Abstract
This paper is focused on studying approximate damped oscillatory solutions of the compound KdV-Burgers-type equation with nonlinear terms of any order. By the theory and method of planar dynamical systems, existence conditions and number of bounded traveling wave solutions including damped oscillatory solutions are obtained. Utilizing the undetermined coefficients method, the approximate solutions of damped oscillatory solutions traveling to the left are presented. Error estimates of these approximate solutions are given by the thought of homogeneous principle. The results indicate that errors between implicit exact damped oscillatory solutions and approximate damped oscillatory solutions are infinitesimal decreasing in the exponential form.
1. Introduction
The compound KdV-type equation with nonlinear terms of any order is an important model equation in quantum field theory, plasma physics, and solid state physics [1]. In recent years, many physicists and mathematicians have paid much attention to this equation. For example, Wadati [2, 3] studied soliton, conservation laws, Bäclund transformation, and other properties of (1) with . Dey [1, 4] and Coffey [5] obtained the kink profile solitary wave solutions of (1) under particular parameter values and compared them with the solutions of relativistic field theories. In addition, they evaluated exact Hamiltonian density and gave conservation laws. Employing the bifurcation theory of planar dynamical systems to analyze the planar dynamical system corresponding to (1), Tang et al. [6] presented bifurcations of phase portraits and obtained the existence conditions and number of solitary wave solutions. On the assumption that the integral constant is equal to zero, they obtained some explicit bell profile solitary wave solutions. Liu and Li [7] also studied (1) by the bifurcation theory of planar dynamical systems. In addition to obtaining the same bell profile solitary wave solutions as those given by Tang et al. [6], Liu and Li [7] also presented some explicit kink profile solitary wave solutions. In [8], Zhang et al. used proper transformation to degrade the order of nonlinear terms of (1). And then, by the undetermined coefficients method, they obtained some explicit exact solitary wave solutions. Indeed, the solutions obtained in [6–8] are equivalent under certain conditions.
Dissipation effect is inevitable in practical problem. It would rise when wave comes across the damping in the movement. Whitham [9] pointed out that one of basic problems needed to be concerned for nonlinear evolution equations was how dissipation affects nonlinear systems. Therefore, it is meaningful to study the compound KdV-Burgers-type equation with nonlinear terms of any order given by Much effort has been devoted to studying (2). Applying the undetermined coefficients method to (2), Zhang et al. [8] presented some explicit kink profile solitary wave solutions. Li et al. [10] gave some kink profile solitary wave solutions by means of a new auto-Bäclund transformation. Subsequently, Li et al. [11] improved the method presented by Yan and Zhang [12] with a proper transformation. Utilizing the improved method, they obtained some explicit exact solutions. Feng and Knobel [13] made qualitative analysis to (2) and gave the parametric conditions under which there does not exist any bell profile solitary wave solution or periodic traveling wave solution. Furthermore, they used the first integral method to obtain a new kink profile solitary wave solution. By finding a parabola solution connecting two singular points of a planar dynamical system, Li et al. [14] gave the existence conditions of kink profile solitary wave solutions and some exact explicit parametric representations of kink profile solitary wave solutions of (2).
Although a considerable amount of research works has been devoted to (2), there are still some problems which need to be studied further, for example, in addition to kink profile solitary wave solutions, whether (2) has other kinds of bounded traveling wave solutions? As the dissipation effect is varying, how does the shape of bounded traveling wave solutions evolve? In this paper, we will find that, besides kink profile solitary wave solutions, (2) also has damped oscillatory solutions. In addition, we will prove that a bounded traveling wave appears as a kink profile solitary wave if dissipation effect is large, and it appears as a damped oscillatory wave if dissipation effect is small. More importantly, we will discuss how to obtain the approximate damped oscillatory solutions and their error estimates.
