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Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 807860, 26 pages

http://dx.doi.org/10.1155/2011/807860

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

School of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 27 April 2011; Revised 12 June 2011; Accepted 12 July 2011

Academic Editor: Josip E.Β PeΔariΔ

Copyright Β© 2011 Weiguo Zhang and Xiang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 and 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 , it appears as a monotone kink profile solitary wave solution; that if , 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 is the physical model describing the long-wave propagating in nonlinear media with dispersion-dissipation [1], where , is any real number and is any positive integer. It is very important in theory and application. If , (1.1) becomes KdV-Burgers equation 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 [2β8]. If , (1.1) has no dissipation effect and becomes generalized KdV equation Obviously, (1.1) is KdV equation [9β11] if , while it is MKdV equation [12β15] if . The stability of solitary wave solutions for (1.3) was studied in [16β22], and the results that the solitary wave solutions are stable if while that they are unstable if were obtained. Pego et al. summarized the discussion of [4, 20] in [1] and gave the following conclusion.

If , (1.1) has a unique traveling wave solution when , where satisfies where is the unique positive solution of . If , monotonically decreases. If , approaches in oscillatory form as ; while as , satisfies where .

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

About the damped oscillatory traveling wave solutions of (1.1) when , 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 , 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, Theoremββ1.1] when 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 in the case of ,, and , respectively. More importantly, we will give approximate damped oscillatory solution and its error estimate in the case of when 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 , (1.1) at most has two bell profile solitary wave solutions or two kink profile solitary wave solutions and that, if , (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 . We find out two critical values and 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 , while it appears as a damped oscillatory wave if ; 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 , while it appears as a damped oscillatory wave if . 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 is a traveling wave solution of (1.1), and satisfies where is the wave speed. Integrating the above equation once yields where 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 and the asymptotic values and (, ) satisfy so under the hypothesis (2.3) and (2.4), the traveling wave solutions of (1.1) satisfy

*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 and , then (2.5) can be reformulated as a planar dynamical system

On phase plane, the number of singular points of system (2.6) depends on the number of real roots of . Denote , , . It is easy to know the following results on the real roots of .

(1) If , when is an even integer, has three different real roots , and ; when is an odd integer, has two different real roots and .

(2) If , when is an even integer, only has one real root ; when is an odd integer, has two different real roots and .

Since system (2.6) has and only has one singular point when , and is an even integer, there does not exist traveling wave solutions in (1.1). We will not consider this case. We use
to denote the Jacobian matrix of system (2.6) at singular points . Therefore, determinant of , that is, , is . Use to denote the discriminant of characteristic equation at , . It is easy to know , ; if is an even integer, .

In the following, we employ the theory and method of planar dynamical systems [27β29] to discuss the type of singular points of system (2.6) and give the global phase portraits.

##### 2.1. In the Case of

In this case, system (2.6) has the first integral

It is easy to see that is a saddle point if and that is a center if . The types of singular points are shown as follows:(1). (i) If is an even integer, system (2.6) has three singular points and , . Since (2.8) holds and , , are centers. (ii) If is an odd integer, system (2.6) has two singular points and . Since (2.8) holds and , is a center;(2). (i) If is an even integer, system (2.6) has three singular points and , . Since , are saddle points. (ii) If is an odd integer, system (2.6) has two singular points and . Since , is a saddle point;(3). If is an odd integer, system (2.6) has two singular points and . Since , is a saddle point;(4). If is an odd integer, system (2.6) has two singular points and . Since (2.8) holds and , is a center.

##### 2.2. In the Case of

It is easy to see that is a saddle point if , while that is an unstable singular point if , where is an unstable node point if and is an unstable focus point if . The types of singular points are shown as follows:(1). (i) If is an even integer, since , , are unstable node points if , and , are unstable focus points if . (ii) If is an odd integer, since , is unstable node point if and is unstable focus point if ;(2). (i) If is an even integer, since , , are saddle points. (ii) If is an odd integer, since , is a saddle point;(3). If is an odd integer, system (2.6) has two singular points and . Since , is a saddle point;(4). If is an odd integer, system (2.6) has two singular points and . Since , is unstable node point if and is unstable focus point if .

