We made linearization stability analysis on traveling wave solutions of KdV-Burgers equation. Numerical results indicate that traveling waves are dynamically stable for positive-dissipation case, while they are dynamically unstable for negative-dissipation case. Then we presented a finite difference scheme, which is conditionally stable, for long-time evolution of perturbed traveling waves. Numerical results also show that traveling waves are dynamically stable as positive-dissipation is held. Our results modify and improve conclusions given in relative literatures.

We obtained a class of solitary wave solutions of modified Benjamin-Bona-Mahony (mBBM) equation with kink-antikink structure by using hybolic-function expansion method. Solitary wave solution reduces to a kink-like solution or bell-like solution under different limitations. We analyzed structures of solitary wave with double kinks. Dynamical stability is investigated numerically with a finite difference scheme. The scheme is implicit and it is absolutely stable in linearization sense. It indicates that single soliton with double kinks is stable under different disturbances. Meanwhile,collision of two solitary waves is numerically simulated. It was found that collision between two solitary waves can be either elastic or inelastic.