Zhang, Weiguo; Li, Xiang Approximate damped oscillatory solutions for generalized KdV-Burgers equation and their error estimates. (English) Zbl 1228.35211 Abstr. Appl. Anal. 2011, Article ID 807860, 26 p. (2011). Summary: 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 \(\lambda_1\) and \(\lambda_2\) 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 \(\alpha \geq \lambda_1\), it appears as a monotone kink profile solitary wave solution; that if \(0 < \alpha < \lambda_1\), 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. Cited in 5 Documents MSC: 35Q53 KdV equations (Korteweg-de Vries equations) 35C07 Traveling wave solutions 35C08 Soliton solutions Keywords:Korteweg-de Vries-Burgers equation; traveling waves; kink PDFBibTeX XMLCite \textit{W. Zhang} and \textit{X. Li}, Abstr. Appl. Anal. 2011, Article ID 807860, 26 p. (2011; Zbl 1228.35211) Full Text: DOI References: [1] R. L. Pego, P. Smereka, and M. I. Weinstein, “Oscillatory instability of traveling waves for a KdV-Burgers equation,” Physica D, vol. 67, no. 1-3, pp. 45-65, 1993. · Zbl 0787.76031 [2] D. J. Benney, “Long waves on liquid films,” vol. 45, pp. 150-155, 1966. · Zbl 0148.23003 [3] H. Grad and P. N. Hu, “Unified shock profile in a plasma,” Physics of Fluids, vol. 10, no. 12, pp. 2596-2602, 1967. [4] J. L. Bona and M. E. Schonbek, “Travelling-wave solutions to the Korteweg-de Vries-Burgers equation,” Proceedings of the Royal Society of Edinburgh A, vol. 101, no. 3-4, pp. 207-226, 1985. · Zbl 0594.76015 [5] R. S. 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