The remainder of this paper is organized as follows. In Section 2, the theory and method of planar dynamical systems are applied to study the existence and number of bounded traveling wave solutions of (2). In Section 3, the influence of dissipation on the behavior of bounded traveling wave solutions is studied. It is concluded that the behavior of bounded traveling wave solutions is related to four critical values , , , and . In Section 4, according to the evolution relations of orbits in the global phase portraits, the structure of approximate damped oscillatory solutions traveling to the left is designed. And then, by the undetermined coefficients method, we obtain these approximate solutions. To verify the rationality of the approximate damped oscillatory solutions obtained in Section 4, error estimates are studied in Section 5. The results reveal that the errors between exact solutions and approximate solutions are infinitesimal decreasing in the exponential form. In Section 6, a brief conclusion is given.
2. Existence and Number of Bounded Traveling Wave Solutions
Assume that (2) has traveling wave solutions of the form , where is the wave speed; then (2) is transformed into the following nonlinear ordinary differential equation: Integrating the above equation once with respect to yields where is an integral constant. To find traveling wave solutions satisfying where are the zero roots of the following algebraic equation, let on both hand sides of (4); then we have . Hence, the problem is converted into solving the following ordinary differential equation:
Let and ; then (7) can be equivalently rewritten as the following planar dynamical system: It is well known that the phase orbits defined by the vector fields of system (8) determine all solutions of (7), thereby determining all bounded traveling wave solutions of (2) satisfying (5). Hence, it is necessary to employ the theory and method of planar dynamical systems [15, 16] to analyze the dynamical behavior of (8) in phase plane as the parameters are changed. Denote that Since the number of real roots of determines the number of singular points of (8), and at least two singular points determine a bounded orbit, it is easily seen that (8) does not have any bounded orbits under one of the following conditions: (I) , (II) is an even number, and , and (III) is an even number, and . Hence, the global phase portraits under the above conditions are neglected in this section. For clarity and nonrepetitiveness, we only present the global phase portraits in the case .
(i) is an even number (see Figures 1–4).
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(ii) is an odd number (see Figures 5–9).
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Remark 1. (i) , (), () in Figures 1–9 represent the singular point , singular points (), and singular points at infinity on -axis, respectively. (ii) When , the regions around () are hyperbolic type. When , is an even number, the regions around () are elliptic type. When , is an odd number, the regions around () are parabolic type. (iii) When is an even number, , , , and , the bounded orbits are similar to those shown in Figure 1. When is an odd number, , the bounded orbits are similar to those shown in Figures 5–9. (iv) When is an even number, , , , , and , there exist four possible global phase portraits described by Figures 3(a), 3(b), 3(c), and 3(d). If , then only Figures 3(a), 3(b), and 3(d) describe the orbit distribution. If , then only Figures 3(a) and 3(d) describe the orbit distribution. If , then only Figures 3(a), 3(c), and 3(d) describe the orbit distribution. Figures 6(a), 6(b), 6(c), and 6(d) can be explained similarly.
In addition to Figures 1–9 shown above, we have the following theorem.
Theorem 2. (i) When , , is an even number, and , , , and satisfy none of the following conditions: (I) , (II) and , and (III) and , (2) has either two bounded traveling wave solutions or four bounded traveling wave solutions.
(ii) When , , is an odd number, and , , , and do not satisfy , (2) has either one bounded traveling wave solution or two bounded traveling wave solutions.
3. Behavior of Bounded Traveling Wave Solutions
To study dissipation effect on behavior of bounded traveling wave solutions, we denote that and quote the following lemma [17–19].
Lemma 3. Assume that , , , , and for all , holds. Then, there exists satisfying such that the necessary and sufficient condition under which problem has a monotone solution is .
By the above lemma, we can prove the following theorems.
Theorem 4. Suppose that , is an even number, , , and .(i)When , (2) has a monotone decreasing kink profile solitary wave solution satisfying , , and a monotone increasing kink profile solitary wave solution satisfying , . These solutions correspond to the orbits in Figure 1(a), respectively.(ii)When , (2) has two oscillatory traveling wave solutions . One satisfies , , and the other satisfies , . These solutions correspond to the orbits in Figure 1(b), respectively.
