Some Dynamical Properties Of BBM And Perturbed KdV Equation

Dynamical systems have become increasingly common as models in physics,chemistry, biology,economics and sociology. The theory of dynamical systems plays an important role in understanding qualitative behavior of these models. In the first part, by using the theory of bifurcation of dynamical systems to the generalized Benjamin-Bona-Mahony equation, the existence of solitary wave solutions, infinite smooth and non-smooth periodic wave solutions are obtained under different parametric conditions. It can be shown that the existence of singular curves in a travelling wave system is the reason why smooth waves converge to cusp waves, finally. In the second part, we study the singular perturbed KdV equation. Based on the relation between solitary wave solution and homoclinic orbits of ordinary differential equations, the persistence of the solitary wave solution for the singular perturbed KdV equation is investigated using Melnikov method. We show that the solitary wave solution exists when the perturbation parameter is sufficiently small. We also show the existence of chaotic motion of the perturbed KdV equation.