Competition between geometric dispersion and viscous dissipation in wave propagation of KdV-Burgers equation

In this paper the competitive relationship between the geometric dispersion and the viscous dissipation in the wave propagation of the KdV-Burgers equation is investigated by the generalized multi-symplectic method. Firstly, the generalized multi-symplectic formulations for the KdV-Burgers equation are presented in Hamiltonian space. Then, focusing on the inherent geometric properties of the generalized multi-symplectic formulations, a 12-point difference scheme is constructed. Finally, numerical experiments are performed with fixed step-sizes to obtain the maximum damping coefficient that insures that the scheme constructed is generalized multi-symplectic, and to study the competition between the geometric dispersion and the viscous dissipation in the wave propagation of the KdV-Burgers equation. The competition phenomena are comprehensively illustrated in the wave forms as well as in the phase diagrams: for the KdV equation (a particular case of the KdV-Burgers equation without dissipation), there is a closed orbit in the phase diagram; and the closed orbit is substituted by a heteroclinic one with the appearance of the viscous dissipation; moreover, the heteroclinic orbit changes from the saddle-node type to the saddle-focus type with an increase of the damping coefficient.