𝜔0,−1)(𝜔𝜔(+−∞1))^𝑝=−11,=0(3.12) Since (3.12) is the same as (3.9) when 𝑝 is an even integer, from the discussion about (3.9) we know there exists 𝑟^∗=−2√𝑝𝑐, such that when −𝛼≤𝑟^∗=−2√𝑝𝑐, namely, 𝛼≥2√𝑝𝑐, (3.12) has a monotonically increasing solution.
From 𝑉(𝜉)=(1/2)𝜔(𝜉), we can see that 𝑉(𝜉) increases when 𝜔(𝜉) increases. So if 𝛼≥2√𝑝𝑐, the traveling wave solution of (1.1) 𝑢(𝜉) satisfying condition (𝐵) monotonically decreases and appears as a monotonically decreasing kink profile solitary wave solution.
Next we will prove Theorem 3.1(2). Namely, if 0<𝛼<2√𝑝𝑐 the traveling wave solution of (1.1) is damped oscillatory. We take the traveling wave solution corresponding to focus-saddle orbit 𝐿(𝑃₁,𝑃₀) in Figure 7(b), for example, and for 𝐿(𝑃₂,𝑃₀) in Figure 7(b), it can be proved similarly. From the theory of planar dynamical systems we know that system (2.6) corresponding to (1.1) has three singular points on the plane, where 𝑃₀(0,0) is a saddle point and 𝑃₁(𝑥₁,0) and 𝑃₂(𝑥₂,0) are unstable focus points. Moreover, 𝐿(𝑃₁,𝑃₀) tends to 𝑃₁ spirally as 𝜉→−∞. The intersection points between 𝐿(𝑃₁,𝑃₀) and 𝑥 axis at the right hand side of 𝑃₁ correspond to the maximum points of 𝑢(𝜉), while the ones at the left hand side of 𝑃₁ correspond to the minimum points of 𝑢(𝜉). Consequently, (3.1) and (3.2) hold. In addition, as 𝐿(𝑃₁,𝑃₀) tends to 𝑃₁ sufficiently, its property is close to the property of linear approximate solution of system (2.6). Thus, the frequency of the orbit rotating around 𝑃₁ tends to √4𝑝𝑐−𝛼²/4𝜋. And then, (3.3) holds.

Since a traveling wave solution 𝑢(𝜉) keeps the shape and speed unchanged when parallel shifting on 𝜉 axis, without loss of generality, we assume that 𝜉̂₁=0 and 𝜉̌₁=0. The portraits of the oscillatory traveling waves described by Theorem 3.1(2) are shown in Figure 13.

(a)

(b)

Figure 13 

(𝑝 is an even integer, 𝑏>0, and wave speed 𝑐>0, 0<𝛼<2√𝑝𝑐).

We can prove Theorem 3.3 in the similar way.

Theorem 3.3. Suppose that 𝑝 is an odd integer and wave speed 𝑐>0. (1)If 𝛼>2√𝑝𝑐, (1.1) has a monotone kink profile solitary wave solution 𝑢(𝜉), satisfying 𝑢(−∞)=𝑥₁,𝑢(+∞)=0. If 𝑏>0, 𝑢(𝜉) monotonically decreases, corresponding to the orbit 𝐿(𝑃₁,𝑃₀) in Figure 9(a); if 𝑏<0, 𝑢(𝜉) monotonically increases, corresponding to the orbit 𝐿(𝑃₁,𝑃₀) in Figure 11(a).(2)If 0<𝛼<2√𝑝𝑐, (1.1) has a damped oscillatory traveling wave solution 𝑢(𝜉) satisfying 𝑢(−∞)=𝑥₁ and 𝑢(+∞)=0.𝑢(𝜉) corresponds to the orbit 𝐿(𝑃₁,𝑃₀) in Figure 9(b) and Figure 11(b), respectively.

In the following, we consider the case of wave speed 𝑐<0. We can prove next two theorems.

Theorem 3.4. Suppose that 𝑝 is an even integer, 𝑏<0, and wave speed 𝑐>0. (1)If 𝛼>2√−𝑐, (1.1) has a monotonically increasing kink profile solitary wave solution 𝑢(𝜉), satisfying 𝑢(−∞)=0,𝑢(+∞)=𝑥₁. Meanwhile, (1.1) has a monotonically decreasing kink profile solitary wave solution 𝑢(𝜉), satisfying 𝑢(−∞)=0,𝑢(+∞)=𝑥₂. Here, 𝑢(𝜉), respectively, corresponds to the orbit 𝐿(𝑃₀,𝑃_𝑖),𝑖=1,2 in Figure 8(a).(2)If 0<𝛼<2√−𝑐, (1.1) has a damped oscillatory traveling wave solution 𝑢(𝜉) satisfying 𝑢(−∞)=0 and 𝑢(+∞)=𝑥₁. Meanwhile, (1.1) has an damped oscillatory traveling wave solution 𝑢(𝜉) satisfying 𝑢(−∞)=0 and 𝑢(+∞)=𝑥₂. Here, 𝑢(𝜉) corresponds to the orbit 𝐿(𝑃₀,𝑃_𝑖),𝑖=1,2 in Figure 8(b).