Applying PoincarΓ© transformation to analyze singular points at infinity of system (2.6), it is clear that there only exists a couple of singular points , at infinity on axis, where lies on the positive semiaxis of and lies on the negative semiaxis of . There, respectively, exists a hyperbolic type region around if is an even integer and . There, respectively, exists a elliptic type region around if is an even integer and . There, respectively, exists a parabolic type region around if is an odd integer. Moreover, the circumference of PoincarΓ© disk is orbits.

For system (2.6), owing to , by Bendixson-Dulac's criterion [27β29], the following proposition holds.

Proposition 2.2. *If , then system (2.6) does not have any closed orbit or singular closed orbit with finite number of singular points on phase plane. Further, as , 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 ,(i) is an even integer,(ii) is an odd integer;(2) Global phase portraits in case of ,(i) is an even integer,(ii) is an odd integer.

From Figures 1β12, we can derive the following propositions.

Proposition 2.3. *(1) If , then except , , , , and , the nonperiodic orbits of system (2.6) are unbounded. Moreover, the coordinate values of points on these orbits tend to infinity. **(2) If , then except and , , the nonperiodic orbits of system (2.6) are unbounded. Moreover, the coordinate values of points on these orbits tend to infinity.*

*Proof. *(1) When , then except , , , , and , 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 or as , the ordinate values on 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
as . However, cannot keep bounded as 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) . If is an even integer, there exists two homoclinic orbits or two heteroclinic orbits and in system (2.6) (see Figures 1 and 2); if is an odd integer, there exists a unique homoclinic orbit or in system (2.6) (see Figures 3 and 6). **(2) . If is an even integer, there exists two heteroclinic orbits in system (2.6) (see Figures 7 and 8); if is an odd integer, there exists a unique heteroclinic orbit 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 1β12, we derive the following theorem.

Theorem 2.5. *(1) . If is an even integer, (1.1) has two bell profile solitary wave solutions (corresponding to the homoclinic orbits in Figure 1), or two kink profile solitary wave solutions (corresponding to the heteroclinic orbits and in Figure 2); if is an odd integer, (1.1) has a unique bell profile solitary wave solution (corresponding to the homoclinic orbit in Figures 3β6). **(2) . If is an even integer, (1.1) has two bounded traveling wave solutions (corresponding to the heteroclinic orbits or in Figures 7 and 8); if is an odd integer, (1.1) has a unique bounded traveling wave solution (corresponding to the heteroclinic orbit in Figures 9β12).*

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

We firstly consider the case of wave speed .

Theorem 3.1. *Suppose that is an even integer, , and wave speed . *(1)*If , (1.1) has a monotonically decreasing kink profile solitary wave solution , satisfying . Meanwhile, (1.1) has a monotonically increasing kink profile solitary wave solution , satisfying (, resp., corresponds to the orbit in Figure 7(a)).*(2)*If , (1.1) has a damped oscillatory traveling wave solution satisfying and . This solution has maximum at . Moreover, it has monotonically decreasing property if , while it has damped property if . That is, there exist numerably infinite maximum points and minimum points on axis, such that
hold. corresponds to the orbit in Figure 7(b).**Meanwhile, (1.1) has a damped oscillatory traveling wave solution satisfying and . This solution has minimum at . Moreover, it has damped property if , while it has monotonically increasing property if . That is, there exists numerably infinite maximum points and minimum points on axis, such that
**
and (3.3) hold. corresponds to the orbit in Figure 7(b).*

*Proof. *We make use of the transformation to (2.5) and note that and . Then we have
If is an even integer, (3.5) is equivalent to
Obviously, if is an even integer, , , and are the singular points of the system which (3.5) corresponds to. They correspond to singular points , , of system (2.6), respectively. Since the linear transformation keeps the properties of singular points, the results on , given in Section 2 also hold for , under corresponding conditions.

In the case that is an even integer and , from the qualitative analysis carried out in Section 2, we have the fact that (2.5) does not have the bounded solution satisfying . It only has the bounded solution satisfying the following alternative:,
. Namely, a bounded solution of (3.5) satisfies one of the following two cases:,
.

We will use the following lemma in the proof.

