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

We focus on studying approximate solutions of dampedoscillatory solutions of generalized KdV-Burgers equation and their error estimates.The theory of planar dynamical systems is employed to make qualitative analysis tothe dynamical systems which traveling wave solutions of this equation correspondto. We investigate the relations between the behaviors of bounded traveling wavesolutions and dissipation coefficient, and give two critical values λ1 and λ2 whichcan characterize the scale of dissipation effect, for right and left-traveling wave solution, respectively. We obtain that for the right-traveling wave solution if dissipationcoefficient α≥λ1, it appears as a monotone kink profile solitary wave solution;that if 0<α<λ1, it appears as a damped oscillatory solution. This is similarfor the left-traveling wave solution. According to the evolution relations of orbitsin 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 estimatesfor these approximate solutions by establishing the integral equations reflecting therelations between exact and approximate solutions. The errors are infinitesimal decreasing in the exponential form