Proof. (i) Substituting the transformation
into (7), it is obtained that
Evidently, , , and corresponding to , , and are the singular points of the planar dynamical system associated with (14). It can be proved that the properties of , , and are the same as those of , , and , and the results obtained in Section 2 also hold for (14). Since when is an even number, , , , and , (7) only has two bounded solutions satisfying(A), ,(B), ,respectively, (14) must only have two bounded solutions satisfying), ,(), ,respectively.
Using the transformation , (14) becomes
Therefore, the solution of (14) satisfying (A′) corresponds to the solution of (15) satisfying
Let for all ; then we have . Since , , , and for all , from Lemma 3, there exists satisfying
such that, when , (15) has a monotone solution satisfying (16).
, we have
where . Since holds for all and , we have for all . Consequently, we have for all , which indicates that monotonically decreases in . Furthermore, we have
Combining (19) with (17), we obtain . Therefore, from Lemma 3, it is concluded that when , (15) has a monotone increasing solution. According to the relation between and , it is easily seen that when , (2) has a monotone decreasing kink profile solitary wave solution satisfying (A).
Now, let us consider the solution of (14) satisfying (B′). Substituting the transformation into (14), we obtain (15) as well. Therefore, the solution of (14) satisfying (B′) corresponds to the solution of (15) satisfying (16). Similarly, we can prove that when , (15) has a monotone increasing solution. According to the relation between and , it is easily seen that (2) has a monotone increasing kink profile solitary wave solution satisfying (B).
(ii) By the theory and method of planar dynamical systems, it is easily obtained that when , there exist an orbit connecting the unstable focus and the saddle point in and an orbit connecting the unstable focus and the saddle point in . Owing to the fact that the orbits tend to spirally as , the corresponding bounded traveling wave solutions are oscillatory. One satisfies , , and the other satisfies , .
Remark 5. denotes an orbit whose limit set is , and limit set is .
Similar to Theorem 4, the following theorems can be proved.
Theorem 6. Suppose that , is an even number, , , .(i)When , (2) has a monotone decreasing kink profile solitary wave solution satisfying , , and a monotone increasing kink profile solitary wave solution satisfying , . These solutions correspond to the orbits in Figures 3(a) and 3(c), respectively.(ii)When , (2) has two oscillatory traveling wave solutions . One satisfies , , and the other satisfies , . These solutions correspond to the orbits in Figures 3(b) and 3(d), respectively.(iii)When , (2) has a monotone increasing kink profile solitary wave solution satisfying , , and a monotone decreasing kink profile solitary wave solution satisfying , . These solutions correspond to the orbits in Figures 3(a) and 3(b), respectively.(iv)When , (2) has two oscillatory traveling wave solutions . One satisfies , , and the other satisfies , . These solutions correspond to the orbits in Figures 3(c) and 3(d), respectively.
Theorem 7. Suppose that , is an even number, , , .(i)When , (2) has a monotone increasing kink profile solitary wave solution satisfying , , and a monotone decreasing kink profile solitary wave solution satisfying , . These solutions correspond to the orbits in Figure 4(a), respectively.(ii)When , (2) has two oscillatory traveling wave solutions . One satisfies , , and the other satisfies , . These solutions correspond to the orbits in Figure 4(b), respectively.
Theorem 8. Suppose that , is an even number, , , and .(i)When , (2) has a monotone decreasing kink profile solitary wave solution satisfying , , and a monotone increasing kink profile solitary wave solution satisfying , .(ii)When , (2) has two oscillatory traveling wave solutions . One satisfies , , and the other satisfies , .