Theorem 3.5. Suppose that 𝑝 is an odd integer and wave speed 𝑐<0. (1)If 𝛼>2√−𝑐, (1.1) has a monotone kink profile solitary wave solution 𝑢(𝜉), satisfying 𝑢(−∞)=0,𝑢(+∞)=𝑥₁. If 𝑏>0, 𝑢(𝜉) monotonically decreases, corresponding to the orbit 𝐿(𝑃₀,𝑃₁) in Figure 12(a); if 𝑏<0, 𝑢(𝜉) monotonically increases, corresponding to the orbit 𝐿(𝑃₀,𝑃₁) in Figure 10(a).(2)If 0<𝛼<2√−𝑐, (1.1) has a damped oscillatory traveling wave solution 𝑢(𝜉) satisfying 𝑢(−∞)=0 and 𝑢(+∞)=𝑥₁. 𝑢(𝜉) corresponds to the orbit 𝐿(𝑃₀,𝑃₁) in Figures 10(b) and 12(b), respectively.

The way to prove Theorem 3.4 is the same to Theorem 3.5. For the case of 𝑐<0 and 𝛼>2√−𝑐, the way to prove that traveling wave solution of (1.1) appears monotone is to utilize some transformations and Lemma 3.2; for the case of 𝑐<0 and 0<𝛼<2√−𝑐, we can imitate the proof of Theorem 3.1(2) to prove the traveling wave solution of (1.1) appears damped oscillatory. Because in this case, 𝑃₀ is an unstable focus point, 𝑃_𝑖 are saddle points and 𝐿(𝑃_𝑖,P₀) is focus-saddle orbit. For clarity, we will prove that (1.1) has a monotonically decreasing kink profile solitary wave solution 𝑢(𝜉) satisfying 𝑢(−∞)=0,𝑢(+∞)=𝑥₁ in the case of 𝑐<0,𝛼>2√−𝑐,𝑏>0.

Firstly, according to Theorem 2.5, the unique bounded traveling wave solution of (1.1) 𝑢(𝜉) satisfies 𝑢(−∞)=0,𝑢(+∞)=𝑥₁.(3.13) Substitute 𝑣(𝜉)=(𝑢(𝜉)+𝑥₁)/(2𝑥₁) to (2.5). Then (2.5) and (3.13) become 𝑣^−𝛼𝑣^+2^𝑝𝑐𝑣−12𝑣−21^𝑝−1𝑣(−∞)=12=02,𝑣(+∞)=1,(3.14) Let 𝜔=2(𝑣−1/2), namely, 𝑣=(1/2)(𝜔+1). Then (3.14) is equivalent to 𝜔𝜔(^−−∞𝛼)𝜔=^+0,𝑐𝜔(𝜔𝜔^𝑝(−+∞1))==01.(3.15)

Assume 𝑓(𝜔)=𝑐𝜔(𝜔^𝑝−1), forall𝜔∈[0,1]. It is easy to verify that 𝑓(𝜔)∈𝐶¹[0,1],𝑓(0)=0,𝑓(1)=0, and forall𝜔∈(0,1)𝑓(𝜔)>0. Since 𝑓^(𝜔)=𝑐[(𝑝+1)𝜔^𝑝−1], 𝑓^(0)=−𝑐>0,𝑓^(1)=𝑝𝑐<0. According to Lemma 3.2, there exists 𝑟^∗ satisfying (3.7), such that when −𝛼≤𝑟^∗, (3.15) has a monotone solution. Since forall𝜔∈(0,1)𝑓(𝜔)𝜔=𝑐(𝜔^𝑝−1),𝑓(𝜔)𝜔^=𝑐𝑝𝜔^𝑝−1<0,(3.16) we know 𝑓(𝜔)/𝜔 monotonically decreases in (0,1). So sup_(0,1)(𝑓(𝜔)/𝜔)=lim_𝜔→0(𝑓(𝜔)/𝜔)=−𝑐. And from 𝑓^(0)=−𝑐 and (3.7), we can derive 𝑟^∗=−2√−𝑐. According to Lemma 3.2, if −𝛼≤−2√−𝑐, that is, 𝛼≥2√−𝑐, 𝜔(𝜉) monotonically decreases satisfying 𝜔(−∞)=0,𝜔(+∞)=1. Furthermore, in the case of 𝛼≥2√−𝑐, 𝑣=(1/2)(𝜔+1) also monotonically decreases. Since if 𝑐<0,𝑏>0, 𝑥₁=(𝑐(𝑝+1)/𝑏)^1/𝑝<0, and 𝑢(𝜉)=2𝑥₁𝑣(𝜉)−𝑥₁. We know if 𝑝 is an odd integer, 𝑐<0 and 𝑏>0, when 𝛼≥2√−𝑐, 𝑢(𝜉) monotonically decreases and satisfies 𝑢(−∞)=0,𝑢(+∞)=𝑥₁.

