A Class of Approximate Damped Oscillatory Solutions to Compound KdV-Burgers-Type Equation with Nonlinear Terms of Any Order: Preliminary Results

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.


Introduction
The compound KdV-type equation with nonlinear terms of any order   +     +  2   +   = 0, ,  ∈ ,  > 0,  ∈  + (1) 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  = 1.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][7][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   +     +  2   +   +   = 0, ,  ∈ ,  > 0,  < 0,  ∈  + . (2) 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  1 = −√4 1 ( 1 −  2 )/(2 + 1),  2 = −√4 2 ( 2 −  1 )/(2 + 1),  3 = −√−4, and  4 = −√4.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.

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  = 0.
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 () = 0 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) Δ < 0, (II)  is an even number,  > 0 and  < 0, and (III)  is an even number,  = 0 and  < 0. 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  < 0.

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][18][19].
By the above lemma, we can prove the following theorems.
Similarly, we can prove that when  ≤  1 , (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).
Remark 15. (i) When  < 0 in Theorems 4 and 10 is changed into  > 0, 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,  < 0,  > 0, and − 2 (2 + 1)/4( + 1) 2 ≤  < 0. (iii) When  = 0 and  > 0, 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   ( = 1, . . ., 4), (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 ( 1 ± ,  0 ) in Figure 1 The oscillatory traveling wave solution with damped property is called a damped oscillatory solution in this paper.
Proof.(i) By using the theory of planar dynamical systems, it is obtained that  1 − is an unstable focus and  0 is a saddle point.The orbit ( 1 − , 0) tends to  1 − spirally as  → −∞.The intersection points of ( 1 − , 0) and  axis at the right hand of  1 − correspond to the maximum points of (), while the ones at the left hand of  1 − correspond to the minimum points of ().Hence, (20) hold.When ( 1 − , 0) approaches to  1 − sufficiently, its properties tend to the properties of linear approximate solution of ( 8) at  1 − .The frequency of ( 1 − , 0) rotating around  1 − tends to √(4( 2 1 +  1 − ) −  2 )/4.Therefore, (21) holds.(ii) The proof is similar to that of (i).

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  = 0,  ̸ = 0,  ≥ 0, 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  < 0.
(ii)  is an odd number (see Figures 13-16).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  > 0,  is an even number,  > 0,  ≥ 0, and  > 0, the global phase portrait is similar to that shown in Figure 10, and when  is an odd number, the global phase portraits with  > 0 are similar to those with  < 0 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].
(i) When  is an even number, there exist two bell profile solitary wave solutions in the form of  ± 1 () = ±  √ 1 () for (1), where  (ii) When  is an odd number, there exist two bell profile solitary wave solutions in the form of   () =  √  () ( = 1, 2) for ( 1), where  1 () is given by ( 23), and Figures 10, 13, and 15, respectively.Since the global phase portraits as  > 0 are similar to those as  < 0, when  > 0, the bell profile solitary wave solutions given in Theorems 18 and 19 also correspond to the homoclinic orbits ( 0 ,  0 ).
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  = −2 2 ( + 2)/( + 4) 2 for (2); that is,  3 () =  √ 3 (), where It can be proved that when  is an even number, the kink profile solitary wave solutions ± 3 () correspond to the heteroclinic orbits ( 0 ,  4 ± ) in Figure 2(a), and when  is an odd number, the kink profile solitary wave solution  3 () corresponds to the heteroclinic orbit ( 0 ,  4 + ) in Figure 8(a).
Since the focus-saddle orbit ( 1 + ,  0 ) in Figure 1(b) is generated from the break of the right homoclinic orbit ( 0 ,  0 ) in Figure 10, the nonoscillatory part of the damped oscillatory solution corresponding to the orbit ( 1 + ,  0 ) can be expressed by the bell profile solitary wave solution where  1 () is given by (23), while the oscillatory part of the damped oscillatory wave solution can be approximated by where  1 ,  2 , , , and  are arbitrary constants which will be determined later.(27) has both damped and oscillatory properties, due to the fact that  (− 0 ) has damped property and  1 cos(( −  0 )) −  2 sin(( −  0 )) has oscillatory property.
Substituting (27) into (7) and omitting the terms including  (− 0 ) , 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,  0 = 0, and that is, can be chosen as a connective point and connective conditions, respectively.Since (27) tends to  √ 1 as  → −∞,  =  √ 1 .Furthermore, from (28) and (30), it is obtained that No matter what  takes in (31), the value of  1 cos() −  2 sin() is always the same.Without loss of generality, it is assumed that  > 0 throughout the remainder of this paper.Summarizing the above analysis, we have the following theorems.
(ii) When  is an odd number, (2) has a damped oscillatory solution, whose approximate solution is (37).

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.

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.