Theorem 9. Suppose that , is an even number, , , and .(i)When , (2) has a monotone increasing kink profile solitary wave solution satisfying , , and a monotone decreasing kink profile solitary wave solution satisfying , . These solutions correspond to the orbits in Figure 2(a), respectively.(ii)When , (2) has two oscillatory traveling wave solutions . One satisfies , , and the other satisfies , . These solutions correspond to the orbits in Figure 2(b), respectively.
Theorem 10. Suppose that , is an odd number, , , and .(i)When , (2) has a monotone decreasing kink profile solitary wave solution satisfying , , which corresponds to the orbit in Figure 5(a).(ii)When , (2) has an oscillatory traveling wave solution satisfying , , which corresponds to the orbit in Figures 5(b) and 5(c).(iii)When , (2) has a monotone increasing kink profile solitary wave solution satisfying , , which corresponds to the orbit in Figures 5(a) and 5(b).(iv)When , (2) has an oscillatory traveling wave solution satisfying , , which corresponds to the orbit in Figure 5(c).
Theorem 11. Suppose that , is an odd number, , , .(i)When , (2) has a monotone increasing kink profile solitary wave solution satisfying , .(ii)When , (2) has an oscillatory traveling wave solution satisfying , .(iii)When , (2) has a monotone decreasing kink profile solitary wave solution satisfying , .(iv)When , (2) has an oscillatory traveling wave solution satisfying , .
Theorem 12. Suppose that , is an odd number, , , and .(i)When , (2) has a monotone decreasing kink profile solitary wave solution satisfying , .(ii)When , (2) has an oscillatory traveling wave solution satisfying , .
Theorem 13. Suppose that , is an odd number, , , and .(i)When , (2) has a monotone increasing kink profile solitary wave solution satisfying , , which corresponds to the orbit in Figure 9(a).(ii)When , (2) has an oscillatory traveling wave satisfying , , which corresponds to the orbit in Figure 9(b).
Theorem 14. Suppose that , is an odd number, , , and .(i)When , (2) has a monotone decreasing kink profile solitary wave satisfying , .(ii)When , (2) has an oscillatory traveling wave satisfying , .
Remark 15. (i) When in Theorems 4 and 10 is changed into , the similar conclusions can be established. (ii) It is easily proved that the bounded orbits shown in Figures 6 and 7 are just the right ones shown in Figures 3 and 4. Therefore, parts of conclusions in Theorems 6 and 7 hold when is an odd number, , , and . (iii) When and , the bounded orbits obtained as is an even number include those obtained as is an odd number. Hence, similar to Theorems 8 and 9, the relations between the behaviors of the bounded traveling wave solutions and are obtained.
The above theorems indicate that when is less than one of the critical values (), (2) has a bounded traveling wave appearing as a kink profile solitary wave, and when is more than one of the above critical values, (2) has a bounded traveling wave appearing as an oscillatory traveling wave. In fact, the oscillatory traveling waves also have damped property. To this end, we take those corresponding to the focus-saddle orbits in Figure 1(b) as examples. The oscillatory traveling wave solution with damped property is called a damped oscillatory solution in this paper.
Theorem 16. Suppose that , is an even number, , , and . If , then (2) has two oscillatory traveling wave solutions . (i) The one corresponding to the orbit in Figure 1(b) has minimum at . Moreover, it has monotonically increasing property at the right hand side of and has damped property at the left hand side of . Namely, there exist numerably infinite maximum points () and minimum points () on -axis, such that (ii) The one corresponding to the orbit in Figure 1(b) has maximum at . Moreover, it has monotonically decreasing property at the right hand side of and has damped property at the left hand side of . Namely, there exist countably infinite maximum points () and minimum points () on -axis, such that and (21) holds.
Proof. (i) By using the theory of planar dynamical systems, it is obtained that is an unstable focus and is a saddle point. The orbit tends to spirally as . The intersection points of and axis at the right hand of correspond to the maximum points of , while the ones at the left hand of correspond to the minimum points of . Hence, (20) hold. When approaches to sufficiently, its properties tend to the properties of linear approximate solution of (8) at . The frequency of rotating around tends to . Therefore, (21) holds.