Synthesize Theorems 3.1–3.5, We obtain two critical values for generalized KdV-Burges equation (1.1), where 𝜆₁=2√𝑝𝑐 and 𝜆₂=2√−𝑐. For the right-traveling wave of (1.1), if dissipation effect is large, namely, 𝛼≥𝜆₁, the traveling wave solution of (1.1) appears as a monotone kink profile solitary wave; while if dissipation effect is small, namely, 0<𝛼<𝜆₁, it appears as a damped oscillatory wave. For the left-traveling wave of (1.1), if dissipation effect is large, namely, 𝛼≥𝜆₂, it appears as a monotone kink profile solitary wave; while if dissipation effect is small, that is, 0<𝛼<𝜆₂, it appears as a damped oscillatory wave. In the theorems of this paper, there are definite conclusions on monotonicity of traveling wave solution when we find critical values of the dissipation coefficient. However, in [1, Theorem  1.1] it did not include the case of 𝛼=2√𝑝𝑐.

4. Solitary Wave Solutions and Approximate Damped Oscillatory Solutions of (1.1)

4.1. Bell Profile Solitary Solutions of (1.1)

From [33, Theorem  1], we know that (1.1) has bell profile solitary wave solutions in the following forms.

Theorem 4.1. Suppose 𝛼=0 and wave speed 𝑐>0. (1)If 𝑝 is an even integer and 𝑏>0, (1.1) has two bell solitary wave solutions 𝑢^±1(𝜉)=±√^𝑝𝜑₁(𝜉)𝜑₁(𝜉)=𝑐(𝑝+1)(𝑝+2)2𝑏sech²2𝑝√𝑐𝜉−𝜉₀.(4.1)(2)If 𝑝 is an odd integer, either 𝑏>0 or 𝑏<0, (1.1) has bell solitary wave solutions 𝑢₁(𝜉)=√^𝑝𝜑₁(𝜉), where 𝜑₁(𝜉) is given by (4.1).

Remark 4.2. It is easy to prove that 𝑢^±1(𝜉)=±√^𝑝𝜑₁(𝜉) given in Theorem 4.1(1) corresponds to two symmetrical homoclinic orbits 𝐿(𝑃₀,𝑃₀) in Figure 1; 𝑢₁(𝜉)=√^𝑝𝜑₁(𝜉) given in Theorem 4.1(2) corresponds to the homoclinic orbit 𝐿(𝑃₀,𝑃₀) in Figure 3 when 𝑏>0 and corresponds to the homoclinic orbit 𝐿(𝑃₀,𝑃₀) in Figure 5 when 𝑏<0.

4.2. Kink Profile Solitary Solutions of (1.1)

From [33, Theorem  4], we know that (1.1) has kink profile solitary wave solutions in the following forms.

Theorem 4.3. Suppose wave speed 𝑐<0: 𝜑₂(𝜉)=𝐴1+𝑒^{−𝛿(𝜉−𝜉0)}²=𝐴21−tanh𝛿2𝜉−𝜉⁰−12sech²𝛿2𝜉−𝜉⁰,(4.2) where 𝛼²=−((𝑝+4)²𝑐)/(2(𝑝+2)), 𝛿=𝑝𝛼/(𝑝+4), 𝐴=−2𝛼²(𝑝+1)(𝑝+2)/(𝑏(𝑝+4)²). (1)If 𝑝 is an even integer and 𝑏<0, (1.1) has two kink solitary wave solutions 𝑢^±2(𝜉)=±√^𝑝𝜑₂(𝜉);(2)If 𝑝 is an odd integer, either 𝑏>0 or 𝑏<0, (1.1) has kink solitary wave solutions 𝑢⁺²(𝜉)=√^𝑝𝜑₂(𝜉).

Remark 4.4. Since 𝛼²=−((𝑝+4)²𝑐)/(2(𝑝+2))≥4(−𝑐), that is, 𝛼≥2√−𝑐, the conclusions in Theorem 4.3 are consistent to those in Theorems 3.4 and 3.5, and the conclusions are the concrete realization of Theorems 3.4 and 3.5 when 𝛼≥√−𝑐. 𝑢^±2(𝜉)=±√^𝑝𝜑₂(𝜉) given in Theorem 4.3(1) corresponds to two heteroclinic orbits 𝐿(𝑃₀,𝑃_𝑖,𝑖=1,2) in Figure 8(a); 𝑢⁺²(𝜉)=√^𝑝𝜑₂(𝜉) given in Theorem 4.3(2) corresponds to the heteroclinic orbit 𝐿(𝑃₀,𝑃₁) in Figure 10 when 𝑏<0 and corresponds to the heteroclinic orbit 𝐿(𝑃₀,𝑃₁) in Figure 12(a) when 𝑏>0.
By using undetermined coefficients method, we can derive Theorem 4.5 about MKdV equation in the case of 𝛼=0,𝑝=2.

Theorem 4.5. Suppose 𝛼=0,𝑝=2. If 𝑏<0,𝑐<0. MKdV equation has two symmetrical kink solitary wave solutions 𝑢^±3(𝜉)=±3𝑐𝑏tanh−𝑐2𝜉−𝜉⁰.(4.3)𝑢^±3(𝜉) given in Theorem 4.5 corresponds to two heteroclinic orbits 𝐿(𝑃₂,𝑃₁) and 𝐿(𝑃₁,𝑃₂) in Figure 2.