Lemma 3.2. *Assume that , and , holds. Then there exists satisfying
**
such that the necessary and sufficient condition under which problem
**
has a monotone solution is .*

This lemma is quoted from [30β32]. Now we consider the solution of (3.5) which satisfies ().

Let , namely, . And then the solution of (3.5) which satisfies () is equivalent to the solution of
Assume , . It is easy to verify that . Since , and hold.

According to Lemma 3.2, there exists satisfying (3.7), such that, when , (3.9) has a monotone solution. Since
where ( is an even integer). From , we have ,.

Therefore, , namely, monotonically decreases in (0,1). So
According to (3.7) in Lemma 3.2, we know . So when , namely, , there exists monotonically increasing solution in (3.9).

Since , if , the traveling wave solution of (1.1) satisfying condition () monotonically decreases and appears as a monotone decreasing kink profile solitary wave solution.

Consider the solution of (3.5) satisfying condition (). Let , namely, . And then the solution of (3.5) which satisfies () is equivalent to the solution of
Since (3.12) is the same as (3.9) when is an even integer, from the discussion about (3.9) we know there exists , such that when , namely, , (3.12) has a monotonically increasing solution.

From , we can see that increases when increases. So if , the traveling wave solution of (1.1) satisfying condition () monotonically decreases and appears as a monotonically decreasing kink profile solitary wave solution.

Next we will prove Theorem 3.1(2). Namely, if the traveling wave solution of (1.1) is damped oscillatory. We take the traveling wave solution corresponding to focus-saddle orbit in Figure 7(b), for example, and for 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 is a saddle point and and are unstable focus points. Moreover, tends to spirally as . The intersection points between and axis at the right hand side of correspond to the maximum points of , while the ones at the left hand side of correspond to the minimum points of . Consequently, (3.1) and (3.2) hold. In addition, as tends to sufficiently, its property is close to the property of linear approximate solution of system (2.6). Thus, the frequency of the orbit rotating around tends to . And then, (3.3) holds.

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

We can prove Theorem 3.3 in the similar way.

Theorem 3.3. * Suppose that is an odd integer and wave speed . *(1)*If , (1.1) has a monotone kink profile solitary wave solution , satisfying . If , monotonically decreases, corresponding to the orbit in Figure 9(a); if , monotonically increases, corresponding to the orbit in Figure 11(a).*(2)*If , (1.1) has a damped oscillatory traveling wave solution satisfying and . corresponds to the orbit in Figure 9(b) and Figure 11(b), respectively.*

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

Theorem 3.4. * Suppose that is an even integer, , and wave speed . *(1)*If , (1.1) has a monotonically increasing kink profile solitary wave solution , satisfying . Meanwhile, (1.1) has a monotonically decreasing kink profile solitary wave solution , satisfying . Here, , respectively, corresponds to the orbit in Figure 8(a).*(2)*If , (1.1) has a damped oscillatory traveling wave solution satisfying and . Meanwhile, (1.1) has an damped oscillatory traveling wave solution satisfying and . Here, corresponds to the orbit in Figure 8(b).*

Theorem 3.5. * Suppose that is an odd integer and wave speed . *(1)*If , (1.1) has a monotone kink profile solitary wave solution , satisfying . If , monotonically decreases, corresponding to the orbit in Figure 12(a); if , monotonically increases, corresponding to the orbit in Figure 10(a).*(2)*If , (1.1) has a damped oscillatory traveling wave solution satisfying and . corresponds to the orbit 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 and , 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 and , 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, is an unstable focus point, are saddle points and is focus-saddle orbit. For clarity, we will prove that (1.1) has a monotonically decreasing kink profile solitary wave solution satisfying in the case of .

Firstly, according to Theorem 2.5, the unique bounded traveling wave solution of (1.1) satisfies Substitute to (2.5). Then (2.5) and (3.13) become Let , namely, . Then (3.14) is equivalent to

Assume , . It is easy to verify that and . Since , . According to Lemma 3.2, there exists satisfying (3.7), such that when , (3.15) has a monotone solution. Since we know monotonically decreases in (0,1). So . And from and (3.7), we can derive . According to Lemma 3.2, if , that is, , monotonically decreases satisfying . Furthermore, in the case of , also monotonically decreases. Since if , , and . We know if is an odd integer, and , when , monotonically decreases and satisfies .