(ii) The proof is similar to that of (i).
4. Approximate Damped Oscillatory Solutions
4.1. Preliminary Work
In order to obtain approximate damped oscillatory solutions, it is necessary to understand the behaviors of bounded orbits of the planar dynamical system corresponding to (1). By the bifurcation theory of planar dynamical systems, Tang et al. [6] and Liu and Li [7] have investigated (1) under various parameters conditions, respectively. Combining the conclusions associated with , , , which were obtained by Tang et al. [6] and Liu and Li [7], with the analysis of types of singular points at infinity, the global phase portraits can be obtained. For clarity and nonrepetitiveness, we only present the global phase portraits in the case .(i) is an even number (see Figures 10–12).(ii) is an odd number (see Figures 13–16).
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Remark 17. (i) , (), () in Figures 10, 11, 12, 13, 14, 15, and 16 have the same meaning as those in Figures 1–9. (ii) It is easily seen that when , is an even number, , , and , the global phase portrait is similar to that shown in Figure 10, and when is an odd number, the global phase portraits with are similar to those with shown in Figures 13–16.
By the integral method, Tang et al. [6] and Liu and Li [7] presented some bell profile solitary wave solutions. These solutions are equivalent to those obtained by Zhang et al. [8].
Theorem 18. Suppose that , , and .(i)When is an even number, there exist two bell profile solitary wave solutions in the form of for (1), where (ii)When is an odd number, there exist two bell profile solitary wave solutions in the form of () for (1), where is given by (23), and
Theorem 19. Suppose that , , , and .(i)When is an even number, there exist two bell profile solitary wave solutions in the form of for (1), where is given by (23).(ii)When is an odd number, there exists a bell profile solitary wave solution in the form of for (1), where is given by (23).
Theorem 20. Suppose that , is an odd number, , , and ; then there exists a bell profile solitary wave solution in the form of for (1), where is given by (24).
When , it is easy to prove that the bell profile solitary wave solutions , , in Theorems 18 and 20 correspond to the homoclinic orbits in Figures 10, 13, and 15, respectively. Since the global phase portraits as are similar to those as , when , the bell profile solitary wave solutions given in Theorems 18 and 19 also correspond to the homoclinic orbits .
In addition to bell profile solitary wave solutions, Zhang et al. [8] also presented the kink profile solitary wave solution of compound KdV-Burgers-type equation with nonlinear terms of any order. For example, the kink profile solitary wave solution (5.8) obtained by Zhang et al. [8]. Let , , and equal to , , and , respectively, we have the kink profile solitary wave solution with the wave speed for (2); that is, , where It can be proved that when is an even number, the kink profile solitary wave solutions correspond to the heteroclinic orbits in Figure 2(a), and when is an odd number, the kink profile solitary wave solution corresponds to the heteroclinic orbit in Figure 8(a).
4.2. Approximate Damped Oscillatory Solutions of (2)
By the theory of rotated vector field, it is clear that (i) the focus-saddle orbits in Figure 1(b) and the focus-saddle orbit in Figures 5(b) and 5(c) are generated from the break of the homoclinic orbits in Figure 10 and the right homoclinic orbit in Figure 13 under the effect of the dissipation term , that is, , (ii) the focus-saddle orbit in Figure 5(c) is generated from the break of the left homoclinic orbit in Figure 13 as , and (iii) the focus-saddle orbit in Figure 9(b) is generated from the break of the homoclinic orbit in Figure 15 as . We take the focus-saddle orbit in Figure 1(b) as an example to construct an approximate damped oscillatory solution corresponding to it. Other approximate damped oscillatory solutions can be constructed similarly.