4.3. Approximate Damped Oscillatory Solutions of (1.1)

In this section we want to obtain approximate damped oscillatory solutions of (1.1) corresponding to the focus-saddle orbits in Figures 7(b), 9(b), and 11(b). We take the approximate damped oscillatory solutions of (1.1) corresponding to the focus-saddle orbits in Figure 7(b) as example, and other cases can be obtained similarly.

By the theory of planar dynamical systems, it is easy to see that focus-saddle orbit 𝐿(𝑃₁,𝑃₀) in Figure 7(b) comes from the break of right homoclinic orbit 𝐿(𝑃₀,𝑃₀) in Figure 1 under the effect of dissipation term −𝛼𝑢_𝑥𝑥(𝜉) (the dissipation coefficient 𝛼 satisfies 0<𝛼<2√𝑝𝑐 orbit 𝐿(𝑃₂,𝑃₀) comes from the break of left homoclinic orbit 𝐿(𝑃₀,𝑃₀) in Figure 1 under the effect of dissipation term). Hence, the nonoscillatory part of the damped oscillatory solution corresponding to 𝐿(𝑃₁,𝑃₀) can be denoted by the bell profile solitary wave solution of the form 𝑢^∗(𝜉)=√^𝑝𝜑₁(𝜉),𝜉∈𝜉₀,+∞(4.4) which is obtained in Theorem 4.1(1), where 𝜑₁(𝜉) is given by (4.1). To express the oscillatory part of this damped oscillatory solution approximatively, we use the following solution of the form 𝑢(𝜉)=𝑒^{𝛽(𝜉−𝜉0)}𝐴₁cos𝐵𝜉−𝜉₀−𝐴₂sin𝐵𝜉−𝜉₀+𝐶,𝜉∈−∞,𝜉₀,(4.5) where 𝐴₁,𝐴₂,𝐵,𝐶,𝛼 are undetermined constants. The reason why we chose (4.5) is that (4.5) has both damped and oscillatory properties since 𝑒^{𝛽(𝜉−𝜉0)} has damped property and (𝐴₁cos(𝐵(𝜉−𝜉₀))−𝐴₂sin(𝐵(𝜉−𝜉₀))) has oscillatory property.

Substituting (4.5) into (2.5) and neglecting the terms including 𝑂(𝑒^{𝛽(𝜉−𝜉0)}), we have 𝐵²=𝛽²−𝛽𝛼=𝛽𝛼−𝑐+𝑏𝐶^𝑝,𝑏2,𝑝+1𝐶^𝑝−𝑐=0.(4.6)

In order to derive approximate damped oscillatory solution of (1.1), there still requires some conditions to connect (4.4) and (4.5). Since the properties of solutions are unchangeable as translating on 𝜉 axis, we take 𝜉₀=0 as a connective point and choose 𝑑^𝑖𝑑𝜉^𝑖𝑢(0)=𝑑^𝑖𝑑𝜉^𝑖𝑢^∗(0),𝑖=0,1,(4.7) namely, 𝐴₁+𝐶=𝑢^∗(0),𝛽𝐴₁−𝐴₂𝐵=0.(4.8) as connective conditions. 𝜉₀=0 is the extremal point of the bell profile solitary wave solutions, thus (𝑑/𝑑𝜉)𝑢^∗(0)=0 holds.

Since (4.5) tends to 𝑥₁ as 𝜉→−∞, thus 𝐶=𝑥₁. Further, 𝐵²=−14𝛼²𝐴+𝑝𝑐,₁=√^𝑝𝜑₁(0)−𝑥₁,𝐴₂=𝛽𝐴¹𝐵=𝛼2𝐴𝐵¹.(4.9) The value of 𝐴₁cos(𝐵𝜉)−𝐴₂sin(𝐵𝜉) is the same, either value of 𝐵 is positive or negative. Without loss of generality, let 𝐵>0 throughout this paper.

According to above analysis, we have the following theorem.

Theorem 4.6. Suppose 0<𝛼<2√𝑝𝑐, 𝑏>0, and wave speed 𝑐>0.
(1) When 𝑝 is an even integer, (1.1) has a damped oscillatory solution corresponding to focus-saddle orbit 𝐿(𝑃₁,𝑃₀), whose approximate solution is 𝑢(𝜉)≈√^𝑝𝑒𝜑₁(𝜉),𝜉∈[0,+∞),_(𝛼/2)𝜉𝐴₁cos(𝐵𝜉)−𝐴₂sin(𝐵𝜉)+𝑥₁,𝜉∈(−∞,0),(4.10) where 𝜑₁(𝜉) is given by (4.1), 𝐵=(1/2)√−𝛼²+4𝑝𝑐, 𝐴₁=√^𝑝𝜑₁(0)−𝑥₁, 𝐴₂=𝛼𝐴₁/2𝐵.
(2) Similarly, (1.1) has a damped oscillatory solution corresponding to focus-saddle orbit 𝐿(𝑃₂,𝑃₀), whose approximate solution is 𝑢(𝜉)≈−√^𝑝𝑒𝜑₁(𝜉),𝜉∈[0,+∞),_(𝛼/2)𝜉𝐴₁cos(𝐵𝜉)−𝐴₂sin(𝐵𝜉)−𝑥₁,𝜉∈(−∞,0),(4.11) where 𝜑₁(𝜉) is given by (4.1), 𝐵=(1/2)√−𝛼²+4𝑝𝑐, 𝐴₁=−√^𝑝𝜑₁(0)+𝑥₁, 𝐴₂=𝛼𝐴₁/2𝐵.

