Coarse-grained simulations were adopted to investigate interaction between fullerene C₆₀ cluster and lipid membranes. C₆₀ aggregates in water and their interaction with lipid membrane were investigated. It was found that the aggregate can penetrate the membrane directly instead of endocytosis and pinocytosis. The penetration caused transmembrane movement of lipid and water molecules. Interaction between C₆₀ molecules and small lipid vesicle was investigated and dependence on C₆₀ concentration was interpreted explicitly. Conflicting experimental observations due to different preparation methods and processes are also provided. Potential mechanism of the biotoxicity of fullerene was put forward. Our results are helpful in medical applications.

A class of shock wave solutions of Zakharov-Kuznetsov-Burgers (ZKB) equation is obtained with hyperbolic-function-expansion method. Dynamical stability property of shock waves under transverse perturbations is investigated. Firstly, we made linear stability analysis on shock waves. A finite difference scheme with high accuracy is presented to solve numerically the eigenvalue problem. It shows that the shock waves are linearly stable with positive dissipation, while they are linearly unstable with negative dissipation. Secondly, a finite difference scheme is constructed to make long-time nonlinear dynamical evolution of shock waves. The results show that shock waves of ZKB equation are dynamically stable in the case of positive dissipation.

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.