Since the focus-saddle orbit in Figure 1(b) is generated from the break of the right homoclinic orbit in Figure 10, the nonoscillatory part of the damped oscillatory solution corresponding to the orbit can be expressed by the bell profile solitary wave solution where is given by (23), while the oscillatory part of the damped oscillatory wave solution can be approximated by where , , , , and are arbitrary constants which will be determined later. (27) has both damped and oscillatory properties, due to the fact that has damped property and has oscillatory property.
Substituting (27) into (7) and omitting the terms including , it is obtained that To obtain approximate damped oscillatory solution to (2), some conditions to connect (26) and (27) are needed. The properties of traveling wave solutions keep the same as translating on -axis; therefore, , and that is, can be chosen as a connective point and connective conditions, respectively.
Since (27) tends to as , . Furthermore, from (28) and (30), it is obtained that No matter what takes in (31), the value of is always the same. Without loss of generality, it is assumed that throughout the remainder of this paper.
Summarizing the above analysis, we have the following theorems.
Theorem 21. Suppose that , , , , .(i)When is an even number, (2) has two damped oscillatory solutions. The approximate solution of the one corresponding to the focus-saddle orbit in Figure 1(b) is where , , , and are given by (23), (31), and (32), respectively. The approximate solution of the one corresponding to the focus-saddle orbit in Figure 1(b) is where and are given by (23) and (31), respectively, , .(ii)When is an odd number, (2) has a damped oscillatory solution corresponding to the focus-saddle orbit in Figures 5(b) and 2(c), whose approximate solution is (33).
Theorem 22. When , is an odd number, , , , , (2) has a damped oscillatory solution corresponding to the focus-saddle orbit in Figure 5(c), whose approximate solution is where is given by (24), , , and .
Theorem 23. When , is an odd number, , , , , (2) has a damped oscillatory wave solution corresponding to the focus-saddle orbit in Figure 9(b), whose approximate solution is where is given by (24), , , and .
Since the global phase portraits with are similar to those with , Theorems 21 and 22 are also established as , and for the case , , , , , we have the following theorem.
Theorem 24. Suppose that , , , , .(i)When is an even number, (2) has two damped oscillatory solutions, whose approximate solutions are where is given by (23), , , , and where is given by (23), , , and , respectively.(ii)When is an odd number, (2) has a damped oscillatory solution, whose approximate solution is (37).
With the help of MATLAB 7.0, the approximate damped oscillatory solutions (33) and (34) are plotted with , , , , , and in the intervals and , respectively (see Figures 17 and 18). The graphics of approximate damped oscillatory solutions (35)–(37) are similar to that shown in Figure 18, and the graphic of approximate damped oscillatory solution (38) is similar to that shown in Figure 17.
4.3. Effect of Parameters on Frequency and Period of Approximate Damped Oscillatory Solutions
In this section, we aim to analyze effect of parameters , , , , , on frequency and period of approximate damped oscillatory solution (33). For other approximate damped oscillatory solutions (34)–(38), we can obtain similar conclusions.
The approximate damped oscillatory solution (33) in the interval can be rewritten as where , , are given by (31), (32), , . Equation (39) indicates that the variation law of approximates to that of simple harmonic vibration. It is easily obtained that the frequency and period of approximate to and . According to the relation between frequency and period , we only give effect of parameters , , , , , and on period . The results on effect of parameters , , , , , and on frequency are opposite to those on period .
4.3.1. Effect of , on Period
For given values of , , , and , the value of is fixed accordingly. Furthermore, it is concluded that (i) for any fixed , is increasing as decreases, (ii) for any fixed , when , is increasing as increases, and when , is decreasing as increases. Figure 19 displays the relation between , , and .
4.3.2. Effect of , , , and on Period
For any fixed , , it can be proved that as increases, is decreasing. Since the value of is proportional to the value of or and is inversely proportional to the value of or , is decreasing as or increases, and it is increasing as or increases. The graphics of (33) for different values of , , , or are shown in Figures 20, 21, and 22, respectively.