If 𝑝 is an odd integer and 𝑐>0, from Theorem 3.3 we know (1.1) has a unique damped oscillatory solution if 0<𝛼<2√𝑝𝑐. When 𝑏>0, it corresponds to orbit 𝐿(𝑃₁,𝑃₀) in Figure 9(b); when 𝑏<0, it corresponds to orbit 𝐿(𝑃₁,𝑃₀) in Figure 11(b). 𝐿(𝑃₁,𝑃₀) comes from the break of homoclinic orbit 𝐿(𝑃₀,𝑃₀) under the dissipation effect (the dissipation coefficient 𝛼 satisfies 0<𝛼<2√𝑝𝑐. Either in case of 𝑏>0, 𝑐>0 or 𝑏<0, 𝑐>0, the solitary wave solution corresponding to the homoclinic orbit 𝐿(𝑃₀,𝑃₀) has the same expression 𝑢₁(𝜉)=√^𝑝𝜑₁(𝜉) (𝜑₁(𝜉) is given by (4.1)). So we can obtain Theorem 4.7 by the method deriving Theorem 4.6.

Theorem 4.7. Suppose 𝑝 is an odd integer and wave speed 𝑐>0. When 0<𝛼<2√𝑝𝑐 (1.1) has a unique damped oscillatory solution, whose approximate solution can be expressed by (4.10) (𝑝 is an odd integer in (4.10)), where 𝜑₁(𝜉) is given by (4.1).

Synthesizing Theorems 4.6 and 4.7, we can obtain the corollary as follows.

Corollary 4.8. Suppose 𝑝 is any natural number and 𝑐>0. Equation (1.1) has a unique damped oscillatory solution, satisfying 𝑢(−∞)=𝑥₁,𝑢(+∞)=0. Its approximate solution can be expressed by (4.10). Particularly, when 𝑝 is an even integer, (1.1) also has a damped oscillatory solution, satisfying 𝑢(−∞)=𝑥₂,𝑢(+∞)=0. Its approximate solution can be expressed by (4.11).

Thus, we have given the approximate solutions of right-traveling damped oscillatory solutions of (1.1). For left-traveling wave, if we know the kink profile solitary wave solution corresponding to the symmetrical heteroclinic orbit in Figure 2 when 𝑝 is an even integer, and the bell profile solitary wave solution corresponding to the homoclinic orbit in Figures 4 and 6 when 𝑝 is an odd integer, we can get approximate damped oscillatory solutions corresponding to the focus-saddle orbits in Figures 8(b), 10(b), and 12(b) in the case of 𝑐>0 by the same method.

Since we have obtained two symmetrical kink profile solitary wave solutions (Theorem 4.5) in the case of 𝛼=0,𝑝=2,𝑐<0, we can imitate the case of 𝑐>0 to get the following theorem.

Theorem 4.9. Suppose 𝑝=2,𝑏<0, and wave speed 𝑐<0. When 0<𝛼<2√−𝑐, (1.1) has two damped oscillatory solutions corresponding to focus-saddle orbit 𝐿(𝑃₀,𝑃₁) and 𝐿(𝑃₀,𝑃₂). The approximate solution corresponding to focus-saddle orbits 𝐿(𝑃₀,𝑃₁) is 𝑢(𝜉)≈𝑢⁺³−𝐴(₂𝜉),𝜉∈[0,+∞),𝑒^(𝛼/2)𝜉sin(𝐵𝜉),𝜉∈(−∞,0),(4.12) where 𝑢⁺³(𝜉) is given by (4.3), 𝐵=(1/2)√−𝛼²+4𝑐, 𝐴₂𝐵=−√−𝑐/2. The approximate solution corresponding to 𝐿(𝑃₀,𝑃₂) is 𝑢(𝜉)≈𝑢⁻³𝐴(𝜉),𝜉∈[0,+∞),₂𝑒^(𝛼/2)𝜉sin(𝐵𝜉),𝜉∈(−∞,0),(4.13) where 𝑢⁻³(𝜉) is given by (4.3), 𝐵=(1/2)√−𝛼²+4𝑐, 𝐴₂𝐵=√−𝑐/2.

5. Error Estimates of Approximate Damped Oscillatory Solutions of (1.1)

In this section, we investigate error estimates between approximate damped oscillatory solutions and its exact solutions given in Section 4.3. We still take the approximate solution (4.10) and its exact solution corresponding to the focus-saddle orbit 𝐿(𝑃₁,𝑃₀) in Figure 7(b) as example. Other error estimates can be discussed similarly.

