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Approximate damped oscillatory solutions for generalized KdV-Burgers equation and their error estimates. (English) Zbl 1228.35211
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.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35C07 Traveling wave solutions of PDE 35C08 Soliton solutions of PDE
##### Keywords:
Korteweg-de Vries-Burgers equation; traveling waves; kink
Full Text:
##### References:
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