5. Error Estimates and Discussion
In this section, we first study error estimate of the approximate damped oscillatory solution (33) and then make some discussion on the exact damped oscillatory solution corresponding to (33). Similar results can be obtained for other approximate damped oscillatory solutions and exact damped oscillatory solutions corresponding to them.
5.1. Error Estimates of Approximate Damped Oscillatory Solutions
Substituting and into (7), the problem of finding an exact damped oscillatory solution of (7), which is subject to is converted into solving the following initial value problem: where , , and . Simplifying the above initial value problem yields where satisfies
By the thought of homogeneous principle, we obtain the implicit exact solution of (43); that is, where , , , and . Substituting into the above formula and making the transformations and (40), we have where , , , and . Since , , are equal to , , in Theorem 21(1), respectively, the first two terms on the right hand side of (46) are the approximate damped oscillatory solution (33). Hence, (46) reflects the relation between (33) and the corresponding exact solution.
Since the damped oscillatory solution is bounded, there exists , such that . Furthermore, from (46), it is obtained that where , is the upper bound of , and . By Gronwall inequality, the above formula becomes where . Equation (48) is the estimate of amplitude. From (48), it is easily seen that tends to as .
Combining (46) with (48) yields Equation (49) reveals that error between the implicit exact damped oscillatory solution and the approximate damped oscillatory solution (33) is less than . Since the error is infinitesimal decreasing in the exponential form, (33) is meaningful to be an approximate solution to (2).
5.2. Discussion
Equation (48) indicates that the implicit exact damped oscillatory solution approaches to in exponential form . Therefore, for any fixed , the smaller is, the faster the speed of converging to is, and for any fixed , the larger is, the smaller the speed of converging to is. In addition, (48) also indicates that the implicit exact damped oscillatory solution locates between two exponential curves in the region of , and its amplitude is less than . When , the amplitude tends to zero in the form of an exponential function. For any fixed or , except the first peak, other peaks and troughs become more inconspicuous when or becomes smaller. Therefore, the wave is obviously flat when it is not far away from the origin, and the effort of damped oscillatory wave becomes weak rapidly. It is shown that the damped oscillatory solution of (2) under the conditions , , , , , and is an even number still has some properties of solitary wave solution.
6. Conclusion
In this paper, we mainly study approximate damped oscillatory solutions of compound KdV-Burgers-type equation with nonlinear terms of any order (2). Based on the qualitative analysis, we obtained the existence conditions and number of bounded traveling wave solutions (kink profile solitary wave solutions and damped oscillatory solutions). According to the evolution relations of bounded orbits in the global phase portraits, the structure of approximate damped oscillatory solutions traveling to the left is designed. And then, they are obtained by the undetermined coefficients method. In addition, we analyze effect of parameters , , , , , and on frequency and period of approximate damped oscillatory solutions. The results reveal that (i) is decreasing as , , or increases, (ii) is increasing as or increases, (iii) when , is increasing as increases, and when , is decreasing as increases. By the idea of homogenization principle, error estimates of these approximate solutions are derived from the integral equations reflecting the relations between approximate solutions and exact solutions. It is shown that the errors are less than , where is a positive constant. Therefore, (33)–(38) are meaningful to be approximate damped oscillatory solutions of (2).
Wang and Yau [20–22] have employed differential transformation method and hybrid numerical method combining the differential transformation method and the finite difference method to analyze many systems, such as a gyroscope system and united gas-lubricated bearing system. These methods are widely used techniques for solving both linear and nonlinear differential equations due to their rapid convergence rate and minimal calculation error. These methods provide us a new approach to study (2); we will do this work in the future.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This project is supported by the National Natural Science Foundation of China (no. 11471215), Innovation Program of Shanghai Municipal Education Commission (no. 13ZZ118), Shanghai Leading Academic Discipline Project (no. XTKX2012), and the Hujiang Foundation of China (no. B14005).