The propagation of a nonlinear wave packet of dust lattice waves sDLWd in a two-dimensional hexagonal crystal is investigated. The dispersion relation and the group velocity for DLW are found for longitudinalm and transversen propagation directions. The reductive perturbation method is used to derive a s2+1d-dimensional nonlinear Schrodinger equation sNLSEd that governs the weak lynonlinear propagationof thewavepacket. This NLSE isused toinves tigatethe modula tional instability of the packet of DLW. It is found that the instability region is different for different propagation directions.

A nonlinear Schrodinger equation (NLSE) is derived for a nonlinear transmission line in which the nonlinear capac-itance C is a function of voltage. For a linear long wavelength perturbations to a solution of a NLSE, the instability region is given in this paper.

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.

It is well known that the Korteweg–de Vires (KdV) equation can describe small but finite amplitude dust acoustic wavesin adustyplasmas. Inthispaper, weuse thereductive perturbationmethodandderive aKadomtsev–Petviashvili (KP) equation, a modified KP (MKP) equation and a coupled KP equation for unmagnetized, collisionless, cold, and two-ion-temperature dusty plasmas with N dierent species of dust grains. We find that if a solitary wave exist in this system, the smaller grains have larger velocities and propagate longer distances than that of larger particles. The com-parisons are given between the dusty plasma composed by di?erent dust particles and the mono-sized dusty plasma.

By using the potential method and the perturbation method under the condition of small amplitude and shallow water waves, we analytically get the KdV-type equation for a viscous shallow water. It indicates that for one soliton-like solution, its amplitude will decrease as it propagates away due to the viscous eeects of water.