Substitute 𝑉(𝜉)=𝑢(𝜉)−𝑥¹−2𝑥₁(5.1) and 𝜉=−𝜂 (𝜂>0) into (2.5). Consequently, the problem of finding an exact damped oscillatory solution for (2.5), which satisfies 𝑢(0)=√^𝑝𝜑₁(0),𝑢^(0)=0,(5.2) is converted into solving the following initial value problem: 𝑉_𝜂𝜂(𝜂)+𝛼𝑉_𝜂(𝜂)+𝑐𝑉(𝜂)−12𝑓(𝜂)=0,𝑉(0)=^𝑝√𝜑₁(0)−𝑥₁−2𝑥₁,𝑉_𝜂(0)=0,(5.3) where 𝑉(𝜂)=𝑉(−𝜂)=𝑉(𝜉),𝑓(𝜂)=(−2)^𝑝[(𝑉(𝜂)−(1/2))^𝑝−(1/2)^𝑝].

Simplifying above initial value problem, it becomes 𝑉_𝜂𝜂(𝜂)+𝛼𝑉_𝜂(𝜂)+𝑝𝑐𝑉(𝜂)+𝑐𝑉²(𝜂)ℎ𝑉(𝜂)=0,𝑉(0)=^𝑝√𝜑₁(0)−𝑥₁−2𝑥₁,𝑉_𝜂(0)=0,(5.4) where ℎ(𝑉(𝜂)) is the polynomial of 𝑉(𝜂) with 𝑝−1 order, satisfying 𝑐𝑉(𝜂)−12𝑓(𝜉)=𝑝𝑐𝑉(𝜂)+𝑐𝑉²(𝜂)ℎ𝑉(𝜂).(5.5) We use the principle of homogenization to solve the following initial value problem: 𝑉_𝜂𝜂(𝜂)𝑉+(0𝛼)𝑉=_𝜂(𝜂_𝑝)+𝑝𝑐𝑉(𝜂)=−𝑐𝑉²(𝜂)ℎ𝑉(𝜂),√𝜑₁(0)−𝑥₁−2𝑥₁,𝑉_𝜂(0)=0,(5.6) where 𝑉(𝜂) satisfies the initial value problem (5.4). It is easy to prove that the following two lemmas hold.

Lemma 5.1. Suppose that 𝑉₁(𝜂) and 𝑉₂(𝜂) are solutions of the initial value problems 𝑉(𝑉0)_𝜂𝜂=(𝜂)_𝑝+𝛼𝑉_𝜂(𝜂)+𝑝𝑐𝑉(𝜂)=0,√𝜑₁(0)−𝑥₁−2𝑥₁𝑉,𝑉_𝜂(0)=0,(5.7)_𝜂𝜂(𝜂)+𝛼𝑉𝑉_𝜂((𝜂0))+=𝑝0𝑐,𝑉(𝜂)𝑉=_𝜂−𝑐𝑉²(𝜂)ℎ𝑉(𝜂),(0)=0,(5.8) respectively, then 𝑉₁(𝜂)+𝑉₂(𝜂) is a solution of the initial value problem (5.6).

Lemma 5.2. Suppose that 𝑉₃(𝜂,𝜏) is a solution of the initial value problem 𝑉(0)𝑉=0_𝜂𝜂,(𝜂)+𝑉𝛼_𝜂𝑉_𝜂(𝜂)+𝑝𝑐𝑉(𝜂)=0,(𝜏)=−𝑐𝑉²(𝜏)ℎ𝑉(𝜏),𝜂>𝜏(5.9) then ∫^𝜂0𝑉₃(𝜂,𝜏)𝑑𝜏 is a solution of initial value problem (5.8).
It is easy to obtain the solution of the initial value problem (5.7) 𝑉₁(𝜂)=𝑒^𝛼1𝜂𝑐₁cos𝛽₁𝜂+𝑐₂sin𝛽₁𝜂,(5.10) where 𝛼₁=−𝛼/2, 𝛽₁=(1/2)√4𝑝𝑐−𝛼², 𝑐₁=(√^𝑝𝜑₁(0)−𝑥₁)/(−2𝑥₁), 𝑐₂=−𝛼₁𝑐₁/𝛽₁.
Let 𝑡=𝜂−𝜏, and substitute it in to the initial value problem (5.9). Then we have 𝑉₃(𝜂,𝜏)=𝑐/𝛽₁𝑒^{𝛼1(𝜂−𝜏)}sin𝛽₁(𝜂−𝜏)𝑉²(𝜏)ℎ𝑉(𝜏).(5.11)

So 𝑉₂(𝜂)=(𝑐/𝛽₁)∫^𝜂0𝑒^{𝛼1(𝜂−𝜏)}sin(𝛽₁(𝜂−𝜏))𝑉²(𝜏)ℎ(𝑉(𝜏))𝑑𝜏 is the solution of the initial value problem (5.8). Thus 𝑉(𝜂)=𝑉₁(𝜂)+𝑉₂(𝜂) is the solution of the initial value problem (5.6). Because the solution 𝑉(𝜂) of the initial value problem (5.6) satisfies the initial value problem (5.4), from the uniqueness of solutions, we have 𝑉(𝜂)=𝑉(𝜂), namely, 𝑉(𝜂)=𝑒^𝛼1𝜂𝑐₁cos𝛽₁𝜂+𝑐₂sin𝛽₁𝜂+𝑐𝛽₁^𝜂0𝑒^{𝛼1(𝜂−𝜏)}sin𝛽₁(𝜂−𝜏)𝑉²(𝜏)ℎ𝑉(𝜏)𝑑𝜏,(5.12) where 𝛼₁=−𝛼/2, 𝛽₁=(1/2)√4𝑝𝑐−𝛼², 𝑐₁=(√^𝑝𝜑₁(0)−𝑥₁)/−2𝑥₁, 𝑐₂=−𝛼₁𝑐₁/𝛽₁.

