Approximate solutions of Korteweg-de Vries-Benjamin-Bona-Mahony-Burgers equation with dissipative terms

In this paper, the Korteweg-de Vries-Benjamin-Bona-Mahony-Burgers (KdV-BBM-B) equation is investigated which plays an essential role in numerous subjects of engineering and science. Using the theory of planar dynamical systems, we qualitatively analyze the bounded traveling wave solutions of the KdV-BBM-B equation. The conditions about the presence of bounded traveling wave solutions are obtained resoundingly. Meanwhile, the relationships between the waveform of the bounded traveling wave solution and the dissipation coefficient γ are investigated. Furthermore, when the absolute value of the dissipation coefficient γ is bigger than the critical value, the equation has a kink solitary wave solution. Nevertheless, when | γ | is less than the critical value, the solution has oscillatory and damped properties. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, by using undetermined coefficients method, we obtain the approximate damped oscillatory solutions with a bell head and oscillatory tail, and the approximate damped oscillatory solutions with a kink head and oscillatory tail. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between approximate damped oscillatory solutions and their exact solutions. The errors are infinitesimal decreasing in the exponential form. Finally, to better understand the dynamics of the oscillatory damped solution, we give a graphical analysis of the effect of the dissipation coefficient γ on it.