Synthesize Theorems 3.1β3.5, We obtain two critical values for generalized KdV-Burges equation (1.1), where and . For the right-traveling wave of (1.1), if dissipation effect is large, namely, , the traveling wave solution of (1.1) appears as a monotone kink profile solitary wave; while if dissipation effect is small, namely, , it appears as a damped oscillatory wave. For the left-traveling wave of (1.1), if dissipation effect is large, namely, , it appears as a monotone kink profile solitary wave; while if dissipation effect is small, that is, , 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, Theoremββ1.1] it did not include the case of .

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

##### 4.1. Bell Profile Solitary Solutions of (1.1)

From [33, Theoremββ1], we know that (1.1) has bell profile solitary wave solutions in the following forms.

Theorem 4.1. *Suppose and wave speed . *(1)*If is an even integer and , (1.1) has two bell solitary wave solutions *(2)*If is an odd integer, either or , (1.1) has bell solitary wave solutions , where is given by (4.1).*

*Remark 4.2. *It is easy to prove that given in Theorem 4.1(1) corresponds to two symmetrical homoclinic orbits in Figure 1; given in Theorem 4.1(2) corresponds to the homoclinic orbit in Figure 3 when and corresponds to the homoclinic orbit in Figure 5 when .

##### 4.2. Kink Profile Solitary Solutions of (1.1)

From [33, Theoremββ4], we know that (1.1) has kink profile solitary wave solutions in the following forms.

Theorem 4.3. *Suppose wave speed :
**
where , , . *(1)*If is an even integer and , (1.1) has two kink solitary wave solutions ;*(2)*If is an odd integer, either or , (1.1) has kink solitary wave solutions .*

*Remark 4.4. *Since , that is, , 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 . given in Theorem 4.3(1) corresponds to two heteroclinic orbits in Figure 8(a); given in Theorem 4.3(2) corresponds to the heteroclinic orbit in Figure 10 when and corresponds to the heteroclinic orbit in Figure 12(a) when .

By using undetermined coefficients method, we can derive Theorem 4.5 about MKdV equation in the case of ,.

Theorem 4.5. *Suppose ,. If ,. MKdV equation has two symmetrical kink solitary wave solutions
** given in Theorem 4.5 corresponds to two heteroclinic orbits and 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 in Figure 7(b) comes from the break of right homoclinic orbit in Figure 1 under the effect of dissipation term (the dissipation coefficient satisfies orbit comes from the break of left homoclinic orbit in Figure 1 under the effect of dissipation term). Hence, the nonoscillatory part of the damped oscillatory solution corresponding to can be denoted by the bell profile solitary wave solution of the form which is obtained in Theorem 4.1(1), where is given by (4.1). To express the oscillatory part of this damped oscillatory solution approximatively, we use the following solution of the form where are undetermined constants. The reason why we chose (4.5) is that (4.5) has both damped and oscillatory properties since has damped property and has oscillatory property.

Substituting (4.5) into (2.5) and neglecting the terms including , we have

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 axis, we take as a connective point and choose namely, as connective conditions. is the extremal point of the bell profile solitary wave solutions, thus holds.

Since (4.5) tends to as , thus . Further, The value of is the same, either value of is positive or negative. Without loss of generality, let throughout this paper.

According to above analysis, we have the following theorem.

Theorem 4.6. *Suppose , , and wave speed . **(1) When is an even integer, (1.1) has a damped oscillatory solution corresponding to focus-saddle orbit , whose approximate solution is
**
where is given by (4.1), , , . **(2) Similarly, (1.1) has a damped oscillatory solution corresponding to focus-saddle orbit , whose approximate solution is
**
where is given by (4.1), , , .*

If is an odd integer and , from Theorem 3.3 we know (1.1) has a unique damped oscillatory solution if . When , it corresponds to orbit in Figure 9(b); when , it corresponds to orbit in Figure 11(b). comes from the break of homoclinic orbit under the dissipation effect (the dissipation coefficient satisfies . Either in case of , or , , the solitary wave solution corresponding to the homoclinic orbit has the same expression ( is given by (4.1)). So we can obtain Theorem 4.7 by the method deriving Theorem 4.6.