Substituting 𝜂=−𝜉 and (5.1) into (5.12) and making the transformation 𝑡=−𝜏 in the above integral, then we have 𝑢(𝜉)−𝑥₁=𝑒^−𝛼1𝜉𝑐₁cos𝛽₁𝜉+𝑐₂sin𝛽₁𝜉+𝑐𝛽₁^0𝜉𝑒^{−𝛼1(𝜉−𝑡)}sin𝛽₁(𝜉−𝑡)𝑢(𝑡)−𝑥₁²ℎ₁(𝑡)𝑑𝑡,(5.13) where ℎ₁(𝑡)=(1/2𝑥₁)ℎ(𝑢(𝑡)) and 𝑐₁=−2𝑥₁𝑐₁=√^𝑝𝜑₁(0)−𝑥₁,𝑐₂=2𝑥₁𝑐₂=𝛼₁𝑐₁/𝛽₁. Evidently, 𝛽₁, 𝑐₁, 𝑐₂ are equal to 𝐵, 𝐴₁, −𝐴₂ in Theorem 4.7(1), respectively. And 𝑒^−𝛼1𝜉(𝑐₁cos(𝛽₁𝜉)+𝑐₂sin(𝛽₁𝜉))+𝑥₁ is the approximate damped oscillatory solution of (4.10). Therefore, (5.13) shows the relation between the exact damped oscillatory solution and the approximate damped oscillatory solution as 𝜉<0.

To derive the error estimate between approximate solution and exact solution of damped oscillatory solution corresponding to 𝐿(𝑃₁,𝑃₀), we start from (5.12). Since damped oscillatory solution 𝑢(𝜉) is bounded and 𝑉(𝜂)=𝑉(𝜉)=(𝑢(𝜉)−𝑥₁)/−2𝑥₁, there exists 𝑀>0,𝑀₁>0 such that |𝑢(𝜉)|<𝑀, |𝑉(𝜂)|<𝑀₁. Consequently, from (5.12), we have ||𝑉(𝜂)||≤𝐶₁𝑒^𝛼1𝜂+𝑐𝑇²𝛽₁^𝜂0𝑒^{𝛼1(𝜂−𝜏)}||𝑉(𝜏)||𝑑𝜏,(5.14) where 𝐶₁=|𝑐₁|+|𝑐₂|, 𝑇₂=𝑀₁𝑇₁, and 𝑇₁ is the supremum of |ℎ(𝑉(𝜂))|. Since 𝛼₁<0, for any 𝜂₁∈(0,𝜂], we have ||𝑉(𝜂)||≤𝐶₁𝑒^𝛼1𝜂1+𝑐𝑇²𝛽₁𝑒^𝛼1𝜂1^𝜂0𝑒^−𝛼1𝜏||𝑉(𝜏)||𝑑𝜏.(5.15) By using Gronwall inequality, the above formula becomes ||𝑉(𝜂)||≤𝐶₁𝑒^𝛼1𝜂1exp−𝑐𝑇²𝛼₁𝛽₁𝑒^𝛼1𝜂1(1−𝑒^−𝛼1𝜂).(5.16) Since 𝜂₁∈(0,𝜂] is chosen arbitrarily, letting 𝜂₁→𝜂, the above formula becomes ||𝑉(𝜂)||≤𝐶₂𝑒^𝛼1𝜂,(5.17) where 𝐶₂=𝐶₁exp(−𝑐𝑇₂/𝛼₁𝛽₁)

Substituting 𝜂=−𝜉 and (5.1) into (5.17), we obtain ||𝑢(𝜉)−𝑥₁||≤𝐶₃𝑒^−𝛼1𝜉,𝜉<0,(5.18) where 𝐶₃=2𝐶₂𝑥₁. (5.18) is the amplitude estimate of damped oscillatory solution of (1.1). From (5.18), it is obvious that 𝑢(𝜉) rapidly tends to 𝑥₁ as 𝜉→−∞.

From (5.12) and (5.17), we have ||𝑉(𝜂≤)−𝑐𝑒^𝛼1𝜂𝑐₁cos𝛽₁𝜂+𝑐₂sin𝛽₁𝜂||𝛽₁^𝜂0𝑒^{𝛼1(𝜂−𝜏)}𝑉²(𝜏)|||ℎ𝑉(𝜏)|||𝑑𝜏≤𝑐𝑇¹𝐶²²𝛽₁||𝛼₁||𝑒^𝛼1𝜂,𝜂>0.(5.19)

Substituting 𝜂=−𝜉 and (5.1) into (5.19), we have ||𝑢(𝜉)−𝑒^−𝛼1𝜉𝑐₁cos𝛽₁𝜉+𝑐₂sin𝛽₁𝜉+𝑥₁||≤𝑇₃𝑒^−𝛼1𝜉,𝜉<0,(5.20) where 𝑇₃=(2𝑥₁𝑐𝑇₁𝐶²²)/(𝛽₁|𝛼₁|). Equation (5.20) shows that the error estimate between the approximate solution (4.10) and its exact damped oscillatory solution is less than 𝜀₁(𝜉)=𝑇₃𝑒^−𝛼1𝜉. Since 𝜀₁(𝜉)=𝑂(𝑒^(𝛼/2)𝜉), (4.10) is meaningful to be an approximate solution of (1.1) when the conditions in Theorem 4.7(1) hold.

