Acta Mathematicae Applicatae Sinica, English Series
, Volume 28, Issue 2, pp 305324
First online:
Approximate damped oscillatory solutions for compound KdVBurgers equation and their error estimates
 Weiguo ZhangAffiliated withSchool of Science, University of Shanghai for Science and Technology Email author
 , Yan ZhaoAffiliated withSchool of Science, University of Shanghai for Science and TechnologyCollege of Mathematics and Physics, Nanjing University of Information Science and Technology
 , Xiaoyan TengAffiliated withCollege of Foundation, Shanghai University of Engineering Science
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In this paper, we focus on studying approximate solutions of damped oscillatory solutions of the compound KdVBurgers equation and their error estimates. We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdVBurgers equation. We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions. Furthermore, we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation. We obtain two critical values of r, and find that a bounded traveling wave appears as a kink profile solitary wave if r is greater than or equal to some critical value, while it appears as a damped oscillatory wave if r is less than some critical value. By means of analysis and the undetermined coefficients method, we find that the compound KdVBurgers equation only has three kinds of bell profile solitary wave solutions without dissipation. Based on the above discussions and according to the evolution relations of orbits in the global phase portraits, we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method. Finally, using the homogenization principle, we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions. Moreover, we also give the error estimates for these approximate solutions.
Keywords
compound KdVBurgers equation qualitative analysis solitary wave solution damped oscillatory solution error estimate2000 MR Subject Classification
34C05 34C37 35Q51 37C29 Title
 Approximate damped oscillatory solutions for compound KdVBurgers equation and their error estimates
 Journal

Acta Mathematicae Applicatae Sinica, English Series
Volume 28, Issue 2 , pp 305324
 Cover Date
 201204
 DOI
 10.1007/s1025501201475
 Print ISSN
 01689673
 Online ISSN
 16183932
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 compound KdVBurgers equation
 qualitative analysis
 solitary wave solution
 damped oscillatory solution
 error estimate
 34C05
 34C37
 35Q51
 37C29
 Industry Sectors
 Authors

 Weiguo Zhang ^{(1)}
 Yan Zhao ^{(1)} ^{(2)}
 Xiaoyan Teng ^{(3)}
 Author Affiliations

 1. School of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China
 2. College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing, 210044, China
 3. College of Foundation, Shanghai University of Engineering Science, Shanghai, 201600, China