Theorem 4.7. *Suppose is an odd integer and wave speed . When (1.1) has a unique damped oscillatory solution, whose approximate solution can be expressed by (4.10) ( is an odd integer in (4.10)), where is given by (4.1).*

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

Corollary 4.8. *Suppose is any natural number and . Equation (1.1) has a unique damped oscillatory solution, satisfying . Its approximate solution can be expressed by (4.10). Particularly, when is an even integer, (1.1) also has a damped oscillatory solution, satisfying . 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 is an even integer, and the bell profile solitary wave solution corresponding to the homoclinic orbit in Figures 4 and 6 when 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 by the same method.

Since we have obtained two symmetrical kink profile solitary wave solutions (Theorem 4.5) in the case of , we can imitate the case of to get the following theorem.

Theorem 4.9. *Suppose ,, and wave speed . When , (1.1) has two damped oscillatory solutions corresponding to focus-saddle orbit and . The approximate solution corresponding to focus-saddle orbits is
**
where is given by (4.3), , . The approximate solution corresponding to is
**
where is given by (4.3), , .*

#### 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 in Figure 7(b) as example. Other error estimates can be discussed similarly.

Substitute and () into (2.5). Consequently, the problem of finding an exact damped oscillatory solution for (2.5), which satisfies is converted into solving the following initial value problem: where .

Simplifying above initial value problem, it becomes where is the polynomial of with order, satisfying We use the principle of homogenization to solve the following initial value problem: where satisfies the initial value problem (5.4). It is easy to prove that the following two lemmas hold.

Lemma 5.1. *Suppose that and are solutions of the initial value problems
**
respectively, then is a solution of the initial value problem (5.6).*

Lemma 5.2. *Suppose that is a solution of the initial value problem
**
then is a solution of initial value problem (5.8). **It is easy to obtain the solution of the initial value problem (5.7)
**
where , , , . **Let , and substitute it in to the initial value problem (5.9). Then we have
*

So is the solution of the initial value problem (5.8). Thus is the solution of the initial value problem (5.6). Because the solution of the initial value problem (5.6) satisfies the initial value problem (5.4), from the uniqueness of solutions, we have , namely, where , , , .

Substituting and (5.1) into (5.12) and making the transformation in the above integral, then we have where and . Evidently, , , are equal to , , in Theorem 4.7(1), respectively. And 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 .

To derive the error estimate between approximate solution and exact solution of damped oscillatory solution corresponding to , we start from (5.12). Since damped oscillatory solution is bounded and , there exists such that , . Consequently, from (5.12), we have where , , and is the supremum of . Since , for any , we have By using Gronwall inequality, the above formula becomes Since is chosen arbitrarily, letting , the above formula becomes where

Substituting and (5.1) into (5.17), we obtain where . (5.18) is the amplitude estimate of damped oscillatory solution of (1.1). From (5.18), it is obvious that rapidly tends to as .

From (5.12) and (5.17), we have

Substituting and (5.1) into (5.19), we have where . Equation (5.20) shows that the error estimate between the approximate solution (4.10) and its exact damped oscillatory solution is less than . Since , (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 is a natural number, study relations between the behaviors of bounded traveling wave solutions and dissipation coefficient , and obtain two critical values of dissipation coefficient: and . For the right-traveling wave of the equation, if dissipation coefficient , it appears as a monotonically kink profile solitary wave; if , it appears as a damped oscillatory wave. For the left-traveling wave of the equation, if dissipation coefficient , it appears as a monotonically kink profile solitary wave; if , 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 . 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, Theoremββ1.1] when 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 in the case of , , and , respectively. More importantly, we have got approximate damped oscillatory solution and its error estimate in the case of when 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, Theoremββ1.1] if .

Corollary 6.1. *Suppose is a natural number, , and dissipation coefficient satisfies . Then generalized KdV-Burgers equation (1.1) has a unique oscillatory solution, which satisfies (1.4) if , possessing the following properties: *(1)*this solution corresponds to the orbit in Figures 7(b), 9(b), and 11(b);*(2)*this oscillatory solution is damped;*(3)*this approximate oscillatory solution is
where , , ;*(4)*the error between the approximate oscillatory solution and its exact damped oscillatory solution is , .*

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 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).

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