By using similar method, we can get error estimates between other approximate damped oscillatory solutions obtained above and their exact solutions. Their errors are all infinitesimals decreasing in the exponential form.

6. Conclusion and Prospect

In this paper, we make comprehensive qualitative analysis to the traveling wave solutions of generalized KdV-Burges equation (1.1) when 𝑝 is a natural number, study relations between the behaviors of bounded traveling wave solutions and dissipation coefficient 𝛼, and obtain two critical values of dissipation coefficient: 𝜆₁=2√𝑝𝑐 and 𝜆₂=2√−𝑐. For the right-traveling wave of the equation, if dissipation coefficient 𝛼≥𝜆₁, it appears as a monotonically kink profile solitary wave; if 0<𝛼<𝜆₁, it appears as a damped oscillatory wave. For the left-traveling wave of the equation, if dissipation coefficient 𝛼≥𝜆₂, it appears as a monotonically kink profile solitary wave; if 0<𝛼<𝜆₂, it appears as a damped oscillatory wave. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, by using undetermined coefficients method, we obtain the approximate damped oscillatory solutions with a bell head and oscillatory tail, and the approximate damped oscillatory solutions with a kink head and oscillatory tail when 𝑝=2. Furthermore, by the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between approximate damped oscillatory solutions and their exact solutions. The errors are infinitesimal decreasing in the exponential form. It can be seen throughout this paper that we have obtained all the results in [1, Theorem  1.1] when 𝑝 is a natural number, as well as obtained the existent number of bounded traveling wave solutions and relations between the behaviors of bounded traveling wave solutions and the dissipation coefficient 𝛼 in the case of 𝑏>0,𝑐<0, 𝑏<0,𝑐>0, and 𝑏<0,𝑐<0, respectively. More importantly, we have got approximate damped oscillatory solution and its error estimate in the case of 𝑐>0,0<𝛼<2√𝑝𝑐 when 𝑝 is any natural number.

According to Theorems 3.1, 3.3, and 4.7 and discussion in Section 5, we can give a corollary to the oscillatory solution of generalized KdV-Burgers equation (1.1) referring in [1, Theorem  1.1] if 𝑏=1.

Corollary 6.1. Suppose 𝑝 is a natural number, 𝑐>0, and dissipation coefficient satisfies 0<𝛼<2√𝑝𝑐. Then generalized KdV-Burgers equation (1.1) has a unique oscillatory solution, which satisfies (1.4) if 𝑏=1, possessing the following properties: (1)this solution corresponds to the orbit 𝐿(𝑃₁,𝑃₀) in Figures 7(b), 9(b), and 11(b);(2)this oscillatory solution is damped;(3)this approximate oscillatory solution is 𝑢(𝜉)≈𝑢̃(𝜉)=⎧⎪⎨⎪⎩𝑐(𝑝+12)(𝑝+2)sech²2𝑝√𝑒𝑐𝜉^1/𝑝,𝜉∈[0,+∞),_(𝛼/2)𝜉𝐴₁cos(𝐵𝜉)−𝐴₂sin(𝐵𝜉)+𝑥₁,𝜉∈(−∞,0),(6.1) where 𝐴₁=((𝑐(𝑝+1)(𝑝+2))/(2))^1/𝑝−𝑥₁, 𝐵=(1/2)√−𝛼²+4𝑝𝑐, 𝐴₂=𝐴₁𝛼/2𝐵;(4)the error between the approximate oscillatory solution 𝑢(̃𝜉) and its exact damped oscillatory solution 𝑢(𝜉) is 𝜀(𝜉)=𝑂(𝑒^(𝛼/2)𝜉), (𝜉→−∞).

The following should be pointed out.

(1) This paper gives a method of finding approximate damped oscillatory solutions of nonlinear evolution equations with dissipation effect. Firstly, we make qualitative analysis to the equation. Secondly, we obtain its solitary wave solutions without dissipation effect. Finally, according to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, we obtain its approximate damped oscillatory solutions. This method can also be applied to find approximate damped oscillatory solutions of other nonlinear evolution equations.

(2) Since we have not got the kink profile solitary wave solution corresponding to the heteroclinic orbits in Figure 2 when 𝑝≠2 and the bell profile solitary wave solution corresponding to the homoclinic orbit in Figures 4 and 6, we cannot obtain the damped oscillatory solutions with dissipation effect evolving from above orbits. This problem can be studied deeply in future.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (no. 11071164), Shanghai Natural Science Foundation Project (no. 10ZR1420800), and Leading Academic Discipline Project of Shanghai Municipal Government (no. S30501).