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The rectification phenomena and the phase locking for an overdamped twodimensional Frenkel-Kontorova model system with commensurate structure under ac-and dc-driven external forces are demonstrated. As a function of longitudinal dc amplitude, the longitudinal velocity increases in a series of steps of height jν0. In the steps, rectifications in the transverse direction occur though there are no net dc force in the transverse direction. Our results can be applied to vortices in superconductors, classical electron motion, biomolecules moving through 2D arrays of posts, and especially in tribology. In these active ratchets the rectified velocities can be controlled by jν0.

A Zakharov–Kuznetsov (ZK) equation, a modified ZK equation and a coupled ZK equation for small but finite amplitude dust acoustic waves in a magnetized two-ion-temperature dusty plasma with dust size distribution have been obtained in this paper. It seems that the soliton velocity for a dusty plasma with power law dust size distribution is larger than that of monosized dusty plasmas. The solitary waves with opposite polarity particles are investigated as well. It is found that rarefactive and compressive solitary waves exist in this system which depend on different conditions. On the other hand, the solitary wave is unstable at certain regions when the wave numbers of the transverse perturbations satisfies the conditions of 0<K<Kc and 0<L<Lc.

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